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Lecture Notes in Mathematics Edited by A. Dold and 6.Eckmann Subseries: Department of Mathematics, University of Maryland Adviser: M. Zedek
Anatole Katok Jean-Marie Strelcyn with the collaboration of F. Ledrappier and F. Przytycki
Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Authors
Anatole Katok Mathematics 253-37, California Institute of Technology Pasadena, CA 91125, USA Jean-Marie Strelcyn Universite Paris-Nord, Centre Scientifique et Polytechnique D6partement de Math~matiques Avenue J.-B. CI6ment, 93430 Villetaneuse, France Fran£ois Ledrappier Laboratoire de Probabilit6s, Universit6 Paris VI 4 Place Jussieu, ?5230 Paris, France Feliks Przytycki Mathematical Institute of the Polish Academy of Sciences ul. Sniadeckich 8, 00-950 Warsaw, Poland
Mathematics Subject Classification (1980): Primary: 28 D 20, 34 F 05, 58 F 11,58 F 15 Secondary: 34C35, 58F08, 58F 18, 58F20, 58F22, 58F25 ISBN 3-540-17190-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-1 ? 190-8 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specificallythose of translation, reprinting, re-useof illustrations,broadcasting, reproduction by photocopyingmachineor similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "VerwertungsgesellschaftWort", Munich. © Springer-Vertag Berlin Heidelberg 1986 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 214613140-543210
TABLE OF CONTENTS Introduction PART
I.
E X I S T E N C E OF INVARIANT WITH S I N G U L A R I T I E S
MANIFOLDS
(by A. KATOK
STRELCYN)
and J.-M.
i.
Class
2.
Preliminaries
3.
Overcoming
4.
The Proof
5.
The F o r m u l a t i o n of Pesin's Manifold Theorem
6.
Invariant
(1.1) 7.
PART
of T r a n s f o r m a t i o n s
-
of Lemma
10
Topics
Abstract
19
Invariant 24
for Maps
Satisfying
Conditions
(1.3)
25
ABSOLUTE
Properties
of Local
Stable 35
CONTINUITY
41
and J.-M.
STRELCYN)
i.
Introduction
2.
Preliminary
3.
Some Facts
4.
F o r m u l a t i o n of the A b s o l u t e a Sketch of the Proof
5.
Start of the Proof
6
The F i r s t M a i n Lemma
41 Remarks
and N o t a t i o n s
from M e a s u r e
Theory
42
and Linear A l g e b r a
Continuity
Theorem
55 62 65
7
Start of the Proof Projection
and C o v e r i n g
9
Comparison
of the V o l u m e s
- II
79 Lemmas
i0
The Proof of the A b s o l u t e
ii
Absolute
Continuity
12
Infinite
Dimensional
13.
Final
88 107
Continuity
of C o n d i t i o n a l
Theorem
Measures
Case
(by F. L E D R A P P I E R Introduction Preliminaries
3.
Construction
4.
Computation
130
154
THE E S T I M A T I O N OF E N T R O P Y FROM B E L O W T H R O U G H LYAPUNOV CHARACTERISTIC EXPONENTS
2.
117
138
Remarks
i.
46
and
- I
8
IIIo
I
of S i n g u l a r i t i e s
3.3 and Related
Manifolds
(by A. K A T O K
PART
Singularities
5
Influence
Some A d d i t i o n a l Manifolds II.
with
F O R SMOOTH MAPS
and J.-M.
and F o r m u l a t i o n
157
STRELCYN)
of the Results
157 162
of the P a r t i t i o n of E n t r o p y
167 175
IV
PART
IV.
THE E S T I M A T I O N OF E N T R O P Y FROM ABOVE LYAPUNOV CHARACTERISTIC EXPONENTS (by A. K A T O K and J.-M.
1
Introduction
2
Preliminaries
3
Construction
4
The Good and Bad E l e m e n t s
of P a r t i t i o n s
The Main Lemma The E s t i m a t i o n
PART V.
Introduction
2.
Terminology
193
199
SYSTEMS
199 and N o t a t i o n
The M a p p i n g
5.
The A p p l i c a b i l i t y Ergodic T h e o r e m
Billiards. ¢.
200
Generalities
The C o m p u t a t i o n
Set.
201
of
of the O s e l e d e c
d#
207
Multiplicative 222
6.
The S i n g u l a r
7.
The B i l l i a r d s of Class ~ . lld¢II and lld2~ll
8.
Proof of Lemma ations
7.4.
P r o o f of Lemma Inequality
7.4.
REFERENCES
DYNAMICAL
196
STRELCYN)
The Plane
2.
184 189
3.
APPENDIX
~t
of E n t r o p y
4.
Final
183
{~t}t~l of P a r t i t i o n
PLANE B I L L I A R D S AS SMOOTH WITH SINGULARITIES
i.
10.
I BO
E S T I M A T I O N OF E N T R O P Y OF SKEW P R O D U C T F R O M ABOVE T H R O U G H V E R T I C A L L Y A P U N O V C H A R A C T E R ISTIC E X P O N E N T S
(by J.-M.
9.
of the R e s u l t
181
6
i.
180
STRELCYN)
and F o r m u l a t i o n
5
APPENDIX
THROUGH
The B i l l i a r d s
of Class
229
P
The rate of G r o w t h
Part One:
Elementary
237
Configur249
Part Two:
P ro o f of the Main 258 273
Remarks OSELEDEC
of
MULTIPLICATIVE
ERGODIC
THEOREM
276 279
i.
INTRODUCTION During the past t w e n t y - f i v e years the h y p e r b o l i c p r o p e r t i e s of
smooth dynamical systems
(i.e. of d i f f e o m o r p h i s m s and flows) were
studied in the ergodic theory of such systems in a more and more general framework [Rue]2,3).
(see
[AnO]l,2,
[Sma],
[Nit],
[Bri],
[Kat] I,
[PeS]l, 3,
The d e t a i l e d h i s t o r i c a l survey of the h y p e r b o l i c i t y and
its role in the ergodic theory up to 1967 is given in
[Ano]2, Chapter ].
One of the most important features of smooth dynamical
systems
showing b e h a v i o r of h y p e r b o l i c type is the e x i s t e n c e of invariant families of stable and unstable m a n i f o l d s and their so called "absolute continuity".
The m o s t general theorem c o n c e r n i n g the
e x i s t e n c e and the absolute c o n t i n u i t y of such families has been proved by Ya. B. Pesin
([PeS]l,2).
The final results of this theory give a partial d e s c r i p t i o n of the ergodic properties of a smooth dynamical an a b s o l u t e l y continuous
system w i t h respect to
invariant m e a s u r e in terms of the L y a p u n o v
c h a r a c t e r i s t i c exponents.
One of the m o s t striking of the many
important consequences of these results d e s c r i b e d in
[pes]
is the 2,3 so called Pesin entropy formula which expresses the entropy of a smooth d y n a m i c a l system through its L y a p u n o v c h a r a c t e r i s t i c exponents. Our first m a i n purpose is to g e n e r a l i z e Pesin's results to a
broad class of d y n a m i c a l systems with s i n g u l a r i t i e s and at the same time to fill gaps and correct errors
in Pesin's proof of absolute
c o n t i n u i t y of families of invariant m a n i f o l d s
([Pes] I, Sec.
3).
We
followed Pesin's scheme very closely and this may at least partly e x p l a i n the length of our p r e s e n t a t i o n and heaviness of details, e s p e c i a l l y in Part II. (and unstable)
Parts I and II c o n t a i n the theory of stable
invariant m a n i f o l d s
c o r r e s p o n d to the context of
in our more general s i t u a t i o n and
[Pes] I.
At the end of Part II we also
prove an infinite d i m e n s i o n a l c o u n t e r p a r t of Pesin's results [Pes] i" The m o t i v a t i o n
for our g e n e r a l i z a t i o n lies in the fact that some
important d y n a m i c a l systems o c c u r r i n g in classical m e c h a n i c s example,
from
(for
the m o t i o n of the system of rigid balls w i t h elastic
collisions)
do have singularities.
the example mentioned) Briefly speaking,
Some of these systems
(including
can be reduced to s o - c a l l e d b i l l i a r d systems.
a b i l l i a r d system describes the m o t i o n of a point
mass w i t h i n a R i e m a n n i a n m a n i f o l d w i t h b o u n d a r y w i t h r e f l e c t i o n from the boundary. Our general c o n d i t i o n s on the s i n g u l a r i t i e s f o r m u l a t e d in Sec. 1 of Part I g r e w out of an attempt to u n d e r s t a n d the nature of s i n g u l a r i t i e s
in the b i l l i a r d problem.
VI
Since
a Poincare
flow u s u a l l y
has
singularities
In Part
whose
(first-return
may also provide
and c o n t i n u o u s
formula.
map
singularities,
time d y n a m i c a l
essential
III we prove the b e l o w
to
in smooth
case
tion
from below.
of Parts
line.
It seems
with
of d i s c r e t e
for the Pesin
changes
time
This
I and
Recently
II.
ingenious
completely simpler
that Mane's
entropy
the paper
[Pes]2, 3.
very
avoids
and it is s u b s t a n t i a l l y
Sinai-Pesin
estimate
an a l t e r n a t i v e
His proof
treatment
with minor
[Sin] 1 and
way the results
gave
manifolds
a unified
for a smooth
transformations
systems.
This part r e p r o d u c e s
idea goes back
map on a section)
considering
[Led]
i' in an
proof uses R. Mane
proof
([Man]l)
of the estima-
the use of i n v a r i a n t
than the p r o o f
method
along
can be applied
the to our
case. The above e n t r o p y
estimate
proved
in Part
IV is largely
independent
of the rest of the book. In
[Pes]
P e s i n derives from his results on invariant stable and 3 m a n i f o l d s the d e s c r i p t i o n of ergodic p r o p e r t i e s of a smooth
unstable dynamical
system
on the i n v a r i a n t
All his a r g u m e n t s property
with
literally
Bernoulli Jacobian
property
apply
from
the g r o w t h
of p e r i o d i c satisfies
points
Sec.
1 of Part
continuous class convex with
invariant measure
includes
the extra
finite
all c o m p a c t
and concave
IV the P e s i n
not know w h e t h e r nents
holds
Let us notice proof
that
exponents
arcs
every
C3
for s o - c a l l e d
of such
of the
is s a t i s f i e d
systems.
M. W o j t k o w s k i
Sinai-Bunimovich
through
measure
This
of
line intervals,
of Parts
of III and
for such billiards.
estimate
invariant
from
absolutely
convex arc has the t a n g e n c y By the results
that
I.
by a finite number
and s t r a i g h t
and
assuming
to the natural
class
bounded
the above e n t r o p y
recently
II. entropy
the s i n g u l a r i t i e s
with respect
of class
formula
between
1 of Part
of
of the
in Part
in our s i t u a t i o n
from Sec.
for a b r o a d
for an a r b i t r a r y that
estimate
and show that the c o n d i t i o n s
all its tangents.
entropy
stronger
the c o n n e c t i o n
regions
assumption:
order w i t h
that the proof
in great detail
I are s a t i s f i e d
exponents.
of B e r n o u l l i
It seems
also hold
for plane b i l l i a r d s
Lyapunov
of his proof
than the one o b t a i n e d
the c o n d i t i o n s
In Part V we study map
a somewhat
map
[Kat] 2 c o n c e r n i n g
the m e a s u r e
Poincare
to our case.
requires
of the P o i n c a r ~
Results
set: w i t h n o n - z e r o
the sole e x c e p t i o n
We do
the L y a p u n o v
expo-
for such a billiard.
([WOJ]l, 2) found billiards
an easy
the L y a p u n o v
are non-zero.
Resuming,
one can say that
lower r i g h t corner
in the p r e s e n t
of the f o l l o w i n g
diagram,
book we c o m p l e t e d
the
VII
The theory of A n o s o v systems
The theory of b i l l i a r d s of
and of the r e l a t e d systems
Sinai and B u n i m o v i c h
as A x i o m A systems,
etc.
i
I
Pesin Theory of m a p p i n g s w i t h
Pesin Theory of diffeo-
> singularities
m o r p h i s m s of compact manifolds
A concise resume of the m a i n results of the p r e s e n t book can be found in
[Str].
Other p r e s e n t a t i o n s of Pesin's theorem c o n c e r n i n g the e x i s t e n c e of i n v a r i a n t m a n i f o l d s were given later by D. Ruelle A. Fathi, M. Herman and J.-C. Yoccoz
([Fat]).
several g e n e r a l i z a t i o n s of that theorem a class of i n f i n i t e - d i m e n s i o n a l maps
([Rue] 1 ) and
D. Ruelle has d e v e l o p e d
(non-invertible smooth maps,
([Rue]2,3)).
R. Ma~e has found
another i n f i n i t e - d i m e n s i o n a l v e r s i o n of Pesin's t h e o r e m The authors w o u l d like to point out in the p r e p a r a t i o n of this book. w r i t t e n by the second author.
([Man]2).
their unequal p a r t i c i p a t i o n
A l m o s t all the text was a c t u a l l y
The first author suggested the general
plan of the w o r k and w o r k e d out the arguments w h i c h allow us to overcome the p r e s e n c e of s i n g u l a r i t i e s
in the c o n s t r u c t i o n of i n v a r i a n t
m a n i f o l d s and in the above entropy estimate.
Naturally, we d i s c u s s e d
together numerous q u e s t i o n s c o n c e r n i n g p r a c t i c a l l y all subjects treated in the text. The first draft of the theory d e s c r i b e d in the p r e s e n t book was p r e s e n t e d by the second author in D e c e m b e r 1978 at the Seminar of M a t h e m a t i c a l Physics at IHES
(Bures-sur Yvette, France).
The m a t e r i a l
of this book r e p r e s e n t s a part of the "Th~se d' Etat" of the second author, d e f e n d e d 30 April 1982 at U n i v e r s i t y Paris VI
(France).
Our n o t a t i o n s are very similar to those used by Pesin, but they are not the same. C o n c e r n i n g the e n u m e r a t i o n of formulas,
theorems,
etc, the first
number indicates the section in which the given formula, is contained.
The lower Roman numeral
In the interior of the same parts, Despite all our efforts,
theorem,
etc.,
indicates the part of the book.
the Roman numerals are not marke~.
some m i s t a k e s can remain.
g r a t e f u l to the readers kind enough to point them out.
We will be
Viii
Acknowledgments.
This book owes very m u c h to Dr. F. L e d r a p p i e r
(CNRS, U n i v e r s i t y Paris VI, France)
and to Dr. F. P r z y t y c k i
(Mathemati-
cal Institute of Polish A c a d e m y of Sciences, Warsaw). Besides being a c o - a u t h o r of Part III, F. L e d r a p p i e r made numerous useful remarks c o n c e r n i n g other topics treated in the book.
In partic-
ular he played a very i m p o r t a n t role in the e l a b o r a t i o n of the infinite d i m e n s i o n a l case. The role of F. Przytycki can hardly be overestimated.
We owe him
the final f o r m u l a t i o n of conditions c h a r a c t e r i z i n g our class of maps w i t h singularities.
In the previous v e r s i o n s conditions on the growth
of the first d e r i v a t i v e as well as of the growth of the two first d e r i v a t i v e s of the inverse m a p p i n g near the s i n g u l a r i t i e s w e r e assumed. Using ideas of F. P r z y t y c k i we were able to dispose of these conditions in Parts I-III and c o n s e q u e n t l y to extend the class of m a p p i n g s under consideration.
We thank sincerely both of them.
We also thank Dr. G. B e n e t t i n M. Brin
(University of Maryland,
nique, Palaiseau, Poland),
France),
Dr. Ya. B. Pesin
of Dijon, France), France)
(University of Padova, USA),
Dr. P. Collet
Dr. M. M i s i u r e w i c z (Moscow, URSS),
Dr. J.-P. T h o u v e n o t
and Dr. L.-S. Young
very useful discussions.
Italy), Dr.
(Ecole P o l y t e c h -
(University of Warsaw,
Dr. R. R o u s s a r i e
(University
(CNRS, U n i v e r s i t y Paris VI,
(Michigan State University,
U.S°A)
for
In p a r t i c u l a r the first author d i s c u s s e d the
early v e r s i o n of the theory d e s c r i b e d in this book w i t h Ya. B. Pesin who made several useful remarks. i m p o r t a n t formula
(4.10) v.
G. B e n e t t i n c o m m u n i c a t e d to us
M. M i s i u r e w i c z
found the c o u n t e r e x a m p l e
d e s c r i b e d in Sec. 7.8 v. We thank also Dr. R. D o u a d y Dr. M. Levi
(Boston University,
of Wroclaw,
Poland)
(Ecole P o l y t e c h n i q u e , U.S.A.)
Palaiseau,
and Dr. T. N a d z i e j a
France),
(University
for their help in the final editing of the text.
We would e s p e c i a l l y like to a c k n o w l e d g e the advice and gentle c r i t i c i s m of Dr. R. Douady,
whose careful reading of the m a n u s c r i p t
enabled us to m a k e m a n y c o r r e c t i o n s and improvements. It is our p l e a s a n t o b l i g a t i o n to express our g r a t i t u d e to the D e p a r t m e n t of M a t h e m a t i c s of the U n i v e r s i t y of M a r y l a n d and the N a t i o n a l Science F o u n d a t i o n for the support given to the second author for his trips to the U.S.A.
(NSF Grant MCS79-030116).
The
second author also thanks the D e p a r t m e n t of M a t h e m a t i c s of the Centre S c i e n t i f i q u e et P o l y t e c h n i q u e of U n i v e r s i t y Paris XIII, e x c e l l e n t w o r k i n g conditions.
for
PART I EXISTENCE
OF I N V A R I A N T
MANIFOLDS
A. K a t o k
1.
CLASS
i.i
OF T R A N S F O R M A T I O N S
In S e c t i o n
formulate
basic
consideration moderately Let following (A) of c l a s s
is "thin"
conditions contains
There
such that
we denote
b)
(TyV, II'IIy)
respectively,
every
Obviously compact
We w i l l equal
to
m
s(e)
satisfying
an o p e n s m o o t h
the
(at l e a s t m.
C < i, R < l, g >_ 1
x
and
w h e r e by
to the set
of the e x p o n e n t i a l
such t h a t
w = exPxly
p(x,y)
and
q
p(x,X)
X c M, and by map
eXPx:
< Rv(X),
linear maps ÷
T x V + V.
one has
lld(exPx I) (Y)II < q.
n o r m of the
(or s h o r t l y
Here
d eXPx(W) : (TxV,II'IIx)
(TxV, ll'iIx),
N'II) d e n o t e s
the n o r m in
TxV
metric.
e > 0
where
condition
there p(x,y)
exists
w = exPxl(y) and
say that the m e t r i c
V
re > 0
such that for
< m i n ( r E , R v ( X ) ) def R E ( x )
(B) is a l w a y s
manifold
and
satisfied
o n e has
lld(exPxl(y)]l <_ 1 + e. when
M
is a s m o o t h
is a s m o o t h o p e n s u b m a n i f o l d
space
Y
with metric
d
of
M.
has a c a p a c i t y
if
l i m sup log s(e) e -+ 0 1 log where
constant
y E V
satisfying
Riemannian
p
x 6 V
II'IIx
lid eXPx(W) ll < 1 + ~
of the m a p g r o w
of f i n i t e d i m e n s i o n
d(exPxl) (y) : (TyV,N'lly)
For e v e r y
x, y 6 V
subset
V
f r o m the p o i n t
i n d u c e d by the R i e m a n n i a n c)
of the m a p u n d e r
(B).
of i n j e c t i v i t y
where
and
on m a p s u n d e r c o n s i d e r a t i o n .
space w i t h m e t r i c
manifold
the o p e r a t o r and
for the p r o b l e m
>_C(min(R, [0(x,M\V) ]g)) def Rv(X )
lld eXPx(W)II <- q, w h e r e
+
metric
(A) and
for e v e r y
II • !I d e n o t e s
setting
and t h a t the s e c o n d d e r i v a t i v e s
as an o p e n d e n s e
the d i s t a n c e
the r a d i u s
the g e n e r a l
(1.1)-(1.3)
exist positive
R(x,V)
SINGULARITIES
set.
for every point a)
SINGULARITIES
Strelcyn
t h a t the set of s i n g u l a r i t i e s
C 4) R i e m a n n i a n
(B)
R(x,V)
insure
be a c o m p a c t
M
WITH
assumptions
n e a r this
M
and J.-M.
1 we d e s c r i b e
These conditions
FOR SMOOTH MAPS WITH
denotes
b a l l s of r a d i u s
e.
= m
the m i n i m a l
cardinal
of a c o v e r i n g
of
Y
by o p e n
As follows border)
from Sec.
of d i m e n s i o n C)
The
space
Let us note
3 of
w hich
N
is a
M
with m e t r i c
that c o n d i t i o n
Let to
~
diffeomorphism
be a Borel
the
Let set
following
A = M\N.
L
(i.i)
N(~-I(B))
of
M
There
positive
V
N
= ~(B)
U E (L)
respect
exist p o s i t i v e
manifold
(with
m.
is of finite only play
capacity.
a role
in Parts
II
IV. ~:N + V
between
N
be a m a p p i n ~
and
its image,
i.e.,
V.
measure
on
for any Borel
conditions
to
and let
into
probability
Let
with
of
p
in Part
c P ( p ~ 2)
~, i.e.,
assume
(C) only
subset of
embedding
equal
(Bc) will
be an open
a diffeomorphic
e v er y c o m p a c t
has a c a p a c i t y
and III and the c o n d i t i o n Let
[Kol],
m
(1.1)-(1.3)
M
invariant
set
B c M.
concerning
be the open
M,
%
e-neighborhood
to the metric
p.
constants
and
cI
a
with
We will and
respect always ~.
of the sub-
such that
for every
e, a
(U e(A))
<- ClE
Obviously,
.
condition
that
~(N)
= i.
that
~(N)
= i, w h e r e
of
~
are defined.
We will
denote
at the p o i n t
(1.1)
F r o m the
x.
N = Thus
by
implies
that
¢-invariance ~ r=-~ N
~r(N)
~(A)
= 0
or e q u i v a l e n t l y
of the m e a s u r e is the set w h e r e
is always
~
one obtains
all
iterates
non empty.
dk# i or dk#Z(x) x suppose that
the k-th d e r i v a t i v e
of
~£
We will
MIl°g÷IId*x lld<÷ )
121
I log+jld~xllld ~ < + M
where,
by definition,
operator These plicative
n o r m induced conditions Ergodic
log+x = m a x ( l o g
quarantee
Theorem
of the
is clear
that
injectivity R(x,N)
and the n o r m
metric
(see A p p e n d i x 2,cf. also
(Ba)
= min(R(x,V),p(x,A)).
is the
of the O s e l e d e c lOse],
let us d e n o t e
of the e x p o n e n t i a l
II'JJ
p.
the a p p l i c a b i l i t y
[Led] 1 and [Rue]2,3). For x E N, like in c o n d i t i o n radius
x,0)
by the R i e m a n n i a n
by
Multi-
[Pes] 2,
[Rag],
R(x,N)
the
map
eXPx:
TxN ÷ N.
Thus
in p a r t i c u l a r
It
R(x,N)
~ C min(R, [p(x,A) ] g ) d ~ f ~ ( x ) .
Let us d e n o t e ~0x
is w e l l
for
defined
is a n e i g h b o r h o o d can not
We w i l l
now
in a n e i g h b o r h o o d
of
be d e f i n e d
-i ~0x = exp,(x)
x 6 N,
O
6 T¢(x)N.
for all
formulate
h
o ¢ o exp x.
U
of
O
Nevertheless,
6 TxN,
the m a i n
The m a p p i n g
6 TxN
and
~0x(U)
in g e n e r a l ,
~0x(h)
llhN < RN(X)-
condition
concerning
the
growth
of
9.
(1.3)
There
x 6 N, h
exist
6 TxN,
positive
constants
IIhll < ~ ( x )
and
c2 ~ 1
%0x(h)
and
b
is d e f i n e d
such then
that
if
o n e has
Hd2~0x(h) N < c 2 P ( e X P x ( h ) , A ) -b
Without the
loss
constant The
of g e n e r a l i t y g
comes
conditions
purpose
of P a r t s
manifolds
of
respect To D r o v e
following
There
every
x E N
cial
because
derivative. To a c h i e v e
validity
the
¢-i
the
estimation
positive
we w a n t This
that,
constants
(1.4),
to w o r k
is o n l y
with
that
stable
for
all
of
for
of c o n d i t i o n s
from above
d2¢0x
when
M
stable
g-l).
(i.i)
we w i l l
such
d >_ 1
(1.1)-(1.4) on
and
need
that
supposed
that
of P a r t s space.
I -III
of
(1.3)
the
for
of the
artifi-
second
linear
spaces.
d2¢x . smooth
manifold,
the
of the c h o i c e
of the
the
choice
of m e t r i c
is a c o m p a c t
metric
space.
influ-
c 3.
it is s u f f i c i e n t
In this
restrictive.
condition.
case,
somewaht
between
is i n d e p e n d e n t
In this
M
looks
"quadratic part
instead
and
above
(1.3)
is a c o m p a c t
cI
is s o m e w h a t
and
for the m a p s
M.
general
p
the
possible
we c o n s i d e r also
metric
c3
condition
the c o n s t a n t s
(1.3)
sufficient
(i.e.,
entropy
metric
more
are
-d
to c o n d i t i o n
for the p u r p o s e
where
necessary.
of the
Riemannian
a complete
b > 2g
for the c o n s t r u c t i o n
ones
ences
It was
that
(Ba) .
(i.i)-(1.3)
counterparts
are not
of c o n d i t i o n s
only
assume
assumption.
exist
Let us n o t e
and
as u n s t a b l e
that
lldCxll < c3P(x,A) Compared
always
in p a r t i c u l a r
as w e l l
to
extra
(1.4)
(B)
I -III,
~
can
f r o m the c o n d i t i o n
(A),
It is r e m a r k a b l e with
one
more
It can
general
to s u p p o s e
setting
be r e p l a c e d
In fact, M
to be
the c o n d i t i o n
by the
following
(1.3')
There
~or e v e r y B0x(h)
exists
x 6 N
make
positive
and
a sense
h
~ TxN
one
in case
such
C I, c 2
that
and
b
such and
IIh!I < RN(x)
that
such
that
has
lld2~0x(h) II ~ m a x ( C 1
Clearly,
constants
c2P(x,A) -b)
of c o m p a c t
M,
the
conditions
(1.3) and
(1.3')
are
equivalent. Nevertheless, an a l r e a d y to the
to a v o i d
sufficiently
case
in some ially
examples
larger
quarantee
(see P a r t
than
the
conditiors
ing c o n d i t i o n compact
set
set
in the u s u a l
(1.3) a n d
submanifolds of
~
an a b s o l u t e l y of the h
form
is a b o u n d e d
Even
for
Finally,
dropped.
1.2
The
is d e r i v e d
conditions
has
in w h a t
sense,
with
together
~.
However, substant-
in o r d e r
to
with
as
case
we can
i.e.,
a = 1
near
A
of a
of
induced
set
number
N = M\A
density,
volume
growth"
a measure
by in
is
p
and
(i.i).
it is s t i l l
a < i.
it will second
concern
is the u n i o n
to
of
and w i t h
a finite
codimensions
a bounded
for some
the
A
satisfy-
number
boundary
along
its r e s t r i c t i o n
"moderate
that
I-III
of
be p r o v e d
condition
Part
I, the
in Sec.
12ii,
(1.2)
can be
of
invertibility
for
of
inessential. of t r a n s f o r m a t i o n s
from
the v e r y
discussed are
satisfied
system
in a r e g i o n
in the E u c l i d e a n (i.e.
whose
with
detailed
in P a r t
region
tic c u r v e s
in
ourselves
of a s i t u a t i o n
of p o s i t i v e
that
In this
to be true
of P a r t s
class
billiards
h
such
(perhaps
is the R i e m a n n i a n
function.
Moreover,
is a l s o
~
set"
of a f i n i t e
codimensions;
measure
where
(i.i)
restrict
to be c h o s e n
example
glued
submanifolds
let us n o t e
the p u r p o s e s
¢
C1
has
is the u n i o n
of p o s i t i v e
if the d e n s i t y
possible
M
MI,...,M r
continuous h~,
complications
(1.4).
m ~ 2
is a m e a s u r e
A
important
Namely,
manifolds
number
as a " s i n g u l a r
the
of
finite
we w i l l
V),
all of d i m e n s i o n
MI,...,Mr;
text,
formal
singular
angles) C1
A
the m o s t
(i.i).
purely
M.
to c o n s i d e r
N o w we d e s c r i b e
smooth
complicated
of c o m p a c t
It is n a t u r a l
additional
V.
singularities
study
of
singularities
In P a r t V it is s h o w n
for an a p p r o p r i a t e boundary plane
described
is not
bounded
of r e a l - a n a l y t i c
Poincar4
too d e g e n e r a t e ,
by a f i n i t e
images
of c l o s e d
of p l a n e
that map
number
above
all
above
of the b i l l a r d e.g.
a compact
of r e a l - a n a l y -
intervals).
For the class of t r a n s f o r m a t i o n s prove
the c o u n t e r p a r t
there
is no doubt
singularities are still general
which
true
since
Let us note
I-III
actua l l y
for 2.
remain
~
satisfying
condition
(1.3)
[PeS]l_ 3.
examples,
Nevertheless
the Pesin's
[pes]l_ 3 are based on very
(1.1)-(1.3),
4.2).
with results few
settings.
is the c o m p o s i t i o n
the a s s u m p t i o n s (see Sec.
i.i we will
of t r a n s f o r m a t i o n s
to various
for any map w h i c h
true
in Sec.
This
class
one because,
the a s s u m p t i o n s
of a finite
all
results
of maps
of
is
for example,
one can
(1.1)-(1.3)
such that
fails.
PRELIMINARIES
2.1
Recall
that
the tangent TxV
if
space
V
to
is a smooth manifold, V
at the point
the set of all t a n g e n t
denotes
the t a n g e n t
Let denote
E,F
by
between
E
where
that
r(E,F)
subspaces
N
defined
the r e s t r i c t i o n
extension
the O s e l e d e c
of
¢.
support
in
X.
denotes by TV
in a E u c l i d e a n
and
of
angle
space
H.
We
and the a p e r t u r e
i.e.
H
F(E,F)
family of
linear
F(E,F)
=
vectors
parallel
to
# F(F,E).
sup (inf lie-vli), e(E' v(F' Nell=l e
and
f
E
and
F
It is clear
and w h e r e respectively. that
II'II (1.2)
is the of
i.i
is
of the d e r i v a t i v e Condition Ergodic
(1.2)
d%
shows
Theorem
To see this,
#~invariant,
and we can
to the set that
T~N
as a
the a s s u m p t i o n s
of
(see A p p e n d i x 2)are satis-
one can c o n s t r u c t
a measurable
isometries
{ax:(TxN, II'IIx) ÷
restriction
in Section
Multiplicative
fied for this extension.
cocvcle
with
the m i n i m a l
the angle b e t w e e n
in general
T V x X c V, we denote
If
V
then
~ y(E,F).
consider
dition
x.
to
x E V,
V.
inf 7(e,f) 0~e(E' 0~fEF' denote
The set
where
of
respectively,
are linear
2F(E,F)
linear
F
=
us note
2.2
and
and
y(e,f)
E', F'
bundle
vectors
be two linear m a n i f o l d s
y(E,F)
y(E,F)
Let
from
than the p r e v i o u s
a map
¢ o ~,
the proofs
that
from
classes
from others
can be a d a p t e d
satisfying
larger
construct
of P e s i n
for many o t h e r
may arise
ideas w h i c h
number of maps Parts
of results
that
introduced
( m m, H'II) }x(N'
standard
d~
becomes
to
T~N
Euclidean
the c o n d i t i o n
in the O s e l e d e c
n o r m in
to the d i r e c t
theorem.
~m,
product
of i n t e g r a b i l i t y
and t r a n s f e r N × ~m.
the
Then con-
for the m a t r i x
To o b t a i n choose
the f a m i l y
one p o i n t
{~}
x.
one can p r o c e e d in e v e r y
as follows.
connected
First
component
V.
1
of
1
V, and for e v e r y p o i n t
ing
x
w i t h the p o i n t
Then i d e n t i f y
TxV
xi
with
(which is an i s o m e t r i c ~ i : ( T x v, II.Irx.) + 1 1 By the O s e l e d e c set
A c ~ (i)
choose
such that Tx V 1
map).
Yx
depends
measurably.
by the p a r a l l e l Finally
(mm,%rl)
Theorem
Vx
a smooth path
choose
on
x
translation
connect-
along
Yx
isometries
i : 1 ..... k
there exists
a
¢-invariant
x 6 A
exists
measurable
sub-
such t h a t ~(i)
(ii)
x 6 Vi
= i,
for e v e r y p o i n t
there
a
d%-invariant
~-mea-
surable decomposition
s(x) T M x
=
~ H. (X) , i=l ±
such that u n i f o r m l y finite
lim n+_+co Xl(X)
loglld Cn (v)II n
< X2(x)
<...<
The n u m b e r s of
#
(iii)
v
( Hi(x),IIvll
= i, one has the
are c a l l e d
x
with
characteristic
d. (x) d~f d i m H. (x) i l ~(x). x 6 A,
times;
exponents
i.e.
the f o l l o w i n g
at
the f i r s t k2(x )
with
x E A, let f l ( x ) , . . . , f (x) be an a r b i t r a r y m such that for e v e r y i,l ~ i ~ m, lim ~IIlog d ~ ( f i ( x ) ) l l Then,
if
m a 2,
p , Q c {1,2 .... ,m}
is c a l l e d
let us d e n o h e by
characteristic
ki(x )
Xl(X),
the L y a p u n o v
and
for e v e r y p o i n t
is r e p e a t e d
i = 1 ..... s(x),
X s ( x ) (x) ,
Xi(x)
all L y a p u n o v
coincides for IRm
Xi(X),
of the e x p o n e n t
I m(x)
Xi(X)
=
at the p o i n t
plicity
_<
for all v e c t o r s
limit
ll(X) x,
s 12(x ) ~...
where
kl(X ) X2(x) ,
any e x p o n e n t
of t h e m etc.
f a m i l y of b a s e s
= li(x).
for e v e r y
two n o n - e m p t y
we have:
n÷~lim ~i log y ( d ~ ( E p ) , d ~ ( E Q ) )
= O,
exponents
the m u l t i -
disjeint
subsets
Now, of
where
Ep
vectors
and
EQ
denote
{fi(x)}i~p
(ii)
and
and
(iii)
such
imply
E 1 @ E 2 = TxN.
lim n÷±~
(ii)
regular
and
of
IRm
spanned
b y the
respectively.
x { A
and every
E2 c T N x
k,l
~ k < s(x),
be an a r b i t r a r y
let u s
linear
subspace
called
regular.
Then
= 0. n
The points If
subspaces
easily
for e v e r y p o i n t k E1 = ~ H. (x) . L e t i=l 1
that
linear
{fi(x)}i~Q
(iv) note
the
x
(iii)
in t h e
such hold
future.
that
(ii)
but only
for
and
(iii)
n ÷ ~,
hold
are
the point
is c a l l e d
2.3
Let
us
Let
Z(x)
fix
T _< 0
be the
×£(x) (x)
(usually,
largest
in the
positive
applications,
integer
such
one
takes
T = 0~.
that
< T <- ×~ (x) +l (x) .
L e t us d e f i n e l(x)
= exp
X Z ( x ) (x)
(2.1)
U(x)
= exp
XZ(x)+l(X).
(2.2)
Clearly,
EIx
%(x)
=
< I.
Let
furthermore
H. (X) i
e i=l
(2.3) s(x) = • E2x i=£ (x)+l
For
the p o s i t i v e
c A
r,s A
H. (x). l
numbers
= {x
r,s
( A:
Obviously, For
~(x)
every
x ( Ar, s,
depending
only
on
Furthermore,
set
let r
r e x p ( 1 0 0 e r , s)
s, r < i, r < s,
let us d e f i n e
= exp(-e(x)) .
From
(2.5)
s
and
it f o l l o w s
< ~'(x)
following
morphisms
is m e a s u r a b l e and r,s e(x) = er, s be a so s m a l l
¢-invariant. positive
number
that r e x p ( 1 0 0 e r , s)
(2.5)
< s.
(2.6) (2.7)
exp(-3e(x))
~(x)
0 < ~(x)
A
and
< 1
(2.4)
5 r < s ~ ~(x)}.
exp(3e(x))
~' (x) = ~(x)
case.
and
let
l'(x) = l(x)
The
r
by
that
< ~'(x)
theorem
of c o m p a c t
(2.8)
< ~(x),
is p r o v e d
manifolds.
~'(x) in Sec.
The proof
(2.9)
< I. 1.3
of
remains
[Pes] 1 for d i f f e o valid
also
in our
THEOREM
2.1.
o__n Ar, s
There
exists
such that
if
a measurable
positive
v E d ~ m E l x , then
fox every
(iid~nmxVii)¢ sign n ~ L ( x ) r n e x p ( e r , s ( n
i_~ ~
v E d ~ m E 2 x , then (Hd~mxVll)sign
and the an@le (i.e.,
+
defined
n ( (2.1.1)
for e v e r y n (
y(¢m(x))
0 ~ v E d#m(E2y))
L
imi))iivN;
n >_ (L(x))-Is n e x p ( - e r , s ( n
the s m a l l e s t
function
between
an~le between satisfies
+
Iml))ilvN;
the s u b s p a c e s
d~m(Elx )
the v e c t o r s
(2.1.2)
and
d%m(E2x )
0 ~ u E d#m(Elx )
and
for e v e r y m ~
(L(x)) -I exp(-er, s ImI) _< y(¢m(x)) . For
£ > 0, let us d e f i n e
AZ = {x E A s; L(x) r,s r, obviously,
if
Zl > £2
Let us d e n o t e
for
d i m EIx = k(x),
(2.1.3)
now the set
~ £}.
then
£i s D AZ2 Ar, r,s
and
U Ai = i . Z>0 r,s r,s
0 ~ k ~ m
A k = {x E A;k(x)
= k},
Ak,r, s = A k N Ar,s I (2.10)
and
A k,r,s £
The sets LEMMA
Ak
= A k N Ar,s £
and
2.1.
integer
are c l e a r l y
For any integer
~ = ~(q,~,s)
Theorem 1.5.1
~,r,s
such that
2.2 f o r m u l a t e d
formulated
the d e f i n i t i o n
in
¢-invariant.
Z ~ 1 there ) c As fq(A Z r,s r,s
q
and
below coincides
almost
[Pes] 1 for d i f f e o m o r p h i s m s
of the sets
A£ r,s
and w o r k s
exists
verbatium
and the p r o o f
a positive
with Theorem uses o n l y
in our case w i t h o u t
any
change. THEOREM A
2.2.
There
(i e., a family
exists
a measurable
of scalar
products
Riemannian < .,. >'
•
x ~ A, w h i c h
metric
on the set
on the spaces
T N,
x
depends
on
x
in a m e a s u r a b l e
way)
x
and a m e a s u r a b l e
10
function
A(x)
following
is true:
A(@nx) sup
such
~ A(x) A(x)
that
for e v e r y
x ( A
and
any
integer
n
the
exp(s(x) Inl) ,
= AZ
<
+
(2.2.1) (2.2.2)
~t
rts
x(A £ rts
IId#xlEl xH'
< X'(x) (2.2.3)
Pld®;lrE2xll ' where
IIAII' d e n o t e s
between II'II' y
two
The
scalar
(2.2.5) the
tanqent
generated
(2.2.4) the
-
b_y the
product
The
ll.llx
The
metric
ll-IIx
that
products
~nd
are
provided
orthogonal
on
to the n o r m
with
II.IIx
by the
following
is s o m e t i m e s
called
Lyapunov
( Ak,
that
dard
scalar
T'
X
respect
t_oo
generated
by
inequalities:
ll'IIx
is the
norm.
set of all p o i n t s
we can d e f i n e
Let us The
transforms
product
OVERCOMING
x
a measurable
( A
such
family
of
in
the
that
invertible
product
T' 2x'E2x
< .,"
>'
÷ ~m-k
into
e D
and
e,Y
y
will
such play
I
~
the
stan-
X
OF S I N G U L A R I T I E S
fix n u m b e r s sets
scalar
'
~m.
INFLUENCE
following
T' :Elx + ~ k lx
that
0 < e < i, y > 0.
an i m p o r t a n t
role
in s u b s e q u e n t
considerations. (3.1)
acting norms
>'. y
N
A(X)
A
with
transformations
such
3.1
of an o p e r a t o r
X
': T M ÷ a m T' = T' x ' Tx x ' x lx T2x'
3.
< ","
E2x
is r e l a t e d
Ak c A
x
norm
subspaces)
>'.
D
<
ll.llx
d i m E l x = k. For p o i n t s linear
H.llx
!
norm
Recall
!
norm
~
Elx
< .,-
operator
(o__[rt h e i r
scalar
subspaces
Riemannian
]
the u s u a l spaces
= {x ( N ; p ( ~ n ( x ) , A )
l
~ ye Inl
for all
integer
n}.
11 LEMMA
3.1.
Proof.
lim ~ ( Q y÷0
y) = i.
Since
N\~
= {x ( N ; p ( ~ n ( x ) , A )
a,Y
< y~ In[
for some i n t e g e r
n}
we have ~(g\9,y)
Evidently Since
~
[ n=-~
.ff
~({X
6 N;p(}n(xl,A)
{x 6 N ; p ( ~ n ( x ) , A ) preserves
< y~ Inl } = ~-n(u
~, we have
~(~-n(uy~inl (A))NN)
y~Inl}).
<
by
= ~(~-n(u
¥~inlChl) ~. n
(i.i), inl (A)) = ~(U
y~
inl (A))
< clya~alnl
y~
and c o n s e q u e n t l y ,
2Cl Ya uCM\~
,y) = ~(N\Q
,y) l-e a
In the future stant
R
come
we will
consider
from c o n d i t i o n
PROPOSITION
3.1.
(3.2.1)
#(~
(3.2.2)
for e v e r y
d¢nIT ~
N: T~
is a c o n t i n u o u s
(Ba)
exclusively f r o m Sec.
y ~ R, w h e r e
the con-
i.i.
,y) c ~ ,~y, integer
n
the r e s t r i c t i o n
N ÷ TN
map.
Proof. (3.2.1) If y = ¢(x), x 6 p(~n+ix,A) ~ y n+l ~ y~. n. (3.2.2) uous fore
Let
x ( ~ ,y.
in some n e i g h b o r h o o d de n
is d e f i n e d
, then by
The d e r i v a t i v e ,
of e v e r y p o i n t
and c o n t i n u o u s
(3.1)
de
¢ix,
p(¢ny,A)
is d e f i n e d i = 0,...,n
in a n e i g h b o r h o o d
of
=
and c o n t i n - i. x.
There-
12
It f o l l o w s
immediately
for any i n t e g e r s
3.2
We p r o c e e d For
x 6 N
nx
=
exp
The m a p p i n g origin
in
a fixed
~
d% n
is c o n t i n u o u s
to the p r o o f of some b a s i c and
~nx = e x p - 1 %n+Ix
~-i
that
-i ~nx
x E •
~-i
o
o
exp~ n+l
(3.2)
xj
~
is w e l l d e f i n e d in some small n e i g h b o r h o o d of the nx T ~ n x N , ~ n x ( 0 ) = O E T n+l x. Our first a i m is to d e f i n e for the small n e i g h b o r h o o d s
A
n
or e v e n
for all
careful description
of the set
There
exists
a
~
such t h a t
for e v e r y
function
C(x,6)
1 Proof.
e-61nl
~nx
.
Unfortunate-
T h u s n o w we w i l l
A kZ , s , r , a , y
c A k~ , r , s
N
easy
for all give the
N ~ ,y
on w h i c h
subset
A 0 c ~,
lemma.
measurable
~, 0 < 6 < i,
defined
on
A0
there
exists
such that
a positive
for e v e r y
one has
Iivll ~ IldCnx(0) (v)II ~ C(x,d) edinl
The c o n d i t i o n
such that
can be o b t a i n e d .
~,y
invariant
H(A 0) = 1
( T
,r,s
N ~
the f o l l o w i n g
measurable
x ( A 0, n ( Z, v
N #n x
can be o b t a i n e d .
F i r s t we w i l l p r o v e
3.2.
0 ( T
that we n e e d can not be o b t a i n e d
x ( A~
estimations
of
(x)
sup Iid~-ix (h)I[, sup IIdCnx (h) II, h(An(X) h6 An+ 1 (x)
of
ly, the k i n d of e s t i m a t i o n
LEMMA
estimates.
let us d e n o t e
sup Iid2~nx (h)II , sup lid2~-~ x(h)!I h( An (x) h 6 A n + 1 (x)
the n e e d e d
~m~
o ~ o exp n ' ~ x
the good e s t i m a t i o n s
x 6 N
on the set
m,n.
(1.2)
implies
that
Theorem
applied
to
llviI,
I] l°gJidCxiid~I
< + ~"
Thus
M
by B i r k h o f f deduces
Ergodic
that there
exists
~(A 0) = i, such t h a t
a measurable
for e v e r y
x 6 A0
log ¢
Iid~xlI - log invariant
Iid¢~(x) II one
subset
A 0 c ~,
13
logiid¢¢nxI[ lim n÷±~ Let by
x
~ N.
¢(x,v).
L e t us d e n o t e
from
v
IIdCx(V) H, w h e r e
E T N
and
x
!lvlI
=
1
Since
1 < <~(x,v) Iid~xlij then
(3.3)
= 0.
n
(3.3)
< Jid<~xN
one obtains
immediately
¢(~n(x),v))
lim ~( sup n+±~ n v6T N ~n
that
for
x E A0
= 0
rlvII=l Thus
for e v e r y
a n d all
v
~ > 0
6 T
N,
there
exists
N(x)
such
that
InI >_ N(x)
for
ljvll = i, o n e h a s
~n x e
-n6
Now,
for
there
<_ ¢(x,v) (3.4)
exists
every
n
<_ e
n5
(3.4)
one deduces
a measurable
E ~,
v
6 T
N
that
for e v e r y
positive one
~ > 0
function
and every
C(x,6)
such
x 6 A0
that
for
has
~n x
1
-~Inl
C(X,6--------~e
From
Ijvll ~
Ild¢ nx(V)II
the d e f i n i t i o n
(3.2)
mapping,
one
d$
follows
from
sees
that
~n x
of
t > 2
l e t us d e f i n e
1 A.t,6 0 = {x 6 A0; ~ < C(x,~)
For
d
AS has
if
tI < t2
a given
= log ~
follows
@nx'
(v) = d¢
as
nx
(3.5)
d eXPx(0)
(0) (v)
and
is t h e
thus
Lemma
identity 3.2
(3.5)
~or every
Clearly,
~ c(x,~)e6n!iv!I-
~,
then
the
tl,~ A0
from Lemma
condition
3.3,
of
A0
by
< t}. t2,6 c A0
and
0 < e < 1 , let us d e f i n e
(cf.
t,~ A0
subset
lim ~(A t+~ ~
>
0
'~) = i.
by
(1.3))
for e v e r y
(3.6)
x
t,6 ~ A0
, n
~ ~,
v
( T n
N
x
one
14
~blnl
t
t
llVII ~ lld¢nx(0) (v)I! ~ ~
For
x
(3.7)
Hvll.
( ~t~,y' w h e r e
t,6 ~t~,y = q ~,Y n i 0 and i n t e g e r and
n
(3.8)
let us i n t r o d u c e
the f o l l o w i n g
W n ( X ) , V n ( X ) c W n ( X ) , of the p o i n t V
neighborhoods
Vn(X)
%n(x) :
(x) = {y ~ M;
p(y,¢n(x))
< (Q(x)~2blnl)2},
(x) = {y ( M;
p(y,%n(x))
< Q(x)~2blnl},
(3.9)
n W
(3.10)
n where
Q(x)
O(x)
Here
= Q(t,~,y)
= min
(Y, CRag, 6tq
the c o n s t a n t s
condition
is d e f i n e d
yb 2tc2 (4) b ) "
from c o n d i t i o n
Let us recall
that
for
injectivity
of the e x p o n e n t i a l of
From
(3.11), (Ba),
eXPx:TxN
by
as
x ( ~
and
<_ C min(R,
[p(~nx,M\V)]g)
the fact that
On the other
÷ N.
hand,
b > 2g
the radius
Since
(see Sec.
y <_ R, for every
C(y~Inl) g) < C min(R,
N
of
is an
Q(x) since
< CR < C(cf. x 6 ~
< R(~nx,N)
V n ( X ) c W n ( X ) c {y ( M ;
n ( Z
[p(¢nx,A)]g)
and the one has
<-
(3.11)). so that by
<_ 41 yc~Inl < P(~nx,A) .
Q ( x ) ~ 2blnl
1.1),
<_ R(#nx,v)
one has
Consequently
R(x,N)
(3.12)
from the i n e q u a l i t y
<_ min(CR,
O(x) 2blnl_
from
R(x,V)).
Q(x) d 2blnl
We used
map
arise
respectively.
V
= min(p(x,A),
condition
(1.3)
x ( N, we d e n o t e
open
R(x,N)
(3.11)
0 < R, C < l, q ~ i, b and c 2 ~ 1
(B) and
subset
by
p(y,%nx)
and finally <-R(~nx,N)}
(3.11)
o(x) .<_ ¼
15
Thus
-i (Ba) the mapping exp% n(x)
(see Condition
restricted
to
Wn(X) ,
maps this neighborhood diffeomorphically onto the ball M (x) around n the origin in T N of radius Q(x)~ 2blnl (cf. e.g., Section 2 of ~n (x) -i [Che] or the lemma from Section 5.3 in [Gro]). Similarly exp n(x) maps the neighborhood Vn(X) diffeomorphically onto the ball around the origin in T N of radius (Q(x)~2blnl) 2 ~n(x) We will now formulate a lemma saying that for Cnx and ~-i (n_l)x(Cf.
Dings
(3.2)) are well defined on
lemma provides also good estimations of
sup
Ild¢-~x(h) If,
hEAn+l(X)
x E ~t
sup
An(X).
i (x) n the mapThis
sup l!d~nx(h) ll, h6A (x) n
IId2¢nx(h) If,
hEAn(X)
sup
lld2~-~x (h) If.
hEAn+l (x)
The proof of this lemma is given in the next section. LEMMA 3.3. x E Dt
There exist constants for some
¢(Wn(X))
c
c. > i, 5 ~ i S 8, such that if 1
t > 2, then for every
{y E M; p(y,¢n+l(x))
and thus the mapping
¢
n E Z
< R(¢n+l(x),N) }
is well defined on
nx
one has (3.3.1)
Mn(X).
c5t
sup IId~nx(h) II ~ ~ hEM n (x) a ild2~nx(h) II ~ sup hEM (x) n Vn+l(X )
c
c6
(3.3.3)
(3.3.4)
¢(Wn(X))
and thus the mapping
sup
(3.3.2)
lld
hEAn+l(X )
~-i nx
(h) II <
is well defined on
An+l(X).
c7t
'-
c8t3 sup Nd2~-l(h) ll ~ b 4bln 1 hEAn+l(X ) nx y a
Remark.
Although
(3.3.2)
maooings
satisfying all our conditions
(3.3.6)
is true, one can construct the examples of for which no global estimation
16 of the form
lld~xlI
for some constants
[p(x,A)] d
C > 0
and
d > 0
= T' nx(A n(x)).
B n (x)
holds. 3.3•
Let us define
B n(x) = T'nxeXp nx(V n(x)
is a neighborhood of the origin in IRm. Suppose that x ( A k n ~ ,y. We can define a map ~' : nx
B
n
÷ IRm
(x)
in the following way -i -i ~'nx = T'@n+l x o exp n+ 1 x o ~ o exp ~nx o (T' @nx)
(3.13)
It follows from (3.3.1) that ~' is a diffeomorphic embedding nx ! o~ Bn(X) into ~m. Moreover, t h e l i n e a r p a r t o f ~nx at the origin preserves the decomposition ]Rm = ]Rk ~ IRm-k Thus, we can represent
~'n x
in a "coordinate form": (3.14)
~nx(U,V) = (AnxU + anx(U,V),BnxV + bnx(U,V)) where Anx
u c ]Rk , v 6 ]Rm-k, #n+l x = T' o (d~ nXlEl~nx)
Bnx = T'Cn+ix o (d~ n x IE
o
( ~n x) T' -i,,1
) o (T'nx) 2@nx
a
nx and bnx are C P mappings da nx (0,0) = 0, dbnx(0,0) = 0. Since the maps
anx(0,0)
= 0, bbx(0,0)
! Ty, y ( A, are isometries,
= 0,
it follows from (2.2.3)
that
IIAnxll < I' (x)
(llBnZxlI) -I
1
> J
(3.15)
17
Let
tnx =
(anx,bnx).
The f o l l o w i n g
theorem
dure of the r e d u c t i o n manifold
3.1.
There
x E A k N Qt~,y, iIdtnx(Zl)
~emark.
n ~ 0, Zl,Z 2 E B n (x)
b
2.2,
Here
appears
Proof.
for the
into
and the
first
in a c r u c i a l
abstract
proce-
invariant
= {w E T nxN;
The map T n + i x N.
(2.8),
IiZl _ z2II,
A(x)
and
~(x)
come
from
respectively.
the fact that
#
has h i g h e r
smooth-
way.
Ilwll n x < (e(x)~2blnl) 2
An(X) ~
nx
is a c o n v e x
is d e f i n e d
We have
for
neighborhood
on
A (x) n
and t r a n s f o r m s
(z I) - d ~ n x
' (z 2) - d~nx' (0)) II = IId #nx
= Iid~nx (w I) - d ~ n x
of the o r i g i n
in
this
set
Zl,Z 2 E Bn(X)
IIdtnx(Zl ) - d t n x ( Z 2) II = II(d~nx ' - (d~nx
for all
one has
functions
time
such that
stated,
and c o n s e q u e n t l y , N.
H > 0
< yb~(x)HA(x) [~bu(x)]n
and f o r m u l a s
As a l r e a d y
An(X)
~n x
in the s u b s e q u e n t
to P e s i n ' s
a constant
_ dtnx(Z2)ii
(]..3) T h e o r e m
T
exists
the c o n s t a n t
ness
the key role
situation
theorem
THEOREM
where
plays
of our
(Zl)
-
d
(0)) ' Cnx
+ (3.16)
(z 2) II =
(w 2) If', J
where
wi =
(T'nx)-l(z i)
E T nxN,
i = 1,2,
and
II~II' means
the o p e r -
I
ator n o r m for the linear ~:
,
(T ~nxN' II'II~n x
N o w we can use the norms From
operator
II" IIx (2.2.5)
.
+
(2.2.5)
generated and
,
(T n+ixN, II II n+l x and
(2.2.1)
)
to relate
by the R i e m a n n i a n
(2.2.1),
it is easy
the norms
metric
to see that
p.
!
If"IIx
with
18
l!~II' <- /2 A ( x ) e x p ( ( n + l )
s(x))II~II
Consequently Iid~nx (w I) - d C n x
(w2) ll' < /2 A(x) exp(
(n+l)e(x))
•
(w 2) II <- /2 A(x) exp(
(n+l)e(x))
•
• Ild~nx (w I) - d~nx
• IiwI - w2II nx
Since
(3.17)
( max lld2¢nx(W) If) • w=lwl+(l-l)w 2 0~I~i
Wl,W 2 (An(X)
and
A (x)
is convex,
the last
factor
in
n
the r i g h t - h a n d sup WEAn(X)
part of
(3.17)
is less
than or equal
to
Ild2~nx(W) ll.
Combining
(3.16)
(3.17)
and u s i n g
lldtnx(Zl)
- dtnx(Z2) l] <
< 2A(x) e x p ( ( n + l ) s ( x ) )
sup yEA n (x)
(2.2.5)
and
(3.3.3)
we o b t a i n
lld2¢nx(W) ll-llZl - z21 } <_
c6 <_ 2A(x) e x p ( ( n + l ) e ( x ) ) Now,
as
v(x)
~
= exp(-e(x)),
IizI - z2I I. we o b t a i n 2c6A (x)
lldtnx(Z I) - d t n x ( Z 2 ) l l
Thus, 3.4
the s t a t e m e n t
be compact. A k£ , r , s , ~ , y
By the L u s i n
for
in such a way that us fix these
is true w i t h
H = 2c 6.
now for
c I(£,k,r,s,e
x
(x) ]n l)Zl - z211
£ ~ 2, the m e a s u r a b l e set £,~ def= AZk,r,s N ~ ,y N A 0 ~ A p r i o r i this
~(I(£,k,r,s,e,y)) o u s l y on
(X) [ b
of the t h e o r e m
Let us c o n s i d e r
I(Z,k,r,s,e,y)
<_ yb
,
theorem y)
one can
such that
and the s u b s p a c e s x 6 Ak,r,s,e, Y. A£ k,r,s,e,y
find a c o m p a c t
~(A Z k,r,s,~,y) Elx
Moreover
c A~+I k,r,s,e,7
sets once and for all.
and
E2x
~
set may not subset
1 (Z - ~) •
depend
one can c h o o s e for
continuthese
~ = 1,2, . . . .
sets Let
19
It is clear
that
lim £ Z÷= ~ (A k'r's'e'Y)
(cf.
= ~(Ak'r's)
(2.10)) .
y÷0 4.
THE P R O O F
4.1
We will
(3.3.1) %(y)
OF L E M M A now p r o v e
Let
and F
W
Let us suppose
arguments
5
We w a n t
~n(x)
(x). n that
RV(¢n+l(x)) 2 }
there
to e s t i m a t e
and
}(F)
exists
p(¢(w),#n+l(x))
TOPICS
3.3.
For this p u r p o s e
joining
the n e i g h b o r h o o d
P(Y'}n+l(x))
the L e m m a
y (Wn(X).
%n+l(x).
geodesic
3.3 AND SOME R E L A T E D
the d i s t a n c e
let us c o n s i d e r
y; such a g e o d e s i c
is not e n t i r e l y (cf. C o n d i t i o n
the u n i q u e
point
lies e n t i r e l y
contained
(Ba)).
w
( F
between
the s h o r t e s t
Thus
in
{y;
by c o n t i n u i t y
such that
RV(~n+l(x))
=
2
and that of
F
Thus
for all
between
in p a r t i c u l a r ,
where
(4.1)
u ( I, u # w, w h e r e ~n(x)
and
by
w, one has
for all
in
u ( I, ¢
I
we d e n o t e
P(~(u)'~n+l(x))
0%n(x)
the i n t e r v a l <
RV(¢n+l(x)) 2
(u) = ;% "w is well nx (u)
defined
u = exp -I (u) ~n(x)
From condition one has
(Bb)
it follows
lld#uN < q211dCnx(U)II
pC~(w) ,¢n+l(x))
< £(F)
immediately
Z(F)
denotes
for e v e r y
u ( I
and
• maxlldCull
.<- q2Z(F)
" maxlld¢nx(U)II u(I
(4.2)
Now,
from the Mean V a l u e
Theorem
u(I where
that
the length
of
F.
one has lldCnx(U) ll < lld~nx(0)II + i(F)
and c o n s e q u e n t l y
from
p(~(w),%n+l(x))
(4.3)
• maxlld2¢nx(Z) ll, z(I
(4.2)
~ q2£(F)
(3.10)
(lld¢nx(O) II + £(r)
Since
x ( R ,y, from
we have
p(z,A)
>_ p(%n(x) ,A) - p(~n(x) ,z) >_
for
• maxl)d2¢nx(Z)II). z(I
z ( I c Wn(X )
(4.4)
20
>
y ~ lnl
-
-
p(~n(x)
Z) ,
> -
~Inl
1
-
~
Moreover, as Wn(X) c {y ; p(y,~n(x)) one obtains that for any z ( I, < c2 (4)b lld2~nx (~) II - ~
5
=
3
~
Y~
I~1
< RN(X) }, from the condition
(1.3),
(4.5)
•
x E ~te,Y
On the other hand, as (4.5) one obtains that p(¢(W) ,~n+l(x))
Inl
Y~
from
q2q(x)~2blnl (#I
(3.7),
(3.10),
Q(x)~2blnl
+
(4.4) and
V ~
) < (4.6)
< 2tq2Q(x)~ blnl Indeed, Q(x)
from (3.11) one has
< CRying < ~ - 6tq2
and
Sec. i.i) one obtains
Q(x)
O(x)
b ~ bc~(¼) ~
< CRYag < cYdg - 6tq2 - 6tq2"
together with
Now,
from (3.11) one has
Thus as
b > 2g (see
that
-2tq2Q(x)~blnl< RN(~n+l(x))3 ~ ~min(R'C(y~Inl+l)g)
This,
.
(4.6) implies
~ ~min(R'C[D(~n+l(x)'A)
]g) ~ I
that
RN(¢n+l(x)) p (¢(w) , ¢n+l (x)) < Since RN(Y) <_ Rv(Y), This proves (3.3.1). (3.3.2) F
Let
h 6 Mn(X),
h = exp
-i y, where ~n (x)
y 6 W
(4.1).
n
(x).
Let From
The proof of (3.3.4) is based on the following equality (5.6.2) from [Car] applied to n = i)
(see
(4.3) one obtains
(3.3.4) formula
i.e.
contradicts
F c Wn(X) o
be the shortest
(3.3.3)
then the last inequality
follows
geodesic
joining
immediately immediately
~n(x)
(3.3.2) with
and
y;
c 5 = 2.
4 b from (4.5); c 6 = c2(~) .
21
f(h+u)
= f(h)
+ df(h)u + [jl (i - t)d2f(h + tu)(u,u)dt,
(4.7)
0 where
f
open,
is a twofold
convex
dean space,
as
of an Euclidean
space
E
mapping
from an
into another
In particular
this implies
Euclithat
- f(h)II ~ llf' (h)ull - (sup lld2f(z)II)IIull2. z6v
applying
x E ~t~,y'
u 6 ~Sn(X)
V
differentiable
h, u E E, h, h + u 6 V.
Nf(h+u)
Now,
subset
continuously
(4.8)
from
c T
f = ~nx' h = 0 6 T n x N
to
(3.7)
N, i.e.,
one obtains
(4.8)
and to
and
(3.3.3)
if
llull = Q(x)~ 2blnl , then
u E M n (x) '
that if
~n x
llCnx(U) ll > ~ blnl _ ~ 1 (t
> Q(x)~2bln]
because
from
into origin implies
(3.11)
I
c6 yb b - ~
The prooof
f
be a
C2
one has
of
Q(x) ~nx
(3.3.5)
mapping
space
and every
E
<
h ~ E
(4.9)
d f ( u ) (h)
follows
>
Thus
~
Mn(X)
contains
An+l(X).
subset
Euclidean
This
An+l(X).
is based on the following
from an open convex
maps origin
nx
lies outside
(~Mn(X))
V
space.
easy inequality. of
0
of
Then for every
one has ( sup O~s~l
IId2f(su) H)IluIIIlhN.
from the obvious
= d f ( O ) (h)
4bln ]
(Q(x) ~2blnl) 2 1
b Y 2tc2(~) b"
into another
lldf(u) (h)H ~ lldf(0)(h)ll-
Indeed,
Q2(x)
c6Q(x) Q(x) e 2blnl b ) > 2t ¥
and the image
some Euclidean u 6 V
2bln
that the image of the ball
(3.3.5) Let
Q(x)
+
(4.9)
identity
(df(su) (h))ds 0
and from From such that
~s(df(su) (h)) = d2f(su) (u,h) . (3.3.4)
one has
~n~(An+l(X)) c Mn(X ) .
Ild~n~(W) (g)!I ~ c7Hgll
for any
Thus any number
w 6 An+l(X)
and
c7
22
g ~ T n + l ( x )N h 6 T
~n (x)
N
satisfy
for a n y
lld%nx(Y) (h)II >- ~71!hll
and v i c e v e r s a .
Indeed
w =
nx
y 6 Mn(X)
(y)
for some
that
for e v e r y
and
y ~ Mn(X). Applying and
(4.9)
h 6 T
to
one o b t a i n s
f = }nx
y
6 Mn(X)
N }n x
Nd~nx(Y) (h) II ~ IidCnx(0) (h) lI -
Now,
using
(3.7)
and
(3.3.3)
one o b t a i n s
~blnl lld~nx(Y) (h)II !
L~L
hFZ-F Q ( x ) ~ 2 b
b b
from
2~
IIhll.
_
Y
-
2tc 6
Thus one can take c7=2t.
2tc2 (4) b"
two t i m e s
y ~ An+l(X) , one o b t a i n s
I < Id2( nx
the f o l l o w i n g
identity
(3.3.6)
follows
from
the
inequality
" (3.3.3)
3 and
(3.3.5).
W e l e a v e the d e t a i l s
to the reader. 4.2
(4.12)
-i o ~nx(Y ) = y
nx
Now,
~bTnl
) Nhll >-
b
~ - -7
O(x)
Differentiating
(3.3.6)
where
(3.11),
that
c6
( 7
because
lld2¢nx(Y) If)llyllllhIl.
( sup h6Mn(X)
[]
We w i l l n o w s k e t c h q u i c k l y
mappings mapping
¢0,...,¢k_1 ¢ = Ck-i o...o
As for the m a p p i n g results Let
satisfies
of P a r t s M,V
satisfying
assumptions
~
implying
the C o n d i t i o n
(1.2)
be as in Sec.
an o p e n s u b s e t
all our a s s u m p t i o n s .
Nj
to
t h e n for the
of L e m m a
is s a t i s f i e d ,
3.3 holds.
t h a n all
~.
i.i. of
that if the
(1.1)-(1.3),
the e x a c t c o u n t e r p a r t
I-III can be a p p l i e d
and
let us c o n s i d e r
¢
~0
the a r g u m e n t s
For e v e r y V
j, 0 ~ j ~ k-l, ~j :N 3. ÷ V
and a m a p p i n g
Let us d e n o t e
Aj = M\Nj
for
0sj~k-l. Let us c o n s i d e r MI,...,Mk_I,
now,
M k = M 0.
k
disjoint
Let us d e n o t e
copies
of
k-i M = U M i=O l
M
noted
M 0,
and let us d e f i n e
23
on
M
the m e t r i c
~(x,y)
~
as
follows
p (x,y)
if
k-i [ d i a m ( M i) + 1
if this
Let
us d e f i n e
T
restricted
measure with
belong
to some Mi, 0 ~ i S k - i
=
x,y
~
on
to
M
~
on e v e r y
Let
us d e f i n e
N0 = NO
the m a p p i n g
we
M. 1 take
¢:M + M
coincides
as
with
n
the case.
follows:
~ ( M i) = Mi+ 1
~., 0 ~ i ~ k-l. As the 1 measure which coincides
the n o r m a l i z e d
M.. Clearly, ~ is 1 the s u b s e t s Ni c Mi,
n ¢01(N I)
is not
~
invariant.
0 ~ i ~ k-l:
(~i o ~ 0 ) - I ( N 2 ) N...N
(~k-2
.....
%0)-l(Nk_l )
N1 = N1 n #~I(N 2) n (¢2 ° ¢I)-I(N3 ) N...n (¢k-i ..... ¢I)-I(N0 )
~k-1 =Nk-1 n~k~l(N0) n (~0o~k_l)-l(N1) n...n (~k-B.....~0O~k_l)-~Nk_l). Let us n o t e all
that
i, t h e n
one
particular, define
and
is w e l l
easily
any
N. 1
~(x)
k-i U Vi, i=0
( V = like
sees
that
= I.
on w h i c h
on e a c h
if
As
~(x)
is a c o p y
mapping
the
set
n},
where j(n)
~
iterates
of
~
{x ( N;
p(~n(x),
Aj (n))
are w e l l
=
I :
for one
the L e m m a for
-i
map,
if
n = pk + i
if
n = -(pk+i)
we d e f i n e
3.3 r e m a i n s
x ( N O , #(x)
and w h e r e
valid
= Tk(x),
the
for
j(n) some
Let
us
defined
V. the
set
defined,
~(N)
= i,
i
y~ Inl
for all
is d e f i n e d
integers
by
p ~ 0, 0 S i S k - 1
for some p ~ 0, 0 5 i S k - !. sets
for any then
>_
for
In V.
is w e l l
let us d e f i n e
all
0 < ~ < I, y > 0
L Like
,y =
of
of
I
and
u(N i) = 1
submanifold
x ~ N,
V. c M. 1 1
of one
Ni"
~(Ni ) = i, 0 ~ i S k-l.
non-empty
Thus
where
in the c a s e
n ~n(~) n(Z
defined
is an o p e n
k-i N = U N. ; ~(N) i=0 i
Now, =
#
our
~t~,Y . x
( ~t
assertion
The
exact
and
counterpart
the m a p p i n g
concerning
of
T.
follows
As
24
f r o m the fact that
it is true for
We
~.
leave the d e t a i l s
to the
reader.
5.
THE F O R M U L A T I O N We
eral
formulate
form than
OF P E S I N ' S A B S T R A C T
now Theorem
in P e s i n ' s
e n c e of s t a b l e m a n i f o l d s the o r i g i n
of
~m
2.1.i
paper.
INVARIANT
from This
[Pes] 1 in s l i g h t l y
THEOREM less gen-
t h e o r e m d e a l s w i t h the e x i s t -
for a s e q u e n c e
and s a t i s f y i n g
MANIFOLD
of s m o o t h m a p s d e f i n e d
some n o n - u n i f o r m l y
near
hyperbolic
con-
ditions. Let
UI,U 2
spaces
~k
be o p e n n e i g h b o r h o o d s
and
~m-k
b o r h o o d of the o r i g i n Suppose maps
that
of the o r i g i n
, respectively. in
Then
such that e a c h
and the d e r i v a t i v e
dfn(0)
f
is a s e q u e n c e preserves
n
preserves
f can be r e p r e s e n t e d n u 6 U I, v 6 U2,
for
fn(U,V) where
=
An,B n
d~(0,0)
=0.
(AnU + a n ( U , V ) ,
Note that
some neighborhood
and
{u ( ~ k ; HuH ~ r}
by
5.1.
{fn}n~0 (i) and
the s t a n d a r d
(ii)
Let ~
~m=~ to
~m ×]Iqm-k.
(3.8).
(5.1)
= 0, bn(0,0)
= 0, dan(0,0)
are also of c l a s s Fn .
The m a p
generally
Euclidean
[Pes] I, T h e o r e m (5.1)
the o r i g i n of
Fn
C P.
is d e f i n e d
depends
= 0,
on
in
n.
B~,
2.1.1).
and in a d d i t i o n
All mappings B n U, 0 < I < min(l,u)
llAnll < l,
and
(cf.
has f o r m
bn by
of the o r i g i n w h i c h
ll'II d e n o t e s
c P ( p ~ 2)
in a f o r m s i m i l a r
an(0,0)
f0
the b a l l
THEOREM
K
an
fn o ...o
Let us also d e n o t e
where
is a n e i g h -
B n V + bn(U,V)) ,
are l i n e a r maps,
Let us d e n o t e
of
the d e c o m p o s i t i o n
In o t h e r w o r d s , Namely,
U = U1 × U2
~m.
{fn }, n = 0,1,2 ....
f : U + ~m n
in the E u c l i d e a n
are
the
invertible
norm Suppose
t h a t the s e q u e n c e
following
is true:
and there exist real numbers
s u c h that for all p o s i t i v e
integers
n,
!IBnlll-i > ~.
tn(U,V)
such t h a t
and e v e r y n o n - n e g a t i v e
=
(an(U,v),bn(U,V)).
There
exist real numbers
I < 9 < i, K > 0, and for e v e r y integer
n,
Zl,Z 2 E U
and
25
Iidtn(Zl) Then
- dtn(Z2)II
for e v e r y
numbers
C = C(<)
Bk c r0 U I,
(5.1.1)
<- K~-nIizl
real
number
(5.2)
- z2! ].
<,~
> 0, r 0 = r0(<)
< < < min(~,9),
> 0
and
a
there
C p-I
map
exist %
such
that
÷ ~m-k ~:B~0
¢(0)
= 0
and
d~(0)
= 0
and
for all
Ul,U 2 E B k r0
'
Iid~(u I) - d ~ ( u 2) II ~ ClluI - u211. (5.1.2) Fn
For
every
u
E Bk r0
is d e f i n e d
on the
granh
Fn(U,~(u))
6 U
IiFn(U,~(u))II
(5.1.3)
Let
defined such
and of
let
integers
n.
for all n o n - n e g a t i v e
F
(u,v)
be c o r r e c t l y
n If t h e r e
exists
C > 0
that E U
and
IiFn(U,V) II < ~ < n
(5.1.4)
There
continuous
i = 1,2,
defined
exist
for all
0 < ~ < K < min(~,~),
n,
then
v = ~(u).
positive
(l,~,~,<)
functions
E ~4
~i = ~i (l'~'~'<)
satisfying
the
inequalities
v < I,
that 1 r 0 = ~ ~l(l,~,~,<) As
follows
not
values
6.1
the m a p
¢,
6 B k , and r0
integers
6.
n,
~ 200
for all n o n - n e g a t i v e
does
integer
and
'(u,v) E U, u
Fn(U,V)
such
any n o n - n e g a t i v e
from
depend
of
<
INVARIANT N o w we
on
and
(5.1.2)
C = K~2(l,~,~,<). and
(5.1.3)
the
function
<:
in o t h e r
words
¢'s d e f i n e d
coincide
in t h e i r
common
domain.
MANIFOLDS
are p r e p a r e d
F O R MAPS
SATISFYING
to f o r m u l a t e
CONDITIONS
and p r o v e
¢
essentially
for d i f f e r e n t
(1.i)-(1.3)
the m a i n
result
of
26
Part
I - the c o u n t e r p a r t Let
us d e f i n e
the
of T h e o r e m
sets
2.2.1
Ak,r,s, ~
=
m and
Ar,s,~
us d e n o t e
[Pes] I. (cf.
Sec.
3.4)
invariant.
Let
A k£, r , s , ~ , y
y>0
= k~iAk,r,s,a"
also
from
U ~>2
A£ r,s,a,y e
All
sets
are
~
m U A~ . k=l K , r , s , e , y
=
and
those
For
fixed
x 6 A
let us d e f i n e
y(x)
= sup
{r ~ R;
D(~n(x),A)
Z(x)
= inf
{i > 2; -
x E A£ k , r , s , ~ , y ( x ) }"
rrs,~
~ r~ Inl
for all
integers
n}
and
As
x
6 A
£(x)
y(%(x))
In the and (6.2)
we
the
when
and
x £ A
(cf.
Theorem
.< Z(x) e - b e x p ( e
2.1
r,s
and
(3.7))
(6.l)
).
or x 6 A~ by y x 6 A£ r,s,~,y k,r,s,~,7 £(x) d e f i n e d above. and y(x)
respectively and
rrSi~
x Bm - k ( x ) r
(3.2.1),
9~(¢(x))
writino
understand
For
Bk ( x ) r and
>_ c~y(x)
future,
£
then
r,s,~,7(x)
in
r > 0
~m
by
let us d e n o t e
the
set
B(x,r)
set
eXPx(T x)-IB(x,r)
Furthermore,
for
by
x
6 A
U(x,r) . and
rts,~
n ~ 0
we
set
(Q(x) 2 b n ) 2 Un(X)
= exp
It f o l l o w s defined
by
We will x 6 A
rtsrd
V(x)
where
n
(T'n)-iB(x' ~ x ~ x from
(2.2.5)
2
that
Un(X)
c Vn(X) , w h e r e
Vn(X)
is
(3.9). construct in the
the
local
stable
manifold
V(x)
of
a point
form
= eXPx(Tx)-l(graph
Cx ) ,
(6.2)
27
and zero
%x:
k(x) ÷ Bm-k(x) B~ (x) 6 (x) '
the
radius
is s u f f i c i e n t l y s m a l l Z set of the f o r m Ak,r,s,~,y"
on any
6(X)
Naturally, equivalent
conditions
Let
and
bounded
TxV(X)
away
= Elx,
from
which
(6.4)
6.1 and
¢: N ÷ V
(1.1)-(1.3).
is
conditions
of the m a n i f o l d s
of T h e o r e m
6.1.
x (V(x)
but
= 0.
properties
formulation
that
following
= 0, d%x(0)
Subsequent
THEOREM
we e x p e c t
to the
Cx(0)
(6.3)
in S e c t i o n
be a map
Let
listed
below
in the
7.
of c l a s s
~ = ~(r,s)
0 < a < i, r . e x p ( 1 0 0 e r s ) < ~
are
V(x)
CP
(p>_2)
be a n u m b e r
such
satisfying that
10b (*)
(6.5)
I
(b c o m e s
from
(1.3)
Moreover, set
i
let
(cf.
rrs
l' (x)
<(x)
E
<(x)
(2.10))
< <(x)
(For e x a m p l e ,
and
is d e f i n e d
r,s
be a such
¢-invariant
follows
(2.5)).
function
defined
on the
that
< K(x) e x p ( 2 S r , s )
as
by
from
< min(~' (x),~b~(x)) .
(6.5),
(2.5)
and
(6.6)
(2.8)
= X ( x ) e x p ( 4 e r , s)
satisfies
these
Then
there
6(x)
<
inequalities.) exists
a positive
measurable
function
Q2(x) 2
(6.7)
(**) defined
on the
%x,X of the
£
form
properties
set
ik,r,s, ~
Cp
a family
of
C p-I
maps
Ak,r,s,c~ (6.3)
satisfying
(6.4)
such
that
the
followinq
additional
hold.
(*)Let us u n d e r l i n e , between ~ and (**)For
and
t h a t this r and s
smoothness
see
is the f i r s t appears.
[Rue]
2,3"
line,
where
a relation
28
(6.1.1) F
For every non-negative = ~' o...o nx
nx
(where
~
ix
Moreover,
y 6 V(x),
p(~nx,~ny) (6.1.2) has
by (3.13)) is d e f i n e d
~x ) c B(x,
for
y
for all n o n - n e g a t i v e ~ Un(X )
%x
and
Q2(x)~4b(n+l)) .
where
V(x)
is d e f i n e d
by
(6.2),
E U(x,6(x)) inte@ers
a n d for s o m e c o n s t a n t
C, one
n
and
0(~n(x),~n(y)) %hen
on the g r a p h of
~ 2 0 0 / 2 A(x) ( < ( x ) ) n p ( x , y ) .
If for s o m e
~ny
n, the map
~' ox
is d e f i n e d
Fnx(graph
integer
<_ C(<(x)) n,
y (V(x). 6(¢nx)
(6.1.3)
~_ d(x) e 2 b n e x p ( - 2 e
rqs
n)
and inf
~(x)
= @Z r,s,~
> 0.
7
x(A ~
r,s,e,7
(6.1.4)
(6.1.5) A
¢(V(x)
D U(x,~(x) e x p ( - 2 s
There exists
r,s
a measurable
)) N U ( ~ ( x ) , 6 ( ¢ ( x ) ) )
function
G
defined
c V(¢(x))
on the set
such that
r,s,e G(x)
~ ~bG(~(x)),
sup
G(x)
= G
x6A
< r,s,~,y
r,s,e,y and
for
y 6 V(x)
F(TxV(X),(d
one has
exPxl)(TyV(X)))~
where
the a p e r t u r e
A(x)
is d e f i n e d To u n d e r l i n e
ponds ber
F(
in T h e o r e m
is d e f i n e d
Characteristic
we will call
in Sec.
2.1 and the f u n c t i o n
2.2.
that the c o n s t r u c t e d
to the L y a p u n o v T ~ 0,
.,. )
G(x) A 2 ( x ) 0 ( x , y )
local
invariant manifold
Exponents
it local T - s t a b l e
bounded
corres-
a w a y by a m e m -
invariant manifold
(TLSM)
29
of
x.
In the l i t e r a t u r e
one c o n s i d e r s
and in this case one
speaks
folds•
in the r e c e n t w o r k
Nevertheless
simply
primarily
about
local
the case
stable
[Led]3,
T = 0
invariant mani-
T L S M for
T < 0 play
a key role. 6.3
Proof
We will
deduce
L e t us fix a p o i n t following
sequence
Theorem
x ~ A k£, r , s , a , 7 "
of m a p s
fn(Z ) = - 4 b ( n + l )
the m a p s
in the some n e i g h b o r h o o d n.
Lemmas
Thus, f
in
All t h e s e m a p s 3.3.1 and 2.2.5
n
(3.14)
(~-4b A
(3.13).
are of c l a s s imply
to
a priori
is dem a y de-
C P.
of the o r i g i n
and w r i t e
(AnU + a n ( U , V ) ,
A n y of such m a p
that all the maps = D
in the f o r m s i m i l a r
fn(U,V) =
by
of the o r i g i n w h i c h
B(x,Q~)
we can use
D
the
(6.8)
are d e f i n e d
in the n e i g h b o r h o o d
and consider
~nx(4bnx)
fined
nx
1 ~ k ~ m
5.1 .
{fn}n~
where
p e n d on
¢'
6.1 f r o m T h e o r e m
fn in
are d e f i n e d ~m.
down the r e p r e s e n t a t i o n
(5.1).
for
Namely
B n V + bn(U,V))
=~
u + ~-4b(n+l)anx(~4bn(u,v))
(6.9)
nx
~-4b B
v + e-4b(n+l)bnx(~4bn(u,v))). nx
PROPOSITION of T h e o r e m by
6.1
If
~ = e(r,s)
5.1 are true
satisfies
for the s e q u e n c e
(6.5),
o_ff m a p s
t h e n the a s s u m p t i o n s {fn}n~0
defined
(6.8), w i t h ~
=
~' ,(x)
~
~
-
~'~ (x)
,
~
=
v(
x) b
and K = K(x)
Here by
b
(2 8) •
Proof.
=
comes •
(i)
As
HA(x) b(x) 4b from H > 2 -
The
"
(1.3), H and
inequality
from Theorem
A(x)
> _
-
1
-
,
K(x)
3.1 a n d > 1
•
v(x)
is d e f i n e d
30
< min(l,~) follows IIAnlI
immediately
and
(ii) (5.2)
from
llBnlII-I
(6.5)
follow
The i n e q u a l i t y
From
satisfies
(6.6)
(2.9).
I < ~ < 1
is an e a s y c o n s e q u e n c e
Remark.
and
immediately
follows
of T h e o r e m
it f o l l o w s
The
f rom
from
(6.5).
(cf.
(6.9)).
3.1
immediately
inequalities
for
(3.15). Inequality []
that the n u m b e r
< _ <(x)
b~--
the i e q u a l i t i e s
I < < < min(~,~). N o w we can r e t u r n Proposition {fn}n~0 and
6.1 we can a p p l y T h e o r e m
defined
< = <(x) 4b "
r 0 = r0(x), without deed,
to the p r o o f of T h e o r e m
by
(6.8), w i t h
The a p p l i c a t i o n
C = C(x),
for this a i m it s u f f i c e s
In v i r t u e
5.1 to the s e q u e n c e
I,~,~,K
defined
of T h e o r e m
and the m a p
any loss of g e n e r a l i t y
6.1.
As
the n u m b e r s
K = K(x)
we can a d m i t
that
to d e f i n e
(5.1.4)
in
of maps
in that p r o p o s i t i o n
5.1 gives
~ = #x"
of
> i, then
C = C(x) a new
> i. C
In-
by
C = K( 1 + ~ 2 ( I , ~ , ~ , < ) ) . Let us set 6(x)
-
r0(x) C(x)
(6.10)
Since < Q2(x) 2
r0
condition (6.4)
(6.3)
also
< 1
follows
follows
from
from
F r o m n o w on the p r o o f of T h e o r e m
2.2.1
s k e t c h y we will, case
from
(5.1.1)
and the d e f i n i t i o n
of
6(x) ;
(5.1.1). follows
[Pes] I.
exactly
a l o n g the lines of the p r o o f
S i n c e the p r o o f
for the sake of c o m p l e t e n e s s ,
in
[Pes] 1 is s o m e w h a t
give the p r o o f
in our
in full detail.
(6.1.1) defined.
Suppose
that
z 6 B(x,$(x))
a n d that
(fn o...o
f0) (z)
is
Then
(fn .... o f0 ) (z) = ~ - 4 b ( n + l ) F n x ( Z ) = -4b(n+l)
=
1
(T,n+lxOeXp-~n+l
o ~ n + l o e x P x o(Tx)-l) (z) x
31 Let now
y (V(x) .
Then by (6.2) ,
Tx(eXPx)-l(y ) = (U,~x(U)) _k(x) u 6 s6 (x)"
for some
From
(5.1.2)
we have for all
n >_ 0,
Fnx(U,~x(U )) c B(x,Q(x) e 4b(n+l)) . Moreover,
(5.1.2)
implies
that,
ll(exp:ixO ~n) (Y) II, nx <- 200(<(x))nl I(u,~(u))II x.
Thus by (2.2.5), P(~nx,~nY)
one obtains
= II(exp :lnx ° ~n) (Y) II%nx <- /2JJ (exp -l~nxo#n) (Y)ll'n x -<
<_ 200 /2(<(x))nlI(:u, ~(u)II x
200 /2 A(x) (<(x))nH(u,~(u))IIx
= 200 /2 A(x) (<(x))np(x,y). (6.1.2)
Suppose
that
y 6 U(x,~(x))
(T x o exPxl ) (y) = (u,v), where Let us assume that Then
Cn(y)
(fn-i o...o fo ) (u,v)
(Un(X)
c U0(x ) .
k (x) u 6 B6(x). for all non-negative
is well defined.
method as in the proof of (6.1.1),
Then
using
Following
(2.2.1),
integers the same
one deduces
the inequality p(~nx,~ny)
<_ C(<(x))n
that II(fn_l o .... f0 ) (u,v)II < A(~nx) (K(X)~-4b)np(~nx,#nY) <- CA(x) (~-4b<[x)exp(gr, s) )n Let us denote the number -4b
<(x) exp(er, s)
It follows
by
from (2.5) and
I <
<-
from
n.
32
Therefore (6.1.3)
we can a p p l y By
(3.2.1)
On the o t h e r K(x)
= 7
Thus,
hand,
b
=
x ( ~
then
as s t a t e d
~,T
that
v = ~(u).
for e v e r y
in P r o p o s i t i o n
n >_ 0, %n(x)
(
~,ya n
6.1,
HA(x) 4b"
(6.11)
n ~ 0, in v i r t u e
HA (~nx) y b nb (x)~
Now, we pass (6.12)
and o b t a i n
~(x) a
for e v e r y
K (~nx)
if
(5.1.3)
<_
of
(2.2.1)
one has
HA (x) exp (2er,s n) y b nb (x) 4 b
to the p r o o f of
(6.1.3).
(6.12)
From
(5.1.4)
and
(6.10)-
we h a v e
6 (~nx)
r0 (~nx) = C (~nx)
C(x) r0(x)
K2(x) K2 (~nx)
T h e last a s s e r t i o n
of
(6.1.3)
a n d the d e f i n i t i o n
of
K(x) .
(6.1.4)
Let us b e g i n w i t h
original
TLSM
V(x)
the
follows
a 2 b n e x p (_2~ r,s n) immediately
following
corresponding
remark.
from
(5.1.4),
Let i n s t e a d
to the p a r a m e t e r s
(2.2.2)
of
l,~,v,<,K
and
Q2(x) we c o n s i d e r a c o n s t r u c t e d T L S M V' (x) c o r r e s p o n d i n g to the 2 ' same p a r a m e t e r s but i n s t e a d of Q2(x)" we take Q2(x)w" w h e r e 2
0 < ~0 < I.
Then
as f o l l o w s
2
from Theorem
5.1 and f r o m the c o n s t r u c t i o n
of T L S M V'(x) Now, when to
= V(x) let
z (V(x)
N U(x,~(x) w).
y (~(V(x)
N U(x,6(x) e x p ( - e r , s ) ) .
~ = e x p ( - E r , s)
for all
n ~ 0
@n(y)
where
one sees t h a t
Then
By the a b o v e
z E V' (x).
y = ~(z),
remark
Therefore
by
applied (6.1.1)
one has
= ~n+l(z ) ~ U ~ + l ( X ) ,
' Un+l(X)
= exp n+l x ( T ' ~ n + l )x - i B ( ~ n + l ( x ) '
F r o m the d e f i n i t i o n for e v e r y
N U(x,6(x)exp(-~r,s)).
n ~ 0
of
one has
Q(x)
and
(6.1)
Q2(x) 4b(n+l) ). 2 e x p ( 2 ~ r , s) a
it f o l l o w s
easily
that
33
U n + l ( X ) c U n ( ~ ( x )) . Consequently ~n(y)
= ~n+l(z ) E Un(#(x)).
On the o t h e r hand, n
(y) = ~
n+l
as
(z) ~ Un+ 1
z 6 V(x),
by
(6.1.i),
for all
n ~ 0
(x)
and = p (~n-i (~(x)), on-! (y))
p(on(x),~n(z))
= NA(x)<(x)(<(x))n-lp(x,z) where,
N =
200/~
5 NA(x) (<(x))np(x,z)
= C(<(x)) n-l,
and
C = NA(x)<(x)p(x,z). Since
y
(6.1.5) 6.1,
6 U(#(x),~(~(x)))
then
F i r s t of all n o t e
it f o l l o w s
that
t h a t for any
(6.1.2) from
v
implies
(5.1.1),
E B(x),
that
(5.1.4)
y
6 V(¢(x)).
and P r o p o s i t i o n
one has
(6.13)
lldCx(V) II <- C(x)IIvll, where
C(x)
= K(I + ~ 2 ( l , ~ , v , < ) )
and
K =
£
HA(x) 4b
y ~(x)~ C(x)
where
L
= A(x) Yb
"
Consequently,
L(X) ,
is a
~-invariant
function
constant
on the set
Ar,s,e
Let v
~ V(x),
(T~oexPxl) (y) =
Let us c o n s i d e r F =
parallel
the l i n e a r m a n i f o l d
(d e X P x I) (TyV(X))
W e w a n t to e s t i m a t e to
such that for
F.
Let
(U,¢x(U)).
c TxN.
F(TxV(X),F') II'II a n d
some c o n s t a n t s
where
F'
is the l i n e a r
II'II' be two n o r m s a > 0, b > 0
subspace
on the s p a c e
and for
0 ~ v
6 H
H
34
allvll' <-]]vH Then
<-bl]vII'
it is easy
dim
to see
E = d i m F,
a F(E,F) b where
one
that
b < a F(E,F)
< F' (E,F) . . . .
F(E,F) ,F' (E,F)
to the n o r m Using
manifolds
E,F
c H,
and
apertures
computed
with
respect
respectively.
(2.2.3)
one
sees
that
in our
situation
<_ /~ A(x) F' (TxV(X),F').
to see t h a t
F' (TxV(X) ,F') Therefore
linear
'
the
]] ]I'
remark
F(TxV(X) ,F') It is e a s y
denote
II'[] a n d
this
for any two
has
using
<_ IId~x(U)If.
(6.13)
F' (TxV(X) ,F')
one
obtains
<_ C(x)[]ul] <- C(x) (]]uN+ll~x(U)[])
< /2 C(x)ll(U,¢x(U))]]
<- /2 C(x)A(x)]]exPxl(y)
I;x
= /2 C ( x ) A ( x ) 0 ( x , y ) . Thus,
we
finally
obtain
F ( T x V ( X ) , F ' ) <_ G ( x ) A 2 ( x ) p ( x , y )
where
G(x)
deduces
= 2C(x).
as
sup
L
is c o n s t a n t
G(x)
¢(x)
6 ~ ,ya , t h e n
one
easily
on the
set
A
rrS,~
from
(2.2.2)
one
< + ~.
has
D
ris,~
The
size
Nevertheless, Ar, s
x 6 ~ ,y,
>_ ~ bG(~(x)) .
Moreover,
6.4
for
that
G(x)
x6A
As
,
such
inequalities
6(x) if
that
of this
~I <(x)
(6.6)
is a
manifold
~-invariant
<
then
V(x) =
for
as f o l l o w s
function
x EAr, s from
V(x,<)
and
(6.1.2)
depends
defined
on
<.
on the
set
satisfies
35
V ( x , < I)
More manner
n U(x,6(x))
generally,
but
{fn}n>0
if one
such that
defined
V(X,<).
=
by
chooses
Theorem (6.8),
5.1
X,~,~,<
one has
of t h e T L S M
depends
o n the c h o i c e
theless,
as
follows
from Theorem
unique.
F r o m n o w on,
and
is a p p l i c a b l e
K
in an a r b i t r a r y
to t h e
the corresponding of p a r a m e t e r s
6.1.2
the
sequence TLSM.
The
I,U,v,<,K.
g e r m o f the T L S M
size
Neveris
for
x ( i~ rrs,~,y' we will
consider
exclusively
the T L S M V ( x ) ,
corresponding
to t h e
parameters
I =
r exp(3er, s ) b ) s e x p ( - 3 e r ,s ) 4b ' ~ = 4b ' ~ = ~ exp(-Sr,s '
< = r e x p ( 4 E r , s) a n d
For
these
parameters
5.1 a r e
satisfied
(6.8).
All
7.
We
of S e c t i o n THEOREM (7.1.1) x,y =
statements
formulate 2.3
HA Z r,s b 4b y ms
by Proposition
for the
SOME ADDITIONAL
7.1
K -
sequence
of T h e o r e m
PROPERTIES this
"
6.1,
assumptions
of m a p p i n g s 6.1
remains
OF L O C A L
section
the
some
STABLE
results
{fn}n~0 true
of T h e o r e m defined
by
for them.
MANIFOLDS similar
to the r e s u l t s
[Pes] I.
7.1. There
exists
~ i~
L = L(Z,r,s,d,y)
,y ~ U ( x , L )
and
y
such ~ V(x),
that
if
t h e n V(x)
n V(y) N U ( x , L ) =
~.
(7.1.2)
There
exists
M = M(Z,r,s,~,y)
x,y
~ A~ r,s,e,y
and
V(y)
n U(x,M)
c V(x).
y
such that
if
~ V(x),
then
(7.1.3)
-If -
x
~ i kZ , r , s , ~ , y '
xi
E i kZ , r , s , ~ , y,i = 1,2 , .... x i
xl t h e n
36
for e v e r y
q :0 < q <
V ( x I)
N U(x,q)
i__nn C 1
topology.
Proof.
(7.1.1)
r,s,~,y'
+ V(x)
We will
N U(x,q)
prove
that
if
2 0 < L <
(Q(x))
1600(A~,s )3 then
V(x)
Then
one
N V(y)
can
From
(6.1.1)
one obtain
N U(x,L)
= ~.
find a point
that
applied
to
for all
p(~n(x),~n(y))
L e t us s u p p o s e
z ~ V(x)
N V(y)
z 6 V(x)
and
that
this
is n o t
true.
n U(x,L). z 6 V(y)
respectively
n ~ 0
~ 0(~n(x),~n(z))
< ~ , s ( K r , s ) n [p(x,z) _ 200/2 A r 800(A~,s)2(Kr,s)nL
+ p(~n(z),~n(y))
+ p(y,z)]
~
(7.1
= C ( < r , s )n 4
The
last
inequality
diameter
diam
follows
U(x,L)
from
~ 2A ~
the
fact
that
the R i e m a n n i a n
L.
rts
Thus, have
from <
we have
= r e x p 4~
r,s
the
second
assumption
of
(6.1.2).
We
(6.5)
We want
to c h e c k
~ny
6 Un(X).
This
verified
together
that
with
< ~ 4b.
r,s
for all
(7.2 n > 0 (7.3
(7.1)
will
assure
that
y (V(x)
which
is a
contradiction. We want %ny
to c h e c k
E U
that
for all
n t 0
(x). n
From
the definition
of
U
n
(x)
and
from Theorem
2.2
it f o l l o w s
that U n(x)
9 {w 6 M ; p ( w , ~ n ( x ) )
< Q2(x) _ - -
4bne-n~r,s}.
2AZ
rrs
Thus n t 0
to f i n i s h
the p r o o f
it is s u f f i c i e n t
to see t h a t
for e v e r y
37
800(A
r•
,s)2L
) n < Q2 (x) - 2A Z r,s
But this
is so in v i r t u e of
(7.2)
(7.1.2)
The proof goes along
We l e a v e
the d e t a i l s
(7.1.3)
By d e f i n i t i o n
E l x ' ÷ Elx.
Thus,
4bn
-ng e
r,s
L.
and of our c h o i c e of
the same lines as the p r o o f
of
(7.1.1).
to the reader. of the sets
by T h e o r e m
A£ r,s,~,~ for e v e r y
6.1,
(cf. Sec.
3.4), 6Z 0 < q < r,s,e,7'
q,
1
and for
i
big enough,
the m a n i f o l d
V. (x) n U(x,q) 1
is of the f o r m { e X P x ( T i ) - l ( u , x i ( u ) ) ;Ilull ~ q}, Xi : Bkq ÷ ~ m - k ,
where
k = k(x)
= k(x i)
for all
i
under
considera-
tion. It f o l l o w s
from
(5.1.1),
exists
such a c o n s t a n t
every
Ul,U 2 6 B k q
(5.1.4)
CZ r,s,~,y one has
and P r o p o s i t i o n
that for
i
6.1 that t h e r e
big e n o u g h and
for
lldxi(u I) - dxi(u2)II < C r,s,~,yIlUl Z - u21 I. This
implies
that the
family
{X i}
be a l i m i t p o i n t of the f a m i l y ip ÷ ~ dxi
one has
(0) ÷ 0
Xip ~ ~
as
in
C1
p ÷ ~, ~(0)
= 0
is c o m p a c t {Xi} , i.e.
in
topology.
Since
and
= 0.
d~(0)
C1
topology.
for some s e q u e n c e Xip(0)
+ 0
P Let
u 6 B k. q
L e t us d e f i n e
Zlp • = eXPx(T~)-l(u,xi p (u))~
!
z = eXPx(Tx)-l(u,~(u)). Let us fix e > 0, for
p
n > O.
J
Because
b i g enough,
x1 ÷ x
one has
and
z.
+ z, for e v e r y
i
P
and
Let
38
p(z i ,z) ~ s, P
P(x i ,x) <_ a, P
p(¢n(z i ),¢n(z)) P
~ s
p ( ¢ n ( x i ),%n(x)) P
<_ s
But
z
V ( x i ), so t h a t P
i
P
o ( ~ n (z i ) , ¢ n ( x i ))
P
P
and consequently
from
p(~n(z),~n(x))
As
~ > 0
.
(6.1.1)
and
(2.2.2)
one has
~ N A ~ , s ( < r , s ) n p (z i ,x i ) P P (7.4)
is a r b i t r a r y ,
C = NA Z
from
5 NA~,s(P(x'z)r
@(%n(z) ,%n(x))
where
(7.4)
(7.5)
(7.5
+ 2~).
implies
that
for all
n { 0, one has
5 C(
(7.6
Moreover
rts
(7.7
z ~ U ( x , ~ r,s, ~ ~,7) because
u
6 Bk q
0 < q < 6Z r,s,e,y
and
Thus,
to f i n i s h
~n(z)
E Un(X).
the p r o o f
it r e m a i n s
to be s h o w n t h a t
for all
na0
This
suffices
(7.8
because
then
(7.6)-(7.8)
and
(6.1.2)
will
imply that
z ~ V(x). Let us fix ~n(z i)
n ~ 0.
E U n ( X i)
the f u n c t i o n s y(x)
From
because y(x)
and
~ lim sup y ( x i)
(6.1.1)
Z(x) and
X,+X 1
Thus, Un
t
as
one has t h a t for any
z i 6 V(xi).
Now,
one sees e a s i l y Z(x)
i > 0,
f r o m the d e f i n i t i o n
of
that
~ l i m inf Z(x i) X,÷X 1
z. ÷ z 1
one obtains
that
and
x. + x, f r o m the d e f i n i t i o n 1
of the
sets
39
~n(z)
(Un(X).
The following of
statement
is an immediate
consequence
of the proof
(7.1.3).
COROLLARY
7.1.
Numbers
6Z r,s,~,y
can be chosen
so small that for
any
x,y
(
i r,s,@,y' £
Y
( U(x,
6 r,s,~,y Z )
one has V(y)
@ U(x,6 £r,s,~,Y)
= {eXPx(T~)-l(u,~y(U))
;u E B k(x)~£
}
r,s,~,~ where :
~y
B k(x) @£ r,s,@,y
÷ Bm-k(x) 6£ r,s,e,y C p-I
is a map of class
such that
max
max
y~A~, s ~,7NU(x,~6~ ,
,
s,e y)
(IICy(U) II+IidCy(U)II) < i.
u6Bk(X)
,
~£
r,s,~,y Let
6Z r,s,~,y
be chosen
according
to Corollary
7.1.
Let
x 6 A k , r , s , a , Y. By the family of local
T-stable
mean the set of all local
manifolds
T-stable
S~ (x) k,r,s,~,y
manifolds
passing
we will
through
all
the points y 6 Ak,r,s,~, Y N U(x,~ Finally
let us formulate
exact analogue x ~ A
rts,~
converge
~ ~r,s,~,7 ) .
and
the following
of Propositions y E V(x).
with exponential
along the trajectory from the linearized
of
2.3.1.1
Since speed,
y
equation
theorem,
and 2.3.1.2
the points
the linearized
may be considered for
~n(x)
¢
which of
is an
[Pes] I. and
equation
for
as an equation
along the trajectory
Let
~n(y)
of
obtained x
40
by a r a p i d l y
decreasing
perturbation
theorems
also
[Rue]2, 3 and
THEOREM
7.2.
(7.2.1)
Let
regular s(x)
(7.2.2)
#n(y)
lld~nvll
(7.2.3) v
~n(y)
Let
= Xi(Y),
ki(x)
= ki(Y),
y E A
r,s
n ~ 0
~ K r n e x p ( e r , s n ) IiVlIy for
~ K-isnexp(-er
and y r,s,e - -
1 lim sup -- logIId~n(v) II < T n÷~ n Y 2.3).
and 17.1.1 of
v 6 V(x).
s u c h t h a t for e v e r y
x E A
remark
and f r o m [Byl],
see
T h e n the p o i n t
y
2.2) and
and let
E T V (x) Y
(cf. Sec.
15.2.1
a n d let
(see Sec.
x E Ai r,s,~,y
K = K(£,r,s,e,7)
F r o m this
one o b t a i n s
x E A~ r,s,~,y
= s(y),Xi(X)
Nd~nvll
(see T h e o r e m s
[Cam])
in the f u t u r e
Let
perturbation.
1 ~ i ~ s(x).
n V(x)
•
There
one has v 6 Ely = T V(x), y
sn) iiviiy for
v E E2y"
E V(x),
for e v e r y v e c t o r
t hen
exists
is
PART ABSOLUTE A. Katok
i.
II
CONTINUITY
and J.-M.
Strelcyn
INTRODUCTION The p r e s e n t
called
absolute
sponding
to the L y a p u n o v
T < 0 (TLSM) Sec.
part c o n t a i n s continuity
a full and very d e t a i l e d
of families
Characteristic
for d i f f e r e n t i a b l e
1 of Part
the proof
in
Pesin
in Sec.
[Pes] 1 c o n t a i n details
After
this
is done,
general
extra
of class
3 of his several
gaps
than fixed
described
proofs
[Pes] I.
in m a n y places
he does
this m a k e s
in clear
the g e n e r a l i z a t i o n
these paper
All
continuity
case of m a p s with
C l+e
fundamental
of his arguments.
sent the proof of a b s o l u t e
more
not bigger
singularities
corre-
in
I.
by Ya.B.
give enough
Numbers
maps with
In the case of d i f f e o m o r p h i s m s given
proof of so
of i n v a r i a n t m a n i f o l d s
necessary
and c o m p l e t e
does
not lead
not to pre-
form.
from the d i f f e o m o r p h i s m
singularities
are However,
case
to
to any
complications. Our proof of a b s o l u t e
plan of Pesin's arguments proof
in his presentation.
is too lengthy;
of the proof would Sections Secti o n s
Probably
it is c o n c e i v a b l e
shorten
due
very c l o s e l y to errors
this
the general
and i n c o m p l e t e
is the reason why our
that more
fundamental
revision
it substantially.
2 - 7 are preliminary;
rough
nuity T h e o r e m is d e v o t e d
the proof
itself
continuity
sketch of the idea of the proof
is given
just after
to one c o n s e q u e n c e
the a p p l i c a t i o n
in ergodic
in infinite
an e x t e n s i o n
of finite
larger
the f o r m u l a t i o n remain
follows
is c o n t a i n e d
in
8 - i0.
A very
somewhat
continuity
proof with m o d i f i c a t i o n s
its f o r m u l a t i o n
in Sec.
of our m a i n results,
theory.
Sec.
Hilbert
dimensional
framework
is also
important
result
very
12 is d e v o t e d
dimensional
class of m a p p i n g of some
of A b s o l u t e
Space
indicated. from
4.2.
Sec.
important
ii
to
to the a b s o l u t e
setting.
described
Conti-
Moreover,
in Sec.
Sec.
[Pes] 2 and
11 to
13 c o n t a i n s [Kat] 2 w h i c h
true in our framework.
The n o t a t i o n s
used
are d e s c r i b e d m o s t l y
in the p r e s e n t
in Secs.
not to read S e c t i o n
2 carefully
with
appears.
it when
a need
Let us note an example for w h i c h
of a
C1
Anosov
and L.S.
diffeomorphism
continuity
fails
They
The reader m a y be advised
at the beginning,
that C. R o b i n s o n
the a b s o l u t e
part are v e r y complicated.
2.2 - 2.6.
Young
but rather
see
[Rob]
constructed
of t w o - d i m e n s i o n a l
for stable
consult
torus
and u n s t a b l e
42
foliations. invariant little
2.
Unfortunately,
measure
doubt
that
2.1.
The basic same
have
as
AND
there
measure.
example
is no f i n i t e However,
of t h a t
sort
there
also
is a
exist.
NOTATIONS
notations
in P a r t
a different
example,
to L e b e s g u e
an a r e a - p r e s e r v i n g
PRELIMINARIES
the
in t h e i r
equivalent
I.
used
here without
Nevertheless,
meaning
than
many
in P a r t
I.
special symbols
But
notification used
it n e v e r
in this leads
are part
to
ambiguities. The
complexity
necessity norms. Part of
of the
As
I are
2.2.
The
concerned, 3.10
proof
Condition for e v e r y
x,y
we w i l l
from
Part
(Bb)
from
Sec.
For
6 V
satisfying
a consequence
Riemannian
formulas,
simply
of the
and L y a p u n o v
theorems,
"formula
etc.
3.101
from
instead
etc.
Continuity
Bc.
every
)Id eXPx(W) ll ~ 1 + e
to the
write
I",
of A b s o l u t e
of C o n d i t i o n
has
is p a r t i a l l y
use of b o t h
far as the r e f e r e n c e s
"formula
ment
of n o t a t i o n s
simultaneous
requires
e > 0
there
p(x,y)
where
the
following
reinforce-
2 I. exists
rE > 0
< m i n ( r e , R v ( X ) ) def
w = exPxl(y)
and
such
that
Re(x)
one
lld(exPx I) (Y) II
~I+~. Obviously smooth of
Starting
mapping
Let s
is a l w a y s
manifold
and
satisfied V
when
is a s m o o t h
M
is a
open
submanifold
always
the m e a s u r e
the n u m b e r s
and
e,0
Sr,s ) < 1
and
er, s)
satisfied. =
we will
that
so small
r exp(50
~,
l, k, r,
< e < i, r exp(100
such
suppose all
s
that
Sr, s)
that
for
conditions
the
from
are
fixed.
L e t us
the c o n d i t i o n s
< s
(2.5) I
< s10b
Then
(6.5) I
we d e f i n e
numbers
I,Z,K
~r,s
= r exp(3er, s) , ~ = ~r,s
Xr,s
= r exp(4er,s),
L e t us n o t e defined
preserving
us a s s u m e
rrs
Ii
this m o m e n t
satisfied.
r exp(100
are
from
~: N ÷ V
11 are
2.4. fix
(Bc)
Riemannian
M.
2.3.
Sec.
Condition
compact
that
in P r o p o s i t i o n
these 6.11
and
v
as f o l l o w s
= s e x p ( - 3 e r , s)
(2.1)
~ = Vr, s = e x p ( - e r , s)
quantities
are d i f f e r e n t
and at the end of Sec.
f r o m the o n e s
61 and d e n o t e d
by
43
the
same
symbols.
L e t us d e f i n e = ~r,s
also
= r e x p ( 7 e r , s ) , ~' - ~ = r e x p ( 8 e r , s ) , (2.2)
^
r exp(12Sr,s) J q = qr,s There
are no p a r t i c u l a r
definition
(2.1)
inequalities from
-
which
(2.5) I and X < ~ < ~' --
<
P
(2.3)
= exp(-7er,s) reasons
(2.3).
What
easily
follow
we
for c h o o s i n g
actually
f r o m our
need
choice
the
constants
is the
in
following
of the c o n s t a n t s
and
(6.5) I. < ~ < ~
(2.4)
q
(2.5)
< ('
< q
(2.6)
~-- < n
2.5.
(2.7)
We assume
following
exp(_Sr,s)
where
p
that
satisfies
(2.5) I,
(6.5) I and
the
def
(2.8)
< 2bp
m - k ~ 1
ll'I[ d e n o t e s
norm
the n u m b e r
inequality:
Let
w
6 A
in
TwN
Let
w
and
the (of.
Sec.
comes
II-]Iw (cf.
from
Euclidean
2.2i). !
and by ( Ai
b
standard
By
norm
II'llw
the L y a p u n o v
Sec.
condition
2.3i).
in
we d e n o t e
norm
From
1.3 I. ]Rm. the R i e m a n n i a n
(See Sec.
the
2 I)
inequalities
(2.2.1) I
rts
and
(2.2.5)
I
one
__I l[vl[ /~ ~n(w)
Sometimes, T' ~n(w )
and
will
and
use
when
w
v 6 ]Am
frequently
similar
the
that
< Hvll~n '
for
We w i l l metric
sees
same
for e v e r y
( A
is fixed, [Iv[[ n
consider
objects symbols
n
and
v 6 T
¢n(w )
N,
< . - A rl, s e x p ( n e r , s)llvll #n(w)
(w)
'
integer
we w i l l
write
instead
of
objects
related
related for t h o s e
T' n
instead
of
I[(T 'n ( w ) )-ivll~ n(w) "
to L y a p u n o v objects
(2.9)
to the
metric.
adding
~
Riemannian Usually above
the
we symbol
44
in t h e
Lyapunov
fixed,
we will
metric
case.
In p a r t i c u l a r ,
when
w
( Ak,r, s
is
denote
Bnl(q)
=
{v ( Ei;llvN n _< q},
]~i(q)
=
{v ~ Ei;llvI[ _< q},
Bn(q)
=
Bln(q)
x B2(q) n
(q)
=
~l(q)
x ~2(q) ,
(2.10)
where
E1
=
1R k
and
Furthermore,
E2
for
=
IRP;
p
v 6 E. ,
=
m
-
k.
i = 1,2,
and
r > 0.
1
B
< and
(v,r)
=
{z
B 1 (v, r)
=
B i(v,r) 0
Bi(v,r)
=
{.z 6 Ei;
for
I
z =
(u,v),
Bn(Z,r
t
2.6.
E Ei;
Nz-vll n <- r} (2 .ii I[z-vll
u 6 El,
-< r}
v ( E 2.
=
Bl(u,r) n
x B2(v,r) n
=
Bl(u,r)
× B2(v,r)
(2.12 B(z,r)
U(w,q)
=
As
in P a r t
(2.13
e X P w (Tw) "I (B (q))
I,
for
n t
0,
and
w
we
( i£ r,s,~,~
will
use
the
notations
~nw
nw'
-i n+ 1
=
exp
=
T' C n + l (w)
Sometimes, and nw
Vn(W)
~'
n
:
o ~ o exp
(w)
when
o exp
w
-i n+ i
( i
n
(3.2) (w)
o % o exp
n
(w)
is
instead
of
{y
p(y,~n(w))
6 M;
%
o
(Tn)-l.
(3.13)
(w)
fixed,
we will
write
instead
of
n
~'
nw
<
(Q(w)e2bn) 2
(3.9)
and (2.14)
I
45
By L e m m a A n ( w ) d-e-f
3.3i,
exp -in
~nw
is a d i f f e o m o r p h i c
(Vn(W))
T n+ 1
into
(w) morphic
embedding
Since then
for ¢-i nw
N,
while
#' nw
(w) of
the m a p
Bn(W) ~-i
= T~(An(W))
is well
into
defined
on
of is a d i f f e o -
~m. Vn(W)
(see L e m m a
3.3 I)
n ~ 0 -i exp~ n(w)
=
Consequently ~-p nw
empedding
def
o ¢-i o exp n+ l(w) "
for
1 ~ p ~ n+l
-1 exp¢ n-p+!(w)
=
we
have
o ~-P o exp
n+ l(w)
¢-i o ~-i (n-p+l) w (n-p+2) w
o
¢-i. nw
In p a r t i c u l a r ~-n (n-l)w
2.7.
Let
~Je d e n o t e
y(E,F)
where are
=
~-i Ow
E,F by
be two y(E,F)
=
denote
in an E u c l i d e a n
angle
between
E
space
and
F,
H. i.e.
F(E,F)
was
the n o r m
(2.15)
the a n g l e of
H
to see that
If we r e p l a c e generating
manifolds
the m i n i m a l
subspaces
It is easy aperture
linear
o ~-i (n-l)w"
min y(e,f), 0#e6E ' 0#f6F'
y(e,f)
linear
o ~-i lw . . . . . . . .
between
parallel always
defined
to
II'II' s u c h
and
f
and w h e r e
and
F
respectively.
2F(E,F)
in Sec.
the E u c l i d e a n
e E
2.11
structure that
> 7(E,F), (cf. in
for e v e r y
also H
where Sec.
E',
F'
the
3.4).
by a n o t h e r
one
v
a!Ivl]' ~< IrvN z b]]vH' then
the n e w a n g l e
satisfies a
the
2-~ 7(E'F)
y' (E,F)
computed
with
respect
to the n o r m
H'II'
inequalities
< Y' (E,F)
~< - - y(E,F).
(2.16)
46
2.8.
As
Int
Y
usual
by
denotes
Y
we
the
denote
interior
the
of
closure
Y.
diam
of
Y
the
set
denotes
Y.
the
When
y c N,
diameter
of
Y. The
cardinality
As
usual
If
f:
@
X ÷ Y
the
restriction
the
standard
Let
3.1.
We
(B)
The
be
assume
A
~
AND
known
#Y.
f IX1
by
this
does
~ cp(~ ,~k) to
FROM
facts
Y
not if
is o f
LINEAR
we
denote
conform ~: ~ ÷
class
to
~k
C p.
ALGEBRA
related
continuous
us
p
p-I
to
the
general
mappings
between
A
by
J(P)
-
0) . X
is t h e
o to
=
0
for
with
probability
B
respectively.
every i.e.
B p
~ 9, is
absolutely has
continuous
~(p(A))
= 0.
if Let
continuous. probability ~p
measure
is a b s o l u t e l y - Nikodym
= d~p du p. p
and for
p ~
of
some
on
-i
are
subset
by
the with
can
which
formula, respect
consider is c a l l e d
absolutely all
of
Borel
absolutely
one
X,
almost
all
~p
continuous
Theorem
defined
- algebra of
i.e.
one
is a c o m p a c t
X
and
is b i m e a s u r a b l e ,
is c a l l e d
1 j (p-l) op
that
mapping,
~(A)
when
1 = +--j
restriction
A
Radon
that
J(p)
and
spaces
- algebras
p
a new
mapping see
suppose
~m. the
to
then
us
A
Thus
the
two measure
~
measurable.
that
measure
function
~1 = +~ Let
on
~.
be
p
are
mapping such
The
of
that
is a b s o l u t e l y
define
is e a s y
on
a measurable
and
6 A
that
Jacobian
is
defined
us
p
the measure
of
THEORY
(Y,~,~)
Let
continuous,
on
and ~
= ~(p(A)).
It
W
by
then
Although
write
absolutely
X ÷ Y
set
assume
admit
c X,
restriction
well
of
p:
the measurable the
some
bimeasurable
Let
to
Xl
set.
whose
and
and
each
~p(A)
and
we will
~
6 A.
bijective
us
denoted
X I.
MEASURE
here
(X,A,~)
Let
for
FROM
be set.
spaces.
Let
-i
will empty
an o p e n
of J a c o b i a n
measures
p
be
mapping
recall
Y
the
to
notations,
FACTS
definition measure
f
y c ~m
SOME
set
is a m a p p i n g
of
is a c o n t i n u o u s
3.
of
denotes
points
a smooth
subsets
of
continuous
X
(we
submanifold X
and
measure
W. Let
called
an
From
x
6 X.
A sequence
x
- sequence
the
Lebesgue
if
{Ar}r~ 1 x
6 Ar
- Vitali
for
Theorem
of
open
any on
balls
r > 1
of and
X
will
be
l i m d i a m ( A r) = 0. r÷~ differentiation (see f o r
47
e x a m p l e Chapt.
l0 of
J(p)(x)
lim r÷~
[Shi] I) it f o l l o w s
that
~(P(Ar)) =
for a l m o s t 3.2.
If
all p o i n t s H
(''')H'
E
dimension,
x 6 X
and for all
is a f i n i t e d i m e n s i o n a l
then by
Let
(3.1)
~ ( A r)
Vol H
and
F
we w i l l
respectively.
Let
scalar
E1 c E
be a l i n e a r m a p p i n g .
vector
denote
{Ar}rt I.
space w i t h a s c a l a r p r o d u c t
the c o r r e s p o n d i n g
be two real v e c t o r
provided with
x - sequences
products
be a l i n e a r
volume
on
H.
s p a c e s of the same f i n i t e (''')E subspace
and of
(''')F E
and
A: E ÷ F
L e t us d e f i n e
VOlFI(A(U)) A E1
= V O l E I (U)
where
U
is an a r b i t r a r y
F1
is an a r b i t r a r y
E1
and
A(U)
Let
X
dimension on and
X
and
and
and Y
c F I.
Y
Y
3.3.
define volume by
is e s s e n t i a l l y
z 6 X
by
E1
manifolds
of the same
as
finite
Riemannian
and consequently ~y
and w h e r e
Idet A I •
diffeomorphism.
and
situation
one has
measures
metrics on
X
respectively.
a standard
=
Idet dT(z) I
one has
= Iyf d~ Y"
change
of v a r i a b l e s
s e c t i o n we w i l l
than usual.
Hilbert
d(~yOT) - - ( z ) d~ X
f ~ LI(y,~y)
'det d T ' d ~ x
For the r e s t of this
9
s u b s e t of
of the same d i m e n s i o n
3.1.
For a n y f u n c t i o n
[Rog]
F
IAIEI
elements
(3.1.2)
Let
C1
~X
For a n y
dimensional (cf.
be a
(3.1.i)
general
of
be two R i e m a n n i a n
T: X ÷ Y
IX (f°T)
This
subspace
We w i l l d e n o t e
w h i c h we d e n o t e
PROPOSITION
open and bounded
linear
Namely,
let
formula.
deal w i t h H
a slightly more
be a f i n i t e or i n f i n i t e
space.
d e n o t e the p d i m e n s i o n a l H a u s d o r f f m e a s u r e in ~P~H P for H a u s d o r f f m e a s u r e s ) . When H is f i n i t e d i m e n s i o n a l
48
restricted P with (cf
the
p
to a
p
dimensional
dimensional
[Rog]).
For
volume
U c ~P,
submanifold
(Lebesgue
by
~
(U)
of
measure)
we w i l l
~P@H
coincides
on this
submanifold
denote
its
p
dimensional
P Lebesgue
measure.
PROPOSITION f: U ÷ H
3.2.
be a
Let C1
U c ~P
be an o p e n
mapping.
If
bounded
set and
suplIdf(v) ll s d, vEU
let
then
£ ~p(U)
~ ~p(graph
f) ~
(i + d 2 ) 2 ~
(U).
(3.2)
P Proof.
If
S c U
and
if
diam(S)
r S diam{(v,f(v));v
where
the d i a m e t e r on
the g r a p h
a cover
of the g r a p h
the a s s e r t i o n 3.4.
If
then
by
E,F
where
are
=
E',
with
f.
from
every
which
to the R i e m a n n i a n cover
are
linear
the
of
/ ~
the d e f i n i t i o n
two c l o s e d
we d e n o t e
respect
Thus
by b a l l s
follows
F(E,F)
F (E,F)
of
then
E S} s / ~ r
is t a k e n
induced
= r,
metric
by b a l l s
times
in a H i l b e r t
between
E
determines
larger.
of the H a u s d o r f f
manifolds
aperture
U
and
F
Now
measure. space
• H,
i.e.
sup ( inf He-vl] ) eEE'~vEF' llell=l
F'
are
linear
subspaces
respectively.
If w e r e p l a c e
one
the n o r m
generating
of
H
the E u c l i d e a n
II'II' such
that
parallel
to
E,
F
structure
in
H
by a n o t h e r
for e v e r y
v
allvll' ~ llvll ~ bllvll' then
the n e w a p e r t u r e
a ~F(E,F)
~ F' (E,F)
Starting are
linear
PK
denotes
subspaces
of
H.
the o r t h o g o n a l
of all
satisfies
the
inequalities
< ~a F ( E , F ) . _
f r o m n o w we w i l l
L e t us c o n s i d e r manifold
F' (E,F)
the
(3.4)
consider
If
K
p - dimensional
the case
is a l i n e a r
projection following
only
of
H
metric
linear
subspace
onto 8
when of
E
and H,
K.
in the G r a s s m a n i a n
subspaces
of
H
F
then
49
8(E,F)
The
=
JJPE - PF H"
following
meaning
lemma
LEMMA
3.1.
3.5.
The main
following LEMMA
(sec Sec.
39 of
[Akh])
explains
the geometric
of t h i s m e t r i c . @(E,F)
and any
goal
of the p r e s e n t
lemma which
3.2.
such that
= max{F(E,F),F(F,E)
For
may
every
for every
also
p 6 ~
}
subsection
b e of
interest
there
exists
two H i l b e r t
Spaces
two
linear
operators
A,B:
and any two
linear
subspaces
El,
H1
H1 ÷ H2
is t h e p r o o f
the
a number
C 1 = CI(p)
and
for a n y
H 2,
with
E2 c H1
of
in i t s e l f .
JJAH ~ a,
> 0
a ~ 1
NBII ~ a
of dimension
p
one has
space
by
IAIEII JBIE2 < ClaP AB + Proof. E2
Replacing
we can assume
subspaces
and
dim H2
Hilbert
± = 0.
of t h e
Now
H2
Indeed,
choice
of
H2
complement
H1
as its
of
H1
to t h e o p e r a t o r s
Replacing
by the
by
the aperture
space
containing
HI
A ( H I)
a
2p
subspace.
in
HI"
from
if n e c e s s a r y
of
spanned
such that
l e t us d e n o t e
containing
H 1 ÷ H2
HI
d i m H 1 s 2p.
subspace
= 2p.
space
the o r t h o g o n a l A,B:
that
is i n d e p e n d e n t
us fix a l i n e a r
BJ
if n e c e s s a r y
HI A
by
UA
them.
Let
B(HI)
c H2
dimensional
L e t us d e n o t e
H2
and
of t w o
c H2'
L e t us e x t e n d to
E1
the o p e r a t o r s
by taking
and
B
I H1
by
by
A I UB
=
where
[
H1 U: H 2
÷ HI
when
H 1 = H 2 = E,
because of
A
is an i s o m e t r y ,
otherwise and
P tensor that will
E
P ~ E
E,
We
we can consider
the
can also
situation assume
A 1 = a-iA
and
to t h e
that
case
a = i,
B 1 = a-iB
instead
B.
L e t us p a s s Let
we can reduce
d i m E = 2p.
to
some
introductory
Euclidean
P A E
denote
and
power
now
be a r e a l
and
P P A E c ® E. be denoted
we
p - fold All
the
u p to the
space
its
remarks. finite
dimension
p - fold direct
exterior scalar
of
power
products
end of this
sum,
respectively. and all
subsection
m.
By
p - fold L e t us r e c a l l
the Euclidean by
(',')
norms
and
respectively. PROPOSITION
3.3.
For
any
p >_ 1
there
exists
a number
C(p)
> 0
Jl'lJ
50
such
that
for all
lleill ~ i,
vectors
IIfill ~ i,
p
1 s i s p,
one
an__~d { f i } l ~ i ~ p ' ei'
~ E,
fi
has
p
e .l -
i=l Proof.
Let
i ~ifi
~ C(P) l.= 1 ei
us c o n s i d e r
T ( h I ......... ,hp) is a
C
Value
Theorem.
onto
{ei}isis p
mapping
-f" I P P T: ~ E ÷ @ E
a mapping
=
~ h i , where h. 6 E, i=l 1 and thus P r o p o s i t i o n 3.1.
given
by the
formula
This
mapping
1 S i S p. follows
from
the M e a n p
•
Let us d e n o t e by z the c a n o n i c a l l i n e a r p r o j e c t i o n of ~ E P P A E g i v e n by the o p e r a t o r of a n t i s y m m e t r i z a t i o n . It is w e l l
known
that
i=~l ei
is e q u a l
the p a r a l l e l e p i p e d a linear
map,
spanned
E1 c E
to the
by
p-dimensional
e I , ......... ,ep.
is a l i n e a r
subspace
with
volume
Thus
if
a basis
in
E
of
C: E ÷ F
is
e I , ....... ,ep
then
i!lCei
Obviously (a) and
of L e m m a
3.2.
follows
from
two
statements
(b) :
-
B I E I I <- C'IIA-BII,
(a)
B E1
-
B E2
(b)
where
Thus
statement
E1
A
Proof
the
(3.5)
C' of by
= C' (p) (a).
(3.5)
Consequently,
_< C " F ( E I , E 2 ) ,
> 0
Let one
and
C" = C"(p)
e I , .......... ep has
using
A E1
=
Proposition
be an o r t h o n o r m a l
P Ae i i=l 3.3.
> 0.
and
one obtains
B El that
basis =
of
P B e .I. i=l
E1 •
51
~<
i!iAei
-<
~p
where
p(p
p )
,~ A e - ® Be. i=l I i=l i
- i!lBei
P~ Ae. - ! l B e i i=l l i
-< ~p C (p)
< -
IIAei - Beill _
C' = PlI~plIC(P).
Proof
of
(b).
For e v e r y pair of s u b s p a c e s
a neighborhood
dU > 0
of
UEI,E 2
such that
(El'E2)
for all
(F1,F 2)
E1
( Gm,p
and
x Gm,p
E2
we will
find
and a n u m b e r
( UE1,E 2
t
B
F1
-
B F2
Obviously because space
(b) follows
F(',.) G2p,p
of all
G2p,p,
where
(3.6)
Gm, p
that
we can o b v i o u s l y
by s t a n d a r d
function
subspaces
let us r e m a r k
F(EI,E 2) > 0
from
is a c o n t i n u o u s
p-dimensional
First,
(3.6)
IX d u F ( F I , F 2 ) .
denotes
of if
compactness
on the c o m p a c t
arguments,
product
the G r a s s m a n n i a n
manifold
~m. E1 # E2
and c o n s e q u e n t l y
find
d > 0
such that
sides
of that
l
I
B
B
E1
Since and
E2
E 2 I< d F ( E I , E 2 ) "
expresslons
at both
continuously,
from some n e i g h b o r h o o d
the
same
of
(EI,E2).
N o w let us c o n s i d e r
UEI
=
{(FI,F 2)
Let us e s t i m a t e dim E 1 = p
Let
the case w h e n
( G 2p,p×G2p,p;
F(FI,F 2)
for
and no v e c t o r
from
then the o p e r a t o r
inequality
PEI
v ~ F I, llvll =
E 1 = E 2.
(FI,F 2)
6 UE •
F1
F2
or
Since
i = 1
E1
F1
and
'
2}
"
dim F 1 = dim F 2 =
~s o r t h o g o n a l
on both
on
F I, F 2
Let
we have by d e f i n i t i o n
1 llv - PE vll-< F(FI,E I) < 100p' 1
depend
for every
1 F(Fi,E I) < lOOp!
is s u r j e c t i v e i;
inequality
holds
to
F 2.
E1,
52
By the s u r j e c t i v i t y PEI w = PEI v.
of
PEI
Furthermore,
on
F2
as
there
exists
w 6 F2
such that
~ 1 + 100p-----T ' 1
PElV
I
IP E I W
1
llwll -<
< i + ~
(3.7)
< 2.
JI-F (F2,E I) 2 Consequently llw- PEIWll -< llwll " F(F2,E I) <
1
50p!
and f i n a l l y 1 llv- PF2Vll ~ Ilv- wll s llv- PEIVll + llw- PEIWll < 2-TpT. " Since
v 6 FI,
r(FI,F2) For in
F1
1
< ~
(F1,F 2)
fl' ........ 'fp not v e r y
let us fix an o r t h o n o r m a l
fi = PF2ei ' i = i, ........ ,p.
is not an o r t h o n o r m a l but as we will
far from the o r t h o n o r m a l value
det[(fi'fj)]i,j=l
Since ei = fi + hi w h e r e one can e a s i l y see that for
(fi,fi) and for
- 1
basis
in
see soon,
basis
e I , ...... ,ep
Of c o u r s e
F2
(except o n l y
it is a basis w h i c h
the is
one.
of the p a r a l l e l i p i p e d
to the a b s o l u t e =
we o b t a i n
(3.8)
E UEI
F 1 = F 2)
The v o l u m e
Jl
is a r b i t r a r y
.
and d e n o t e
case w h e n
equal
IIvll = i,
generated
of the G r a m m
by
fl' ..... 'fp
is
determinant
...... p"
llhill ~ F(FI,F 2) i = i, ..... ,p
then by
< 5F(FI,F2 ) < 5p! 1
(3.7)
and
(3.8)
(3.9)
i # j, i, j = 1 ....... p
I (fi'fj) I ~ 5F(Fl'F2) F r o m the d e f i n i t i o n
1 < 5p!
of the d e t e r m i n a n t
(3.10) one o b t a i n s
53
I
IJ
- 1 I -< 2 p • 5 F ( F I , F 2) + p!(1 + 5 (FI,F2))
in By
Jl # 0
and the v e c t o r s
F N2o.w we can p r o c e e d (3.5)
(3.11)
1 2) < ~ •
< (5- 2 P + 2 p ! ) F ( F I , F
In p a r t i c u l a r ,
• 5F(FI,F2)<
fl' ...... 'fp
to the e s t i m a t i o n
we can e x p r e s s
this q u a n t i t y
of
form a basis
I B IF1
through
-
B F2 ~I I •
Gramm determinants,
namely B F1
=
J0
=
det
ei,Be j
i,j=l ...... P , B F2 =
~Ii
,
where J2 Thus,
=
det[(Bfi'Bfj)]i,j=l
we have,
21JoJl
3.11)
using
B F2
IB F1
=
IJ 0 - J21. Namely,
l~ol, ijll_ij21 ijll
- J2 I ~ 2 1 J l l
31J 0 - J21 + 2(IJol Since
IIBN ~ 1
then
We do that
we have
IIBhi[] -< IIB/I
...... p"
IJo - J2
+ 21J21
IJl
+ IJ 0 - J21 IJ01
~ C(p)
similarly
IJoJz-a21 I jll
~
IJ I - I I
(3.12)
- 1F.
so it is left to e s t i m a t e
to the e s t i m a t i o n
Be i = Bf I + B h l
~
and since
of
IJ 1 - ii above.
IIB}I < i,
Ilhil[ -< F ( F 1 , F 2 ) .
Therefore
(3.13)
I (Bfi,Bf j) - (Bei,Bej) I -< 5F(FI,F2). We will
use the f o l l o w i n g
Y = (Yi,j)i,j=l ..... P
be two
inequality: p × p
let
matrices
X = and
(xi,j)i,j=l,..., p Ixi,j-Yi,jl
< e.
Then Idet Y - det X I _< p ! 2 P ( m a x
(IIXlI,I))p • ~.
(3.14)
54
The n o r m of the m a t r i x a b o v e by a c o n s t a n t
[ ( B e i , B e j ) ] i , j = l ' .... P Hence,
Cl(P).
we can a p p l y
is b o u n d e d
(3.12)
from
and u s i n g
(3.14) we o b t a i n }J0-J2[ with another
constant
we h a v e
(F1,F 2) ~ UE1
for
B F1 where
-
d(p)
B F2 = 3D(p)
Finally
D(p).
-< d(p)
L e t us r e m a r k elegant
(3.15)
-< D ( p ) F ( F I , F 2) by
(3.11)
and
(3.15)
• F(EI,E2)
+ 2(C(p)
+ D(p))
that L e m m a
3.1.
(5 • 2 P + 2p!).
and L e m m a
3.2.
i m p l y the f o l l o w i n g
inequality.
COROLLARY
3.1.
For e v e r y
p 6 I~,
there exists
s u c h that for e v e r y two l i n e a r o p e r a t o r s H,
(3.12),
NAN -< i,
[[BH -< 1
finite dimension
p
and every
A,B
a constant
CI(p)
in the H i l b e r t
two s u p s p a c e s
E, F c H
space
of the same
one has
[
A E 3.6. C
-
B E
-
For a linear operator
by
GC: G C = { (x/Cx)
PROPOSITION
3.4.
F ( G A , G B)
C: E ~ F
let us d e n o t e
the g r a p h of
E E × F; x 6 E}.
F o r a n y two l i n e a r o p e r a t o r s
A,B:
E ÷ F
-< 2([[AI[ + HBH).
Proof.
O n e can see from
assume
in the d e f i n i t i o n
(3.3)
that without
of the a p e r t u r e
F
less of g e n e r a l i t y that
Hvll -< i.
F ( G A , G B) = sup (inf ,,(x,Ax)-(y,By), I. {x6E;HxlI2+IIAx]I2=I}I{y6E;Hy][2+IIByII2-
II(Ax,x)
-
(By,y)[[ -< H (Ax,x)
-
(0,y)I[ + I[(By,y)
[[( A x , x - y ) I[ + HByll -< HAxll + [[x-y[[ + [[BH Thus,
from
(3.16)
one deduces
that
because
-
(0,y)II =
[[YI[ < i.
we can
Thus
(3.16)
55
!
\
|inf IIAxll + Iix-y[l + IIBI[| _< F (GA,G B ) _< sup ! {xEE ;IlxN-
-< IIAII + IIBI[ ÷ 1 - ~ The last inequality valid 4.
for
follows
from the inequality
m
l+x
<
x
CONTINUITY
from Part I (especially
6,7) along with some extra notations
described
from sections
in Section
2 without
notification.
Thus Sec.
_
AND A SKETCH OF THE PROOF OF ABSOLUTE
We will use the notations
special
1
x t 0.
FO~{ULATION
4.1.
-< IIAII + 211BN.
'
7i).
we begin with the family
S~ (x) of TLSM (cf. k,r,s,~,7 for any number q, 0 < q ~ ~ r,s,~,y
Let us define
~Z (x,q) k,r,s,e,y
=
U
V(y) N U(x,q)
(4.1)
y~i i trtstd,Y NU(x,q) where
U(x,q)
is defined
in
(2.13).
As
Al is a closed set (cf. Sec. 3.4i), from k,r,s,e,y Lemma 7.1.31 one sees that A~,r,s,~,y(x,q) ~ is a closed subset of U(x,q). Consider
a smooth
(T~oexPxl(W)
submanifold
(with boundary)
W of U(x,q)
is the graph of a smooth mapping
~
such that
such that (4.2)
t~
(CI(B2(q) ,E 1 ) (cf. (2.10))
Z and such that W intersects any TLSM V(y), y E Ak,r,s,e, Y N U(x,q) at most at one point and that this intersection is transversal. Any submanifold transversal
W
to the family
as above will be called a submanifold SZ (x) k,r,s,~,y
in the neighborhood
U(x,q).
Let us denote IWI
=
max II~(v)II + max lld¢(v)II, v(~2(q) v6B2(q)
(4.3)
56 where
Ild~(v) II denotes
norms on
E1
and
the operator
E 2.
for every submanifold
norm induced by the E u c l i d e a n
From Corollary W
of
U(x,q)
7.11 it is easY to see that satisfying
(4 2) •
then
W
is transversal
Let us consider family W1
S~,r,s,e,y(x)
two submanifolds
W1
S£ (x) k,r,s,e,7
in
U(x,q).
may not intersect
W2
or vice versa.
By Theorem of
to the family
W1
and
W2
and
Of course,
W2
if in
--
2
U(x,q).
transversal
to the
a TLSM which intersects
7.1.31 there exist two open submanifolds respectively,
IWI < ~
'
~l
and
such that the so-called Poincar~
~2 map
^£ Y (x,q) p: ~i D ~£k,r,s,~,y (x,q) + ~2 R Ak,r,s,~, is well defined and is a h o m e o m o r p h i s m and
~2 R ~£ (x q). k,r,s,e,y '
follows:
if
w 6 U(x,q)
p(y)
=
~2 n V(w).
Let us note that
(
This map
n A£ k,r,s,~,y
P~I,~2
between (see Fig. and if
~i R A£ (x,q) k,r,s,~,~ i) is defined as Y = ~i n V(w),
then (4.4)
}l
= P~2,~I"
57
v(w)
\
Fig.
i.
Here
centered note
at
that
on
N.
~
~
and
V(x)
used
represented
intersects
of T L S M w i t h
~i
and
U(x,q)
as a s q u a r e ~2.
Let
us
is n o t
connected.
and the
was
~
always
b e the m e a s u r e
not going
measure
is s y m b o l i c a l l y
intersection
W e do n o t a s s u m e
between we are
x.
the
necessarily
Let
U(x,q)
on
N
that
v(N)
~-invariant
to u s e
induced
in P a r t
are non-empty
I only and
< ~ neither
measure
the m e a s u r e
b y the R i e m a n n i a n
~
~.
explicitely
to p r o v e
henceforth
we
that
assume
In fact,
the
metric
any relation
in t h i s
part,
anymore.
This 1 .t,6 Ak,r, s, A 0
sets
it is p r e s e n t
in o u r
considerations. The
family
absolutely to the P~l,~2
S £ (x) r e s t r i c t e d to U(x,q) is c a l l e d k,r,s,~,y c o n t i n u o u s if for a n y two s u b m a n i f o l d s W I, W 2 transversal
family
is a b s o l u t e l y
This means and
S£ (x) k,r,s,e,y
that
in
U(x,q)
the P o i n c a r 6
map
continuous for a n y
for a n y m e a s u r a b l e
et
two
submanifolds
A c ~i
W1
such that
and
I(A)
W2
= 0
as a b o v e one has
W
v 2(P(A)) W
= 0.
Here,
~W
denotes
the m e a s u r e
induced
o n the
smooth
P
58 submanifold
W
of
N
Let us note by
by the Riemannian V~ k,r,s,~,y
metric
p.
the set of all points
such that for infinitely many
n ( ~,
Poincar6
~(V~,r,s,~, Y) = n(A~, r,s,e,y).
Recurrence
Theorem,
Let us denote by of the family V~ k,r,s,e,y
£ y(x) Rk,r,s,e,
S1 (x) k,r,s,~,y
N U x
to
U(x,q)
the collection of all elements the set
1 Rk,r,s,~,y(x)
continuity of the family
in the same way as for the family
s£ (x). k,r,s,~,y Theorem 4.1 formulated below and proved the main result of this part. absolute continuity
From
"
We define the absolute restricted
( iI k,r,s,~,y"
which intersect
r,s,~,y 2
'
%n(x)
x 6 AZ k,r,s,~,y
theorem
in sections
It is a counterpart
(Theorem 3.2.1.
of
5 - 10 is
of Pesin's
[Pes] I) for
diffeomorphisms. THEOREM and
4.1.
Let us fix
/,k,r,s,~
and
y
satisfyinq
(2.5) I,
(6.5) I
(2.8).
1 Rl (4.1.1) For every x ( Ak,r,s,~, 7, the family k,r,s,~,y (x) restricted to U(x'61r,s,~,y) __is absolutely continuous. 61 £ 0 < q~, s,a,y < r,s,~,y (4.1.2) There exists a number qr,s,a,y' 2 such that for every W1
and
u(x,q
W2
x ( A1 and every two submanifolds k,r,s,a,y to the family S1 (x) restricted to k,r,s,~,y Iw I [ ~ ~1 and IW21 ~ }, the mappinq and such that
transversal
,s,~,y)
P~l,~2
i_{s absolutely
(4.1.3)
Moreover,
is a density point of exist numbers s
r,s,~,y
such that
if
~(A~i - r,s a,y) , ,
Ak,r,s,a, Y,
~ (B) ' qr,s,~,y
(B) > 0,
transversal
continuous.
and
x ( AI k,r,s,e,y
then for every
0 < q~,s,~,~(B)
B > 0
S1 (x) k,r,s,~,7
Iwil < - ~ ,s,~,y(B) Furthermore,
one has for
~ 1 W
there
6~ < r,s,~,72
such that for every two submanifolds
to the family
> 0, i = i, 2.
> 0
and W1
and
W2
(B)) ,s,~,y ~wi(WiNA~ ,r,s,e,y (x,q~ ,s,~,y (B)) > restricted
to
almost all p o i n t s
U(x,q~
59
y E ~i N ~ i (x Z y(B)) k,r,s,~,y 'qr,s,~, Sec.
3.1.)
satisfies
the J a c o b i a n
J(p) (y)
(See
the i n e q u a l i t y (4. 5)
IJ(p) (y) - II -< B. Remarks. It w i l l
i) m
> 0
r,s,~,y Z
follow
from the proof,
such t h a t
for e v e r y
= er,s,a, ~(B)
qr,s,a,N (B)
that t h e r e e x i s t s
B, 0 < B < i,
a constant
one can take
B
- m
r,s,e,y 2) all
If
Z
~(Ak,r, s) > 0
big e n o u g h
Let us e m p h a s i z e to the p r o o f in Sec. 4.2.
and if
a n d all
7
enough,
t h a t the s t a t e m e n t
of c r i t e r i a
of
K-property
is fixed,
then
~ ( A k , r , s , ~ , Y)
> 0.
(4.1.3) given
Let us d e s c r i b e
the g e n e r a l
in the p r o o f of s t a t e m e n t from e s t i m a t e s
obtained
of s t a t e m e n t
Indeed,
roughly
the m a n i f o l d s
is fully s u f f i c i e n t
structure
of the proof.
that all the d i f f i c u l t i e s
(4.1.2).
Statement
(4.1.3)
in the p r o o f of s t a t e m e n t
(4.1.1)
speaking, ~n(~i),
i n s t e a d of w o r k i n g the time
(Secs.
also
follows
for some
i = 1,2,
n
in
directly
easily
(4.1.2).
the m a p p i n g
(4.1.2).
parts w h i c h
to the
family
of T L S M in P a r t
~,
are d e f i n e d
follows The
one can d i v i d e
into c o r r e s p o n d i n g
which
o (~)-i
we w o r k a l m o s t ~'0w = T~(w ),
I, all
o
in some n e i g h b o r h o o d s
~m.
A characteristic frequent verifications
f e a t u r e of the p r o o f that v a r i o u s
a r e in fact w e l l d e f i n e d . paper
with
are c o n c e n t r a t e d
from the p r o o f of big enough,
5 - 9) w i t h the local m a p p i n g s
o e x p ~-i (w ) 6 ~ o eXPw 0
[Pes] 2 and r e p o r t e d
in
s a t i s f y the a s s u m p t i o n s of (4.1.2) w i t h r e s p e c t Cn l (x)) . (Rk,r,s,e,y M o r e o v e r , as in the p r o o f of the e x i s t e n c e
of
for
13.2.
F i r s t of all let us n o t e
proof
e, 0 < e < 1
small
as w e l l
as in P a r t
is the n e c e s s i t y
composition
This problem I, but h e r e
already
of m a p p i n g s , occurs
of etc.,
in the P e s i n
it is one of the l a n d m a r k s
of
the f i r s t half of the proof. To p r o v e T h e o r e m
4.1, we w a n t
to c o m p a r e
~ I(D) W
where
D
is a m e a s u r a b l e
s u b s e t of
of the p r o o f w h i c h g o e s b a c k to Ya.G.
~i
with
~ 2(P(D)), W
N AZ . The m a i n k,r,s,~,y S i n a i (see [Ano] 2) is to
idea
60
deduce the comparison comparison between
between
~
Cn(wl )
VwI(D)
(¢n(D))
and
and
v
~w2(P(D)) ~n(w2 )
from the
(¢n(p(D)))
for
n
big enough. Indeed,
as the submanifolds
family of TLSM
SZ (x), k,r,s,~,y
~n(~l N ~£k,r,s,~,y (x,q)) and closer as
and
n ÷ ~.
between
and
W2
~n(~2 N A~k,r,s,e,y(x,q))
in the integral,
~wI(D)
are transversal
to the
then one hopes that
And really it is so.
of the change of variables comparison
W1
and
become closer
After using the formula one deduces the desired
~w2(P(D)).
Unfortunately,
the
realization
of this idea is not easy. First of all for n big ^Z enough the set ~ n ( ~ i N Ak,r,s,~,y(x,q)) in general does not belong to one local chart.
Thus,
~n(~l N AZ (x,q)) k,r,s,~,y such that for every chart.
Moreover,
to avoid this difficulty,
by sufficiently
i, 1 5 i ~ p(n),
we cover
small pieces
W~n
belongs
{W 1 }-< < in ±_i_p(n)' to one local
the next step consisting in comparison between W1 and Cn(w2) (#n (p(D N in))) is also rather
~ n(wl ) (~n(DN W~n))
complicated. The realization
of the above idea is contained
and is divided between eight Lemmas The proof is sharply divided formed by Secs.
5 - 7, contains
to the proof properly After contains
speaking,
5 - 10
in two parts.
The first part
all necessary tools and preliminaries which is given in Secs.
some important preliminaries
described
among others Lemma I, we prove in Sec.
which can be considered
in Secs.
I - VIII as follows.
as the Main Lemma.
8 - 10.
in Sec.
5, which
6 the Lemma II,
Roughly
speaking,
this
Lemma describes the asymptotic behavior of #n(A), for large where A is a small piece of ~i whose size depends on n. Sec.
7 is mainly devoted
construction mentioned
to the careful description
which leads to the small submanifolds
of the
{W~ } in l~isp(n)
above.
In order to be able to estimate special coverings these coverings In Sec.
n,
of
#n(D).
9 n(w2) (%n(p(D))) ,
We devote Sec.
8 to the construction
and to the proof of their good geometrical
8 we prove Lemmas
III-
V.
as a second main lemma and Lemmas
we need some of
properties.
Lemma III can be considered IV and V are devoted to the con-
61
struction of desired covering.
Unlike all other sections,
Sec.
8
has a true g e o m e t r i c a l character. Sac.
9 is m a i n l y d e v o t e d to the proof of third m a i n lemma
Lemma VI giving the e s t i m a t i o n w h i c h allows to c o m p a r e ~w2(P(A)) 9 n(wl)(¢
n
for
A c ~i
(A)) and
as above,
~ n(w2)(¢
n
~wI(A)
and
if only one knows the m e a s u r e s
(p(A))).
A m o r e global Lemma VII is an
easy c o r o l l a r y of Lemma VI. Finally,
in Sac.
i0 we prove T h e o r e m 4.1.
special coverings whose c o n s t r u c t i o n in Sac. 8.
The proof uses some
is based on coverings c o n s t r u c t e d
The proof uses Lemma VIII w h i c h is the a d a p t a t i o n of
Lemma VII to the elements of just c o n s t r u c t e d coverings. note that the exact values of c o n s t a n t s at the end of Sac. Sac.
i0.
ii contains a very i m p o r t a n t c o n s e q u e n c e of A b s o l u t e
C o n t i n u i t y T h e o r e m 4.1. (see Sac.
Let us
from T h e o r e m 4.1 appear only
ii.i),
N a m e l y we show that the conditional m e a s u r e s
induced on T L S M
~V(Y) N U ( x ' q ~ , r , s , ~ , y ~
~
Z
Y6U(X'qk,r,s,a,y)
by an
N A£
k,r,s,e,7
a r b i t r a r y measure w h i c h is a b s o l u t e l y c o n t i n u o u s with respect to the Riemannian measure measures
~,
are a b s o l u t e l y c o n t i n u o u s w i t h respect to the
induced on these submanifolds by the R i e m a n n i a n metric
The idea of the proof goes back to Ja.G. The role of L y a p u n o v m e t r i c c o m p l e t e l y illusory. the sets
U(x,q)
Sinai
(see
p.
[Ano]2).
in the f o r m u l a t i o n of T h e o r e m 4.1 is
In fact this m e t r i c intervenes o n l y through
and by the q u a n t i t i e s
IWII
and
IW21.
But as on
the set
A£ the L y a p u n o v and the R i e m a n n i a n m e t r i c s are r,s,~,y' u n i f o r m l y equivalent, nothing will change if instead of L y a p u n o v
metric we use to formulate the T h e o r e m 4.1. the R i e m a n n i a n metric. As far as the proof of T h e o r e m 4.1 is concerned, metric plays a substantial role.
the L y a p u n o v
In p a r t i c u l a r this is true for the
Main Lemma II, but also for m a n y other places.
As it was e x p l a i n e d
in Part I, the use of L y a p u n o v m e t r i c allows to reduce the study of mapping
~IA i
to the study of m a p p i n g
satisfying the u n i f o r m
rts hyperbolic conditions Finally,
(see
(2.2.3)i).
let us u n d e r l i n e that A b s o l u t e C o n t i n u i t y T h e o r e m is
not a local result despite the fact that it is c o n c e r n e d w i t h a small n e i g h b o r h o o d of a point S£ (x) k,r,s,~,y
x.
Indeed,
involves the m a p p i n g
the d e f i n i t i o n of the family ¢
on the w h o l e
N.
62
5.
START OF THE PROOF - I
5.1.
Starting from this moment up to the end of the proof of
Theorem 4.1,
x ( AZ k,r,s,e,y
will be fixed once and for all. w 6 Ak,r,s,~, Y n U
Let us consider a point US note
V(w) = (TwI
Let
US
o
exPwl) (V(w)) .
r,s,~,y '
2
~£ (Tw ° exPwl)(Y)(9(w)QB (~r's'~'Y) (see2
As follows from (6.1.i)
I
Let "
£
I y E V(w) N U lw, 6r's'~'Y2 ) "
choose a point
P0 = (u0'v0)=
x
Let
(2.10)). (5.1)
and from (2.9) for every
n > 1
one
has I P n = (Un,Vn) def ( ~
~i) (P0)
= (T~n(w)
o
i)
exp n
(}n(y) )(
(w)
(5.2))
( B n (w) , Bn,w.[)
where
is defined in Sec. 2.6.
Moreover, the proof of (6.1.i) I together with (2.9) gives [ (Un'Vn)N = [I( n-~ 9! .= 1 I~
) (p0) II ~
200(K r t s)nll (u0,v0)ll <
(5.3)
n~l < 100(Ks,r ) r,s,~,y "
In the future,
when t h e p o i n t
(Uo,V 0) ( V ( w )
n U
w,
r , s ,2~ , ~
will be fixed, we will use the notation u and v exclusively n n in the sense defined above, without additional explanation. For a fixed q0 > 0 and 60 > 0 let us note for n t 0 qn = q0(eSbexp(-15Er,s ))n ,
Cn where
=
60(~')n.
~' = r exp(8Sr, s) < i.
(5.4) (5.5)
63
LEMMA
5.1
(5.1.1).
(I) Let
C > 0
and
0 < q0 < I.
n O = n0(l,r,s,~,y,C,q0) every p o i n t
(u0,v 0)
6 V(w)
~ l ( u n , 6 n ) c B l(qn ) n ~q~.
(2.10)
and
such that
(2.11))
B~(qn ) × B~(qn ) c
and
Then
there
for e v e r y
60 ,
and e v e r y
n t no
~ 2 ( v n 6') ' n
c B2n(qn )
exists
a number
0 < 60 < i,
one has (5.6)
and h E ~ m ; llhll <
1
(Q(l,~,y)~2bn) 2
c (5.7)
c Bn(w) d e f Tn' o exp2 I(V n ( w ) ) . (ef.
(3.9) I ,
(5.1.2)
(2.9) and Sec. 2 . 6 ) ) . 8£ L e t q0 r,s,a,y Let
"
~
q0 - -
q0 0 < 60 < - -
and if
400~ and
Proof
(5.1.1).
(5.7).
Now,
then
for e v e r y
x_ff IP(u0,v0)[[~ "
the i n c l u s i o n s
n > 0
4 '
(5.6)
proof
( u 0 , v 0) ( V ( w )
2
(5.7)
hold. We will
The proof of
(5.1.2)
prove
only
of the second are c o m p l e t e l y
Let
u ( B l ( u n , 6 n)'
from
(5.3)
the first
inclusion
of
inclusion (5.6)
of
as well
(5.6)
and
as the
similar.
for some
n ~ 0.
and from the a s s u m p t i o n s
Thus
IIU-Unll ~ ~n' : ~0 (~')n"
of the L e m m a
it follows
that
llull <- llUnll + 6'n <- ll(Un'Vn) ll + 6'n <- 200(
6 rZ , s , ~ , y ( K r , s )n + ~0(~') n <
< 100((
proof
~Z r,s,e,y
(2.4)). F r o m (2.9)
< Q(l,~,y)
one has
of the first
that there
exist
(~,)n)
< 1
that
inclusion
a number
+
(see
BI(~n) of
< 200(~,)n, (6.7) I and b e c a u s e
c B l) (' qn n
(5.6),
Thus
to finish
it is s u f f i c i e n t
n01 = n 0 1 ( Z , r , s , ~ , y , C )
< < ~'
the
to v e r i f y
such that
for
64
n t n01
one has
is so, because
200(~')n
< qn = q0[a8bexp(-15Sr,s )]n
(6.5) I implies
The proof of
(5.7)
that
is evident,
But this
~' = r exp(8Er, s) < aSbexp(-15er,s). because
for
n > n02(Z,s,r,a,y)
one has
]n < _i [Q(Z,a,y)a2bn]2.
[a8bexp(_15Er,s)
/Y Let us note that in fact this proof (under assumptions of Lemma
5.1.2)
as shown
in Fig.
of Lemma
one has
5.1.1)
~2(Vn,6,)n
and for
5.2.
Lemma
permits
and
us
to
~ 2 ( V n , 6n )
B 2 (qn) n
with respect
B2(qn ) n
consider
= Bl(qn ) n
consider
(under assumptions
to the standard
is, generally
speaking,
Euclidean
an ellipsoid.
5.1.i will play in the future only a technical
The role
and
~P,
v ~ B2(Vn,~ ~)
Bn(qn)
to
/
Considered
norm in
n > 0
n t nO
2.
~
2.
that for
c Int Bn(qn)2 = {v 6 ~P; llvll < qn }
/
Fig.
implies
for
any
because
n ~ n O,
for
~'(u,V)n
n ~ no,
~'n
for
role.
It
u 6 Bl(qn)
is well defined
on
x B~(qn ) c Bn(W ) . of
far
(5.1.2) every
v 6 B2(Vu,~ ~)
will n ~ 0
because
be the
completely mapping
for every
analogous. ~(u,v)
n ~ 0,
It
for !
~n
permits
u ( {i (qn)
is well defined
85
on
Bn(qn)
= B ~ ( q n ) × B 2 ( q n ) c B (w). n n
f r o m the fact that for e v e r y =/~q0[ from
6.
8bexp(_15Sr,s)]n (6.7) I b e c a u s e
n
t 0
The last i n c l u s i o n has
< [Q(/,~,y)
/2 qn =
2bn]2.
q0 ....r,s,e,y2 <
follows
Indeed,
this
follows
(Q(l,e,y))2
THE F I R S T M A I N L E M M A
6.1.
Let
C > 0
=q(/,r,s,~,y,C)
q(l,c)
(cf.
(2.1)
Sec.
i.i I,
and
b
be some number.
Let us d e f i n e
r,s,~,y ~-~ 8q(Arl,s+l ) ' F
~n-I ' F(CI_+n)
q(l,C)
=
by
= min
- (2.3)) , w h e r e
q
comes
(6.1)
from the c o n d i t i o n
(Bb)
from
HA Z br,s ' H c o m e s f r o m the T h e o r e m 3.1 I y f r o m c o n d i t i o n (1.3)I. F r o m (2.4) , ~ > ~ and f r o m
F = F(/,r,s)
= 2
^
comes
(2.5),
~
> i.
Thus
q(l,c)
the f i r s t s t e p t o w a r d s Theorem 6.2.
The d e f i n i t i o n of n u m b e r s
of q(l,C) 1 qr,s,e,y(B)
is from
4.1.2.
The
following
first p a r t d e s c r i b e s where
> 0.
the d e f i n i t i o n
W
First Main Lemma the b e h a v i o u r
is a s u b m a n i f o l d
in the L y a p u n o v m e t r i c case w h e n
W
to d e s c r i b e
is v e r y c l o s e the behavior
can be c o n s i d e r e d l-lemma
close
is a b o u t
in o u r
(see for e x a m p l e
p a r t is u s e d
consists
of Cn(w)
to the s u b s p a c e 6'. n
~n(w)
framework [New]
in the p r o o f of
or
n
The
big e n o u g h ,
E2W,
whose
size
The s e c o n d p a r t d e a l s w i t h the
to the s u b s p a c e
of
of two parts. for
for all
E2W.
T h e n we are able
n ~ i.
as a c o u n t e r p a r t [Pal]).
(4.1.1),
This
of so c a l l e d
Like Lemma
the s e c o n d
lemma
5.1,
the f i r s t
in the p r o o f of
(4.1.2). LEMMA
6.1
(6.1.1).
(II) Let
to the f a m i l y
x ~ A k1 , r , s , e , ¥ S kZ, r , s , ~ , y(x)
L e t us c o n s i d e r such that T L S M
V(w)
a point
and let _in _
W
be a s u b m a n i f o l d
U(x,~ / r,s,~, y).
l
w E ilk,r,s,~,y n Int U(x,
intersect
W
at the p o i n t
transversal
z 0.
6r's' ) ~ 2' Y Let us n o t e by
88
(u 0 I v 0) Thus
6 ~m
for
= ~k
£0 > 0
® ~P, small
the point
enough,
there
(u0,v 0) = T'W o e x P w l ( z 0 ) " exists
the u n i q u e
C1
~0: B2(v0'A0) + ~ k such that ~0(v0) = u 0 and that , -i ~2 eXPw o (~w) {(Y0(v),v) ; v ~ (v0,A0) } is a s u b m a n i f o l d Then (where > 0
there
no
comes
such that
of class
exists
C I,
a number
from Lemma
for e v e r y
n I = nl(l,k,r,s,e,y,W)
5.1),
n t nI
~n: ~ 2 ( V n , ~
a number there
) ÷ mk ,
~nl
exist
=
of
W.
t nO
the u n i q u e m a p p i n g
~,n = ~ n I (~,)n
where
mapping
~n
such that
n1 exp nl
o (7 (w)
and that
'nl ~ (w)
for every
)-l{(~nl(v),v) ; v ( ~2
n t nI
one has
~n(Vn)
~' ) } c ~ (vnl'6n 1
= un
(W)
and
{(~n(V),V) ; v 6 B2(Vn,~n) } c B n ( W ) = T'n ° e x P n l ( V n (w))" Thus
the m a p p i n g For e v e r y
~n'
6
is well
0 < 6 nI '
defined
< ~ nI
on the g r a p h
(6"2)n
of the m a p p i n g
~n.
one has nI'
{ (Yn+l(V) ,v) ; v ( ~2 (Vn+l,@n+l) , } c {9'n(~n (v) ,v) ; v 6 ~2(Vn,6n)}
,
(6.3) where
@' = 6 (~,)n n nI
a n
= max iiYn(V) N _< ~n v6~2 (Vn' ~n )
(6.4)
n-n 1 b
n
= max
v~ 2 (6.1.2). where
Let
q(l,C)
lld~n(V) ll -< n
(6.5) n
(Vn,~n) P w 6 Ak,r,s,~, Y. is d e f i n e d
by
Let
(6.1).
0 < C <_ i. Let
= u0
and
0 < q0 <_ q(l,C)
q0 0 < 60 _< -~-
q0 (u0,v 0) ( V ( w ) and let H (u0,v0)II _< 800/2 Let ~0: ~ 2 (v0'60) + ]Rk be a m a p p i n g ~0(v0)
Let
Let
Let
6' = ~0(~') n n
of class
C1
such that
67
max
1 ll~0(v) II <- ~ q0
(6.6)
IidY0(v) ]j -< C.
(6.7)
v6B 2 (v 0 , 6 0 ) max v6B 2 (v 0 , 6 0 )
Then there exists C 1"
class
a unique
, ~n: ~2 (Vn'6n)
{(Yn(V),V) ; v 6 ~2(Vn,6,)]n
and thus the mapping
¢' n
sequence
{~n}ntl
o f mappings
÷ ]Rk , such that for every
n >_ 0
o_~f
one has
c Bn(W ) : T'n o exPnl(Vn(W))
is well defined
(6.8)n
on the qraph of the mapping
Pn' I
{ (~n+l(V) ,v) ; v E ~2 (Vn+l,6n+l) } c (6.9) n c {}n(~]n(V),V) ; v E B2(Vn,6n) },
a
= n
max
II~n(V)II _< (I~ + c ) q0~n
(6.10) n
Iid~n(V) H ~ C~ n.
(6.11) n
vEB2(Vn' 6') n
bn =
max vEB2(Vn,6~)
Let us note that 6.3. Proof. of
(6.1.2)
quickly
We
Proof of
of the reader
~n+l
in all details
the same lines,
the proof of
(6.1.1)
from
(6.1.1).
but is somewhat
scetch this proof after the proof of
convenience steps.
will first prove
follows
(6.9) n define
(6.3) n and
The proof
simpler.
(6.1.1).
~n .
We will
For the
is divided
on four
and
and
Proof goes by induction. (6.1.i)
Step i. we will check
In this step we will determine (6.2)ni,
(6.4)nl
and
(6.5)ni.
n1
nI
68 Let us note by F 0 = {(~0(v),v)
F0
the g r a p h
of the m a p p i n g
6 ~m: v 6 ~2(v0,A0)}.
As
~0
then no v e c t o r from the t a n g e n t space to to ~k in ~ m = ]Rk ® ~ P . For every H'n = ~ o .... o ~ hood
Xn
of
is well (Uo,V 0)
p - dimensional derivatives ~m
= ~k
in some F
in some
(cf. T h e o r e m
d H ~ ( u 0 , v 0) = d~A ...... d ~ ( u 0 , v 0 ) then the t a n g e n t
(Un'Vn)
= H'(u0'V0)n
sufficiently
is the g r a p h
n ~2 (Vn,6n) Yn:
small
of some
÷ ~k,
In p a r t i c u l a r
= un
defined)
6n > 0
where
~n(Vn)
max
(uniquely
the
As the
preserve
the
splitting
to
Fn
space, d e n o t e d
Yn
neighbor-
~m.
is not p a r a l l e l
neighborhood
small
Let us c o n s i d e r of
C I,
(u0,v 0) is p a r a l l e l the d i f f e o m o r p h i s m
sufficiently
6.1i).
i.e.,
is a map of class
F 0 at n ~ 0,
F n = H'n (F0nX n)
submanifold
@ ~P,
the p o i n t
defined
~0'
Ln, to
]Rk.
of the p o i n t smooth
C1
is a s u f f i c i e n t l y
at
Consequently, (Un,Vn),
mapping small number.
and
lld~n(V) ll < +~.
v6B2(Vn,6n ) Now,
we will
estimate
Let us recall
#'n
=
T 'n+l
~n(U,V)
=
I
of
1
also and
llAn(U) II < lllull
every
tn = ~
o (T'n
-i )
,
+ bn(U,V)),u (an,bn).
From
n
>
(3.13) I
0
6 IRk , v 6 ]Rp. (2.2.3) I
(3.14) I
and from the
one has for
u 6 ]Rk (6.12)
llBn(V) II >- ~llvll
PROPOSITION
o ~ o eXPn
(AnU+an(U,v),BnV
Let us d e n o t e definition
that
-i eXPn+l
o
lld~n(Vn) II.
6.1.
n ~ nll
There one has
for
v 6 ]Rp.
exists
nll=
n l l ( l , r , s , e , ¥)
such t h a t f o r
69
lldtn(Un,Vn) II < exp(-10nEr,s). Proof.
As
n ~ 1
(u0,v 0) 6 v(w),
then from
one has
(Un,V n) 6 Bn(W).
3.1 I.
From this theorem,
Theorem obtains
(5.2) one knows that for every
Thus for every
n ~ 1
from
dtn(0,0)
(5.3), as
we can apply = 0
one
IIdt n (Un,V n) II = IIdt n (Un,V n) - d t n (0,0) II -<
2o0.#
I_<
yb
2o0.#
r,s )n (u0,v0)l I < [ b ]n (
r exp(5er, s) ~b
r,s r,s,~,y ybv
Now, from (6.5)i, one deduces that there exists n o such that for n ~ nll one has
nll=
n.
i
nll(/,r,s,e,Y)~
n lldtn(Un,Vn)ll
Now, as (cf. n12
r exp(5er, s) ) ~
~ C(/,r,s,~,y)
~ = r exp(12Er,s ) < ~
(2.5)) and as = nl2(/,r,s,a,
< Y)
rts
< ~
~ nll
(cf. such
(cf.
~ exp(-10n~r,s).
(2.4)),
as
•
I < ~ < 1
(2.4)), then there exists
that
for
all
n ~ n12
one
has
X+exp(-10n~ r,s ) ~-exp(-lOnc
r,
s)
< ~'
(6.13)
400 ~l
r,s,~,y
(
,s
)n < ~n
PROPOSITION
6.2.
a constant
Q = Q(/,k,r,s,~,y,W)
and all
lim lld~n(Un,Vn) II = 0. > 0,
More precisely
there exists
such that for all
n ~ n12
t >_ 0,
IId~n+t(Un+t,Vn+t )I[ -< Q t. Proof.
As all the linear mappings
d~n(Un,V n) = (An +dan(Un,Vn) ,B n +dBn(Un,Vn)),
n >_ 0,
(6.14)
70
preserves
the
d H ~ ( u 0 , v 0)
also p r e s e r v e s
Let us c o n s i d e r tangent
space
the g r a p h
Ln
= ]Rk
n > n12.
+ ~k
~+> IRp ,
then
the linear
mapping
it.
to the g r a p h
L n = { (KnV,V);
Kn = d ~ n ( V n ) : ~ p As
]Rm
splitting
So that
(6.13)
of
at the p o i n t
v E ~P ] We have
Ln+ 1 = d # ~ ( U n , V n ) L n ,
~n
The
is satisfied.
of the linear
(Un,V n)
is
mapping
Ln = dH~(u0,v0)L 0
then
from
(6.14)
one has
o KnV, (B n + db n ( U n , V n))v) ;v E • P } =
Ln+ 1 : {((A n + d a n ( U n , V n ) = { ((An + d a n ( U n , V n ) )
o Kn o (B n + d b n ( U n , V n ) ) - i v , v )
; (6.15)
v ( ]RP}= = { (Kn+lV,V); From
(6.12)
and
v E ]Rp }.
from P r o p o s i t i o n
6.1 one o b t a i n s
that
for
n t n12
llAn+dan(Un,Vn) ll s llAnlI + }Idan(Un,Vn) ll ~ ~ + e x p ( - 1 0 n S r , s ) . Moreover,
i
by the same arguments,
one o b t a i n s
that
for
(6.16)
v E ]Rp,
llBnV+db n ( u n , v n)vli > llBnVlI - !Idbn ( u n , v n)vli >_ >- (~ - exp(-10nSr, s) IIvII,
i.e.
II[B n + db n (Un,V n) ] -iii <-
Now,
(6.15)-(6.17)
(6.17)
1 u - e x p (-10ne r' s)
together
with
(6.13)
implies
that
only to p r o v e
that
n ~ n12 IIKn+!ll ~ IiKnlI~. Consequently
for all
n ~ nl2
and
t > 0
IiKn+tiI-< IIKnlI~t. Thus,
to finish
the proof
Q(l,k,r,s,e,y,W)
it is e n o u g h
= supNd~nl2(Vnl2)II
< +~,
for all
71
where
sup
is t a k e n o v e r all p a r t s
w E iI k,r,s,e,y
N Int U ( x
61 r,s,~,y ' 2 )"
the fact t h a t for a n y f i x e d and the d e c o m p o s i t i o n Indeed,
one can a p p l y
Beginning n
> 0
max
q > O,
But this
the a b o v e
remarks
for all
where
is so in v i r t u e
~q(AS,r,s,a,y)K
TwN = E l w ~ E 2 w
f r o m now,
so small
Z 0 ( W N V(w),
is a c l o s e d
is c o n t i n u o u s to
n > n12
of
on
set
that set.
g = n12. we fix the n u m b e r s
that
]Idtn(~n(V)
I v)]l
< exp(-10ne
r,
(6.18)
s ),
v6!82(Vn,~ n) n-nl2 max
I[d~n(V)[I ~ 2Q~
(6.19)
[I~
(6.20)
vEiB2(Vn,~n) and t h a t max
n
(v) II < in.
vEB2(Vn,6n ) This
is p o s s i b l e
the s e c o n d of
by P r o p o s i t i o n
n t nI
6.2 and
(5.3)
together
n I = nl(/,r,s,e,y,W)
t n12
s u c h t h a t for
one h a s
1200 H A / r,s yb~
(~,)n < e x p ( - 1 0 n s
r,s
)
(6.21)
< ~ - e x p ( - 1 0 n e r,s)
301(~,)n
and
2Q~
nI
This n ~ nI
with
(6.13).
L e t us fix the n u m b e r all
6.1 and
<
(6.22)
< ~n
(6.23)
n12
last condition,
together
with
(6.19)
implies
that
for
one has n-n 1 max
v(~ 2 (v n ,~n )
Iid~n(V) II ~ ~
~ i.
(6.24)
72
In particular, IlV-Vnlll Now,
and thus
from
Clearly,
from that
(5.1.1), (6.24)
(6.24)
one has
n I ~ no, also
one obtains
(6.5)
. (6.4) nI
Step
2.
Beginning
meaning
of
follows
these
Now, allows
~n
symbols
(6.5)
will
have
Lemma
This,
c Bl(Vnl,~nl ) .
that
(6.2)ni
follows
from
we forget
above,
a different
we will
the m a p p i n g
(6.5)n+ I.
proves
introduced
n ~ nI
to c o n s t r u c t
(6.4)n+ 1 and
~n
- ~nl(vnl)II
is true. (6.20).
nI
from this m o m e n t
and of
for every
!I~nl(v)
(~nl(v) ; v E B 2 ( V n l , ~ n l ) }
as
imply
that
n # n I.
that
(6.2) n and
(6.5) n
(6.3)n,
(6.2)n+ I,
satisfying
together
the In w h a t
meaning.
prove
~n+l
completely
for
with
(6.2)ni,
(6.4)ni
and
6.1 by induction.
n1 Thus (6.5)
n
let us suppose
are true.
tn(Yn(V),V)
PROPOSITION
for some
In particular,
is well
6.3.
that
Let
defined
from
for
n > nI
n ~ n I, (6.2)
(6.2)n , (6.4) n and
one knows
n
that
v E B2(Vn,~).
v I ,v 2 E B 2(Vn,~n ' ) .
and let
Then
-lOne lltn(~n(vl),vl) Proof.
- tn(~n(V2),v2)l]
By the Mean Value
Theorem
< e
r'Slivl-v21I.
one o b t a i n s
that
IItn(Yn(vl) ,v I) - t n ( Y n ( V 2) ,v 2) II _<
where
I
(6.25) -< sup Iidtn(Z) II(IiYn(V I) - Yn(V2)ll zEF n
+ Iivl-v211)
Fn = { (~n(V) ,v) ; v 6 ~ 2 ( V n , ~ n ) }.
Let us e s t i m a t e and from T h e o r e m
sup Ildtn(z) llzE~ n
3.11 one gets
From
the proof
of P r o p o s i t i o n
6.1
73
lldtn(Z)N ~ ildtn(Un,Vn) ll + lldtn(Z) - dtn(Un,Vn)ll 200 HArl,s r,s,c~,y
(6.26)
y 600 HA/
61 r,sb r , s , a , y
<
Indeed[, from (6.4) n,
(r e x p ( 8 E r , s ) ) n z
~ Fn
c ~ l ( u n,~ ~)
×
~2(v n
,~)
and thus
r exp(5e
I] Z- (Un,V n) II 5 ¢~" ~'. n
Moreover,
from (2.8),
r,s
)
< ~, =
b
r exp(SCr,s). Again from the Mean Value Theorem and from (6.5) n one obtains that (6.27)
ll~fn(VI) - %'n(V2)ll _< llvl-v211. Thus,
(6.25)-(6.27)
Htn(~n(V I) ,v I)
together with
(6.21) imply
tn(~n (v2) ,v 2) II -<
1200 HA / 61 r,sb r,s,e,y
(r exp(8Sr,s))nllvl-v211
_<
_< exp(-10ner, s)]Ivl-v211.
Step 3.
Let us define the mapping
Tn:
~2 (Vn,~ n )
÷
]lqp
TnV = BnV + bn(Yn(V),v). PROPOSITION
6.4.
For every
vl,v 2 ~ B2(Vn,~ n)
one has
llTnvl - TnV211 >- [llvl-v 211, where Proof.
~ = r exp(12er,s). As
t n = (an,bn),
one obtains that
then from Proposition
6.3 and from (6.22)
74
l]Tnvl -TnV2II >- ]]Bn(vl-v2) [I - lJbn(~n (vl) ,v I) - bn(Yn (v2) ,v2) II > >_ ~IlvI -v2[I - e x p ( - 1 0 n e r , s ) i l v l - v 2 H
=
(~-exp(-10nSr,s)) llvI -v211 > ~IivI -v211.
=
PROPOSITION
6.5.
For every
0 < £
< nI
L = {~(~n(V),V);V
( B2(Vn,6n)}'
is the graph of a m a p p i n g Proof.
By P r o p o s i t i o n
image contains point on
~2
Yn+l:
6.4., T n the ball of radius
TnV n = Vn+ 1.
~2 (Vn+l,@n+l).
¢'n o (~n×Id)
o
Thus,
nI'
N { m k x ~2 (Vn+l,~n+l)}, , (Vn+l'6n+l)
is a
C1
÷
injective
~6~ > ~'6 n' = 6'n+l -i Tnl
in particular,
This allows us to express
T~ 1 ,
where
Id
~k
denotes
of class
C I.
immersion
whose
around
is defined ~n+l
as
the
and
C1
Yn+l =
the identity m a p p i n g
in
~P.
• Thus
(6.3)n+ 1 is fully proved.
construction
of
~n+l'
it follows
Let us note that from the that
~n+l(Vn+l)
= Un+ I.
Step 4. We will now prove (6.5)n+ I. In virtue of (3.14) and (6.4) one can write I n i
# ~ ( ~ n ( V ) , V ) def = (AnOn(V) Let
By t h e
([,~) =
+ an(~n(V) ,v),BnV + bn(~n(V),V)-
v ,v + T E ~2 (Vn+l,6n+ I)
definition
of
T
n
since
and let
9 = T-iV'Vn + ~ = T-l(v+T)'n
we h a v e
T : B n~ + b n ( ~ n ( ~ + ~ ) , v NOW,
(6.28)
%n(Yn(~),~)
(6.29)
+ ~) - bn(Yn(V),V) = (~n+l(V),V),
one has from
(3.14) I
75
f <
Tn+l(V)
= AnTn(V)
+ an(Tn(V),v),
(6.30)
(
Tn+l(V+~)
= AnTn(V+~)
+ an(Tn(V+T),v
+ ~).
It is clear that
< lim sup bn+l - ]ITII~O From
(6.31)n+ I,
v 2 = v + ~, from
[]Tn+l(V+T)-~n+l(V)ll llTil (6.30),
from P r o p o s i t i o n
from the e s t i m a t i o n s
(6.5) n and from
(6.22),
as
for A and n n t n I t n12,
6.3 with
v1 = ~
B given by n one obtains
and
(6.12),
[IYn+l (v+T) - ~n+l (v) II -
II• I]
=
IIAn [Tn (v+?) -Tn (v) ] +
[an (Tn (v+~') 'v'+~) -an (Yn (~) '~) ] II
[]Bn%~+bn (~n (~+T~) '~+%~ -bn (Tn (~) '~) [[
IIAnll • IITn (~+ ~) -Tn (~) ll+IIan (T n (v+~) ,v+~) -a n (T n (v) ,v) l! [[Bn~l-[Ib n (T n (~+%~ ,v+k~) -b n (T n (v) ,v) II
lIT n (v+~) -~n(V) II II "[ II + exp (-10ner, s) - exp (-10ner, s) n-n
lq
l+exp (-10ne r ,s ) p_exp (_10ner, s )
LS
n-n 1 i + e x p ( - 3 n S r , s ) e x p ( - 7 n l S r , s = n
)
~-exp(-10nSr, s)
n-n I l+exp(-10nler, s) n-nl l+exp(-10nler, s) _< ~_exp(_10nlSr,s ) z ~_exp(-10ner,s)
n+l-n I
n+l-n 1 Thus
bn+ 1 ~ n
, i.e.,
(6.5)n+ 1 is true.
We pass now to the proof of Let us estimate
let us note that for We have
an+ 1 =
(6.4)n+ I.
max ~ lldTn+l(V) ll. ~'~) v(B2(Vn+l ' n t z
(u0,v 0) ( V ( w )
one has
First
let
ll(u0,v 011[ <- /2 ~ r,s,~,y
76
~n+l(V) As
= ~n+l(Vn+l)
~n+l(Vn+l)
from
+
= Un+l,
[~n+l(V)
then
(6.5)n+ 1 and from
from
(6.23)
- ~n+l(Vn+l)]. (5.3),
from the M e a n V a l u e
one o b t a i n s
that for
Theorem,
v E ~ 2 ( V n + l , ~)n~+,l
f II~n+l(V) II ~ N~n+l(Vn+l)II
+ II~n+l(V)
=
ll(Un+l,Vn+l)ll
+ II~n+l(V)
- ~n+l(Vn+l)ll
+
6' n+l-nl n+l" ~
<
< ~n+l an+ 1 _ ;
Thus
Finally
line as the proof
of
=
< 200(
~ )n+l + 3°°~r,s,~,y(
i.e.
the proof
- ~n+l(Vn+l)ll
(~,)n+l
< ~n+l
(6.4)n+ 1 is proved.
of
(6.2)n+ 1 p r o c e e d s
(6.2)
described
along
exactly
the same
at the end of Step
i.
nI P r o o f of
(6.1.2).
us to d e d u c e n a 0,
F i r s t we prove
(6.10) 0 .
(6.8) n and
satisfying Step
(6.10) n allows
(6.9) n, 1.
(6.8)n+ I,
to d e f i n e
(6.10)n+ 1 and
from
(6.6),
(3.9) I. q0
ll(~0(v) '
from
Step
2.
in the proof PROPOSITION
~n+l
(6.11)n+ I.
v) II <- q0 - 2 < 61r , s , e , ¥
(6.6)
and
The of 6.6.
(6.7)
inductive
<
0 < 60 ~ -~-,
Indeed, q0
+ ~0 ~ ~ " [Q(Z,~,y)]2 .
(6.10)0
and
(6.11) 0
similar
to that
respectively. step in this proof
is v e r y
(6.1.1). Let
n >_ 0.
For e v e r y
vl,v 2 6 ~2(Vn,$') n -IOE
lltn(~n(V I) ,v I) - tn(~n(V2 ) ,v 2) [[ _< Fq0e
where
allows
for e v e r y
q0
Jl(~0(v),v)rl ~ II~0(v) rl + IFvH < ~
follow
This
that
the m a p p i n g
from the i n e q u a l i t y
(6.1) , (6.7) I and
(2.2.5)i,
Thus
(6.11) 0.
step we prove
The base of induction,
(6.8) 0 follows from
(6.8) 0 and
In the second
(cf. Sec.
6.1)
F -
2HA l r,s b 7~
r'sn[Ivl-v21] ,
one h a s
+
77
Proof.
By the Mean Value Theorem one has
I<
llhn(~]n(vl) ,v I) - hn(~n (v2) ,v 2) II< (6.32) supI1dhn(Z) II(II~n(V I) - ~n (v2) [I + 11vl-v211) • ZEI
where
I
is the interval
in
6Z r,s,~,y< 8
Q2
(l,~,y) 8
As
from
(6.5) I and from the convexity
I
c Bn(W
(~n(vl),v I)
connecting
(~n(V 2) ,v 2) . (6.10) n,
q(l,C) <
~m
(see (6.1) and
of
and (6.7)i),
it follows that
Bn(W)
).
HA Z lldhn(Z)ll <_ lldtn(Z)ll = lldtn(Z)_dtn(0)l I < -7 b (by) r,s n ilzll. But z E I and thus by proof of Lemma 5.1.
(6.10)
one obtains
in the same way as in the
+ - - qo
Nzll -< max {llYn(Vi)N + llviIl} -< ( l + c ) q 0 ~ n i=i,2
I
(~') n _<
2/~
< q0({') n <_ q0[<~8bexp(-15Sr,s)] n
because
~' < e
lldhn(z) II -< Now let
8b
exp(-15Sr, s)
and
0 < C <_ i.
Consequently
one obtains
n HA l -I0£ n -i0~ r,s r,s r,s _ 1 Fq0e b q0 e 2 y
vl,v 2 E ~2(v n,6n).
From
(6.11) n one obtains
(6.33) that
ll~n(V I) - Yn(V2)ll _< cqnllv I -v211 and consequently
that
[[~n(V I) - ~n (v2) ]I + IIvl -v211 -< (CD n + l)Ilv I -v211 !
I
Finally
(6.34)
_< (C+I) IlvI -v211. from
(6.32) - (6.34) and from
0 < C 5 1
one gets
78 -lOne llhn(~n(vl),vl ) - h n ( ~ n ( V 2 ) , v 2 ) H
Step applies
3 of the p r o o f of
to e v e r y
n t 0.
(6.1.1)
< Fq0e
r'SlIvl-v211.
remains
unchanged
In the proof of c o u n t e r p a r t
but now it of P r o p o s i t i o n
6.4, we use the i n e q u a l i t y < ~ - Fq0 which
follows
(6.35)
from
Let us p r o c e e d exactly
(6.1). now to the proof
the same as the proof
w h i c h we are g o i n g thus one o b t a i n s
of
to d e s c r i b e
of
(6.11)n+ I.
(6.5)n+i,
now.
From
This proof
except the final (6.35),
is
step
~ - Fq0 > 0,
and
that
~ll~n+ l(v+T) - ~n+l (v) Jl
l
I]~ll
flT II
-- + F q 0 e x p ( - 1 0 n S r
s)
~ - F q 0 e x p ( - 1 0 n S r , s) l c ~ n + F q 0 e x p ( - 1 0 n e r , s)
<
~- Fq0 ex P( -10nSr, s )
CD n
To finish l+Fq0 < ~-Fq----~ - ~' assumption
(6.11)n+ 1 it is s u f f i c i e n t < ~n-I q0 - F(CI-+~)
or e q u i v a l e n t l y (cf.
The p r o o f Since
the p r o o f of
(6.1)). of
II(u0,v0)]l
Thus
(6.10) n goes q0 800/2
l+F~0exp(-3n£ r s ) ~ - F q 0 e x p ( - 1 0 n e r , s)
bn+ 1
C~ n+l,
similarly
we o b t a i n
which
to see that
is s a t i s f i e d
i .e. , (6.11)n+ 1
to the proof
of
by our is true.
(6.4)n+ I.
79 f sup
an+ 1 -< ll~n+l(Vn+l) ll +
ll~n+l(V) - ~n+l(Vn+l) ll
v(B 2 (Vn+ I, @n+l )
_<
200
<
sup lld~n+ 1 (v) II < 6 ° . ~ ) v(~2(Vn+l ' nt±
<
n+l q0 -~- + bn+l~0(~')n+l
q01 4e )n+l cqn+l q0 ( 8g )n+l < _~_ire r,s + -~- re r,s
_<
1 C)q0~n. ( +C)q 0 r exp(4(n+l)~r, s) < (~+
Finally,
let us prove
From (6.11)n+ 1 one has that
(6.8)n+ I.
ll~n+l(V) - ~n+l(Vn+l)ll < llV-Vn+lll and thus that ,)} {~n+l(V) ;v 6 ~2 (Vn+l , ~n+±
c
~l(Vn+l, 6n+l)"
Now, from Lemma 5.1.2 one obtains START OF THE PROOF
7.
-
(6.8)n+ I.
II
yg Yl ' Y2 E A1r,s,~,y and let p(yl,Y2 ) < -(cf. Condition 4 (Ba) from (Sec. 2.1i). Let us define the smooth mapping I by Yl,Y2 the formula
7.1 •
Let
-
IyI'Y2
= ~' Yl
o
eXpy- iI
o
eXpy 2 o
(~Y2
)- i
This mapping is well defined on the set Indeed, if ball
y 6 AZr,s,~,y'
{U6TyN;llUIly < yg}.
by (2.9)
then the mapping Moreover,
II(T' )-l(h) fly2 < Yg Y2 /2
I P (Yl, eXpy 2
o (T~2)-l(h))
if
(7.1) yg {h 6 ]Rm; 11hll < -~}. eXpy
h ( ]Rm
is defined on the and
< Yg IIhll - T '
and consequently
~ P(Yl,Y2 ) + p(y2,eXpy2O(Ty2)-l(h)) ! < Yg -
T
yg +
--
<
Yg"
-<
then
80
Thus
PROPOSITION
7.1.
nondeereasing q(0)
-I eXpy I o eXpy 2 o (T'y2) (h)
the map
= 0,
For every
function
~(q)
> 0
yl,y 2 ~ A l r,st~,y 7g , IIhll z -~-
l,r,s,~,7
~ = ~(q)
for
q > 0
satisfying
is well
defined.
there
exists, a c o n t i n u o u s yg for 0 z q z -4such that
defined
and such that f'or e v e r y .
p(yl,Y2 ) ~ q
and for e v e r y
h 6 ~m,
one has
ll(dIyl,y 2) (h) - Idll -< ~(q),
where
the d i f f e r e n t i a t i o n
Proof.
of --
(7.2)
I
is taken w i t h
respect
JA~
h.
yl,y 2
Let us note
h P = {(yl,Y2,h)
E Agr,s,a,y x A gr,s,e,Y x ~ m ; p(yl,Y2)
~ 4'[
llhll ~ Y~}AS
Ag r,s,~,7
is compact,
is c o n t i n u o u s q,
and c o n s e q u e n t l y
0 ~ q 5 diam
(q)
P
i£ r,s,~,7
is also compact. uniformly
let us d e f i n e
The m a p
continuous
on
dI P.
J
yl,Y2
(h)
For e v e r y
the n o n - d e c r e a s i n g
function
by
q(q) def --
sup (yl,Y2,h) EP
lldl (h) - d I ~ l , Y 2 (~) II. YI'Y2
(Yl,Y2,h) (P P (YI'Yl) +p (Y2'~2) +[lh-hlI~q
It is clear continuous
that on
0 ~ ~(q)
A r,s, Z ~,y)
< +~
is
and that
[0,diam A Z r , s , ~ , y ]"
N O W our p r o p o s i t i o n h = h.
< ~(diam
_
follows
if one takes
Yl = Y2 = Y2
and
81
7.2.
As the next
step t o w a r d s
us fix a n u m b e r
q(2)
conditions
(7.3)
and
(7.3)
0 < q(2)
(7.4).
in 1 Jw I z ~
= q2(£,r,s,~,y)
satisfying
< ~T g
and
~ : U W
U(x,q(2)) (see Sec.
an a r b i t r a r y 4.1).
Then,
÷ ~k,
where
U
W
let
the f o l l o w i n g
smooth
submanifold
for e v e r y
w 6 U(x,q(2))
is the g r a p h
is an open
subset
W n
of the smooth
of
~P.
The
W
possibility
of such choice of q(2) f Ak,r,s,e,y(cf. (3.4)i).
the sets
Z qr,s,a,y(B)
< ~1 .
0 < ~(q(2))
A~ (Tw, o exp i) (W) n Int B(q(2)) k,r,s,~,y' mapping
of
(7.4).
Consider
such that
the d e f i n i t i o n
follows
from the d e f i n i t i o n
of
Let W = eXPx(~x)-l{(~(v),v);v be a s u b m a n i f o l d
Iwl < 12"
transversal
(7.4')
6 B2(q(2))} 9~ Sk,r,s,e, Y (x)
to the family
such that
We will w r i t e
IITwli =
lldTwlI =
awl
=
sup II~w(V)II, V6Uw sup lld~w(V)II, V6Uw
H~II +
IJTIi =
lldTxlI =
sup liT(v)II, v 6 B 2 (q(2)) (7.5)
sup Nd~(v) II, v6B 2 (q (2))
lld~xll.
%.
7.3.
In this
transversal
s e c t i o n we show how to c o m p a r e
submanifold
respectively. PROPOSITION
in local
We b e g i n w i t h 7.2.
If
charts
the c o m p a r i s o n
IId~xlI s I,
then
the r e p r e s e n t a t i o n
centered of
for every
at
x
and at
lld~xll and
IId~wlI.
w E U(x,q(2))
n A£
k,r,s,a,y 1
IId~wll ~ l - 2 a ( q ( 2 ) ) Proof.
From
(7.3),
[(l+~(q(2)))lrd~xH
condition
+ ~(q(2))].
(Bb) of Sec.
2 1 and
(2.9)
of a w
one has
n
82
p(w,x) < z~2g
Let us denote
I
wtx (u,v) = (HI(U,v),H2(u,v))
(cf. 7.1).
It is easy to see that YwH2(~(v),v)
Let us d i f f e r e n t i a t e =
I
where
(7.6).
We obtain
for
o [DiH2(~(v),v ) o d~(v)
d~w(Z)
z = H2(~(v),v)
6 Uw
+ D2H2(~(v),v)] = (7.7)
DIHI(Y(v) ,v) o d~(v)
DI, D 2
respectively.
f
(7.6)
= HI(~(v),v ) .
denote
the d i f f e r e n t i a t i o n
Furthermore
sup
+ D2HI(Y(v) ,v) ,
we have from
along
and
~k
(7.2) and
]Rp
(2.9)
lldIw, x(h) - Idll =
{h(IR m ; IIhll_<~}
rDIH l(h)
- Idk, D2H1 (h)
=
1 (7.8)
sup
~-g
{h6IR m ; IINII_
DIH 2 (h) ,
~ _< ~(q(2)),
IIDIH 2(~(v) ,v) o d~(v) -< HDIH2(~(v),v)
+ D2H 2(~(v) ,v) - Idpll -<
.]Id~(v) ll + rID2H2(~(v),v)
- Idpll-< (7.9)
-< ~(q(2))Ild~(v)ll
+ ~(q(2))
_< ~(q(2)) (IId~(v) ll + I) -<
_< Q(q(2))(IId~xl I + i) _< 2~(q(2)) because
lld~xll _< 1
In particular,
and because
(7.3)
< 1 implies
2~(q(2))
the linear m a p p i n g
DIH2(Y(v) ,v) o d~(v)
+ D 2 H 2 ( ~ ( v ) ,v) : ]Rp
÷ ]Rp
< i.
83 is invertible. If for a linear operator
A,
IIA-IdlI < i
then
1
IIA-III -< I-IIA-IdH Thus using
(7.7) - (7.9) we obtain
IId~w(Z)]I _< IIDIHI(~(v),v)
o d~(v) + D2HI(~(v),v)II
• I!(DIH 2(Y(v) ,v) o d~(v) + D2H 2(~(v) ,v))-iI] -<
< 1 -i-2~(q(2)) (IIDIHI(~(v),v)l!
1 i-2~(q(2))
[(i+ ~(q(2)))Hd~xll + ~(q(2))].
Proposition COROLLARY
7.1
" lld~(v) I] + IID2HI(Y(v),v)II) <-
7.2 and
(7.3) imply the following
For every
•
w E A£ k,r,s,~,¥
statement.
N U(x,q(2))
one has .-
Iid~wlI _< 2(Iwi+~(q(2))). The next step is the comparison of Let
K = /2 q2A£
where
rts
Sec. i.i I.
As
q ~ 1
PROPOSITION
7.3 •
Zl,Z 2 E U(x,q(2))
and
q
and
then
(Bb) of
i.
K
w E A £k,r,s,e,y N U(x,q(2))
N U(w,q(2))
!
II-IIx.
comes from condition
AZ > I r,s /2
For every
!
H.IIw
and every
one has
I~ 1 -i -i , < ~Iiexp w (z I) - e X P w (z2)II~ ~ IlexPxl(z I) -exPxl(z2)Iix (7.10)
KilexPwl(Zl ) Proof.
since
x
and
exPwl(z2)II~. w
appear symmetrically
to prove the second inequality. Let L be the interval in and
exPwl(z2 ) .
For every
u E L
TwN
in (7.10) it is enough
connecting
we have
the points
exPwl(Zl )
84 Q(eXPwU,W ) s /~ q IIullw' <- 2qq(2) and thus p(exPwU,X) By
(7.3),
condition u.
~ P(eXPwU,W) + P(w,x) s 4qq(2), 4qq(2)
< 7 g.
Thus
p(eXPwU,X)
(Ba) of Sec. i.i I the mapping
By condition
< ¥g,
eXPx I o eXPw
so that from is defined at
(Bb)
-i 2 IId[exp x oeXPw(U)]ll -< q .
and
(7.11)
Henceforth we can write using (7.11)
(2.2.5) I, the Mean Value Theorem
IIexPxl(z I) - exPxl(z 2) IIx <_ Ar,£silexPxl(Zl ) _ exPxl(z2) iix
=
=
A£r, sH (exPxl oeXPw ) o exPwl( Z l ) - (eXPx I oeXPw ) o exPwl( z2) IIx
A£ supHd(exPxloexPw) (u)II " IlexPwl(Zl ) - e X P w l(z 2) IIw r,s uEL
q2Ar,s£ Ilexp~l(zl)_exp~l(z2)iIw
~ KNexp~l(zl) _ exp~l(z2)Iiw.,
7.4. We will now specify several extra conditions on the number introduced in Sec. 7.2. These supplementary conditions on q(2) well as new quantities C. Let
0 < C ~ i.
e
r,s,e,y
will depend on one extra parameter
Let us define the numbers
and q(2,C) = q2(i,r,s,~,y,C) (7.12)-(7.14) hold.
q(2) as
£(C) = £r,s,e,y(C)
such that the following three conditions
1 0 < E(C) < ~ ,
(7.12
0 < q(2'C)
(7.13
< min (q(l'C) ) 3600K ' q(2) -
e(C) + ~(q(2,c))
<_ ~C
(7.14)
85 where
K
comes
from Proposition
From now we will Let us consider
replace
a manifold
7.2.
in
W
(7.5)
of type
the parameter (7.4)
q(2)
such that
(cf.
by
q(2,C).
(7.5)) (7.15)
Let
w 6 U(x,q(2,C))
V(w) (7.12)
n AZ k,r,s,e,y
n W n Int U(x,q(2,C))
and the remark
consists
¢ ~.
following
(4.3)
of one point which we denote
T'w o exPwl(y)
be such a point that
implies
that this
intersection
by
and as in
(5.1)
y,
= (u0,v0).
Let P
qC(W,W)
be the supremum
80 _< q(2,C)4
and
of all numbers
60
~2(v0,d 0) c Uw fl ~2(q(2,C))
such that
where (7.16)
Uw
is defined
c U(x,q(2,C))
in (7.3) and
(see
[eXPwO(7w)-l] (~((u0,v0),60))
c
(2.12)).
As II(T~ o exPxl) (y) - (T x, then from
(7.13)
o
exp i) (w) II < /-2 q(2,C),
and Proposition
7.2 it follows
that
7.17)
ll(u0,v0)H ~ q(l,C) 800/2 Moreover, exactly
as the graph of the mapping
the same arguments
sup
that
lies in
B(q(2,C)
,
imply that
7.18)
ll~w(V)lI s q(l,C)
v~2(v0,~0) Finally
Y
8OO/2
from Corollary
7.1 and from
(7.13)-(7.15)
one obtains
86
sup
(7.19)
lld~ (v)II-< C. w
v(~2 (v0, (S0)
The above for
remarks
(7.17)
q0 = q(l,C)
defined
0 < 8 0 ~ qC(W,W)
and for
- (7.19)
above,
prove
for e v e r y
that ~0
Y0 = TwI~2
for every
0 < C ~ i,
such that all a s s u m p t i o n s
(v0,~0) of L e m m a
6.1.2 are fulfilled.
(~) -1] (~( uo" Vo), (SO))
/~[eXPwO
\
Y
~w U(x,q(2,c) Fig.
3. Here
U(x,q(2,C))
are s y m b o l i c a l l y
and
presented
[eXPwO(r~) -I] (B(u0,v 0) ,(S0))
as squares
centered
at
x
and
y
respectively. Thus
from
of m a p p i n g (6.11) 7.5.
(6.1.2)
~n: B 2 ( V n ' 6 n )
÷ IRk
for e v e r y
satisfying
n >_ 0, conditions
the e x i s t e n c e (6.8) n -
n"
We pass
the proof, in Sec.
n o w to the d e s c r i p t i o n
which
leads
of the m a i n c o n s t r u c t i o n of 1 {Win}l<_i<_p(n) mentioned
to the s u b m a n i f o l d s
4.2.
As in Sec. Y =
one o b t a i n s
[eXPw
o
For any
7.4,
(T')-I]
w
n >_ 0
let
y = V(w)
N W N Int U ( x , q ( 2 , C ) ) ,
(Uo , Vo)" and any
r,
0 < r < qC(W,W)(~,)n,
let us
87 define the following Wn(W,y,r)
p
= {exp n
In particular, submanifolds
for
dimensional
N. (7.20)
o(T' n )-l(~n(V),V) ;v 6 ~2(Vn,r)}. (w) ~ (w)
60 ,
0 < 60 < qC(W,W),
Wn(W,y,6 n)
This definition
submanifold of
one can consider !
where as before
immediately
6n
implies
--
the
60(~') n
that for every
n >_ 1
one
has !
~n(W,Y,6n)
(7.21)
!
c ¢(~n_l(w,Y,6n_l)).
This last property
is very important
for the future.
Let us
emphasize that if one uses the Riemannian metric instead of Lyapunov metric in (7.20), the analog of (7.21) is not true an>~ore. Let us recall that for every Vn(W)
where
= {y EM;
Q(l,~,y) Moreover,
Wn(W,y,6~)
p(y,~n(w))
and
n > 0
< (Q(Z,~,y)a2bn) 2}
is defined by for every
w ~ iI k,r,s,~,y
(3.9)
(3.11) I.
n t 0
one has (7.22)
c ~ ( ¢ n ( w ) , q n ) c Vn(W ) .
The first inclusion
follows directly
second one follows directly
from
from
(2.9),
(5.3)
(2.13)
and and
(6.10) n.
The
(3.9) I. 6o
PROPOSITION Proof.
diam(%-n(Wn(w,y,6~)))
This is an almost
which holds proof of By
7.4.
immediate
in our framework
S /2(I+C)
exp(4nSr,s)"
corollary of P r o p o s i t i o n
for every
n t 0
6.4,
(see step 3 of the
(6.1.2)). (2.2.5) I it is enough to prove that
w
lim diam((~0w)
--i
w
....... (~(n_l)w)-l(Mn(W,Y,6n))
<
n-~
(7.23) 60 (l+C) exp(4nSr,s)
where
'
Mn(W,y,6 n) = {(~n(V),V) ; v 6 B2(Vn,6n)}.
88 Let us denote by ~P .
~'
From P r o p o s i t i o n
the p r o j e c t i o n
of
]Rm
6.4 and from the d e f i n i t i o n
= ~k of
~ ]Rp
onto
Mn(W,y,6 ~)
one has 6 !
z' [(¢(n_l)w)-iMn(W,y ' , 6n) ' ] c ~2 (Vn_ 1 , _~)
, < ~2 (Vn_l,6n_l), 6 !
-i
o (~(n_l)w)-l(Mn(W,Y,6n))]'
c ~2(Vn_2,
c ~2
Finally
one obtains
(Vn-2
c B2(v0,
PROJECTION
8.1.
AND C O V E R I N G
For a fixed
introduce
from the c o n s t r u c t i o n
60 ,
Qn (w,y, 6n) = exp -in
and if
exP n
LEMMAS
then
(w)
z = exp-~
o (T~
n ~ 0,
let us
because
)-i
denote
Mn(w,y,6 ~) = H~l(Wn(w,y,6~)). that
(w)(z). by
H . n
n(w)
on
Let us recall (7.23))
and
(~n(W,y, 6 , ) n
(w)
defined
(cf.
Bn(W)
Wn(W,y,6 n)
notations.
z ~ Wn(w,y,6~)
We denote
c
(v0, exp(4nSr,s )) .
of m a n i f o l d s
0 < 60 < qC(W,W),
the following
6'
(~)
60
~2 c
8.
,6' ~) etc. n-z '
that
I ~' [(¢'w )-I ....... ( ~ n _ l ) w ) - l ( M n ( w , Y , 6 n ) ) ]
Now, (7.23) follows and from (6.10) n
(~)2n---)c
w 6 Ak,r,s,~, Y.
Hnl(y)
= (Un,Vn)
This m a p p i n g
As in Sec.
E ]Rk @ ]Rp
Mn(W,y,6 ~) = {(~n(V) ,v) ; v E ~2(v n,6')}.n
is well
7.5, let us
and that
89
For any Let
~
p
(respectively
p
on
N
z
with
the c l o s e d
respect
in
to the i n d u c e d
z (W~(w,y,@~)
Vz ~ ~2
(7.5))
ball
~)
be the m e t r i c or on
by the s c a l a r
metric
(cf.
z = H-l(z)'n
Wn(W,y,6~)
(respectively
z ~ ~n(w)
will d e n o t e
or
on
p r o v i d e d by the R i e m a n n i a n on mm). When
let
(respectively
<(w,y,6~) metric
z ~ Wn(W,y,@~)
@
on
N
and w h e n
%n(w)
product
z =
(-,-)
cn(w)
q > 0
by q
(~n(Vz),Vz)
where
let us d e f i n e
we
and c e n t e r e d %n(w).
60,
metric
Q(z,q)
on
denote
by the
or by E u c l i d e a n
metric
then we will
on
Mn(W,y,@~))
of radius
Riemannian
induced
at
If
(V n , ~)
For
fixed
i, r, s
60R do
= 12q(Ar, s+l) (8.1)
dn
where
R
d0~n
=
comes
PROPOSITION
,
n
t
i,
from c o n d i t i o n
8.1.
For any
(Be)
If
z ( ~ n ( W 'y , }
~)
(8.1.2)
If
z ( Wn(W,y, ~ 6 n)
3
,
F i r s t of all we will
z (Wn(W,Y,6n) K(z,3dn) where
!
prove
then
~ ( z , $ W n ( W , y , ~3 6')) n
then
p(z,$Wn(W,Y,6n))
> 3d n-
that
for e v e r y
and e v e r y
n t 0
> 3d n
one has (8.2)
c Vn(W )
K(z,q)
(3.9) I • Let
1.1 I, 0 < R < 1.
n ~ 0,
(8.1.1)
Proof.
from Sec.
= {y 6 M;
y 6 K(z,3dn).
p(y,¢n(w))
O(y,z)
~ q}
Thus
~ 3d n + p(z,¢n(w)).
and w h e r e
Vn(W)
is d e f i n e d
by
90 By
(2.9) and
(6.10)
0(z,#n(w))
n
one obtains that
_< /2 [IzlI _< /2 (ll~n(Vz)]! + NVz]I) _<
, /2 ( sup llTn(V)II + 6n) <_ /2 (q0 ~n + 60(~')n ) . v ~ 2 (vn, 6n)
Consequently, ~
P(z'~n(w))
<
_< 3d n + /2(q0 ~n + 60(~')n)
3(d0~n + q0 ~n + 60(~')n)
because
~ < ~',
Now, as2
60 do < ~
~, < 2 b
(cf.
(cf.
<
~< 6q0(~')n ,
q0 60 < -~-.
(8.1)) and
(6.5) I)
and as
6q0 _< 6q(l,C)n <
[Q(l,~,y)] (cf. (6.1) and (6.7) I) one has (Q(l,~,y)a2bn) 2 which proves (8.2). Let us prove now analogous. In virtue of
(8.1.1).
6q0(~')
The proof of
(8.2), for every
(8.1.2)
< is completely
u 6 Q(z,6d n) c K(z,6dn)
~ = H-l(u) E IRm. n From Condition (Bb) of Sec. 21 and from
one can
consider the point
for every
n >_ 0,
the m a p p i n g
H
n and the mapping
with constant
q/2
with constant
qA r~ ,s e
(2.9), it follows that
satisfies the Lipschitz H -I n
satisfies
condition
this condition
n£ r,s
Thus
I flu- ziI < qAri,se n~ r'Sliu - zll~n(w)
_< 3qA ri
e ner'sd ts
< n
(8.3) ^ 7ne ~ 7ne r,s < 4± ~°0re r,s < 41 6'n ' < 3qd0rA r~,se
because from
(8.1) it follows that
1 , I17' (u) - 7'(z) ll -< IIU-zlI < ~ 6 n, projection of then
IRm = IRk
I17' (~)II -< ~i 6'n
.~ IRP
3qd0Ar, s where
onto
7'
IRp.
< 1 ~ 60 .
Consequently
denotes the canonical Now, as
and finally one obtains that
z E ~n(W ,y,~6n )I , , I)7'(~)}I < 3 6'.n
91
In other words: inclusion
c Int ~ 2 ( v n ,~3 6~) .
~' (H l(Q(z,6dn)))
implies
immediately Let
8.2.
The
last
(8.1.1).
F c T
N
be a
k-dimensional
subspace
cn(w) T
of
N
transversal
to the s u b s p a c e
E
~n(w) for n big e n o u g h
n a n20 1 y(F,E n ) ~ £ exp 2% (w) (nSr,s)
where
the a n g l e
(i.e.
~(~,.)
Two p r i n c i p a l orthogonal Section
complement
Let us denote
to
of such
(2.16)
t nF
to the s u b s p a c e
F.
and
q,
e X P ~-I n ( w ) (Q(z,q))
and
H(z,q)
v
Let us note by
in the space
by
(2.15).
subspaces and
and
(2.1.3)i of
E
(cf.
with
onto
ZF = tFn(~) "
let us d e n o t e !
provided
from
respectively).
= {x 6 W(w,Y,6n) ; p(x,z) the ball
(iv)
T #n(w )N
Let us d e f i n e
0 < q ~ 3d n,
B~(v,q)
are the R i e m a n n i a n
E]~n(wi
the p r o j e c t i o n
3 z 6 Wn(W,y, ~ 6')n
8.3.
(8.4)
E26n(wl
with
by
(Z,r,s,F))
is d e f i n e d
examples
2.21 t o g e t h e r
parallel
and such that 2#n(w)
E2#n(w)
For
Q(z,q)
=
<
_ q}. centered
of radius
q,
Riemannian
norm
in
at
T n(w ),
2~n(w) noted
II.IIn. The
stones main
following
about projection
Continuity
Theorem
is one of the c o r n e r proof;
it is like a second
lemma.
LEMMA
8.1.
q,
For e v e r y
(III).
n2(l,r,s,e,Y,e) any
Lemma
of the A b s o l u t e
such that
e,
there
0 < s < i,
for any
any
n ~ n 2,
0 < q s 3d n, and any s u b s p a c e
F c T
exists
n 2 = n2(e)
z E Wn(W,y, 3 6n), N
satisfying
cn(w)
(8.4) , one has B~(ZF' (l-e)q) Remark.
Without
decreasing
c t~(Q(z,q))
^ c ~2n(ZF, (l+e)q) .
loss of g e n e r a l i t y
function
of
e.
we will
assume
that
n2
is a
=
92
Proof.
Step
1.
Let us r e m a r k
n21 = n 2 1 ( ~ , y , e ) 0 < q < _ 3d n,
such
and
e
H(Z, (i - ~ ) q )
Indeed,
that
and
Q(z,q)
Let
n ~ n21,
satisfying
there
exists
z ~ W n (w,y,~
(8.4),
one
~n ) ,
has
(8.5)
c H(z, (I + ~ ) q ) .
8.1 and
0 < q s 3dn,
c ~n(W,Y,~n)
e > 0
e
c Q(z,q)
q,
for e v e r y
for e v e r y
F c T n(w )
from Proposition
n ~ 0
that
7.22)
one
one
obtains
that
for all
has
c Vn(X ) .
n21 = n 2 1 ( ~ , ~ , e )
be d e f i n e d
as a n u m b e r
such
that
for all
n ~ n21
I
Q ( l , ~ , y ) ~ 2bn
)2
( )
-< rain
r_e, ~
_< Re(w)
8 where
r
and
e
Let
R
n ~ n21.
belonging
to
come
from C o n d i t i o n
Let
y E ~Q(z,q)
Q(z,q),
E (i + ~ ) q .
than
e
Thus
connecting
z
(Bc)
and
let
and
y,
-i exp n
the c u r v e
from
Fe
Sec.
be a s m o o t h
whose
(F e)
2.2.
belongs
length
curve
is less
to
Q(z,q),
(Bc)
from
Sec.
q,
because
(w) its ends its
are
z
p -length
0 < e < i. the
first
and is less
This one
account
that
for a n y
has
i(F)
In v i r t u e
than
proves
follows
y,
one
9.
1 + ~
the
second
exactly smooth
~ q
and
the
<
1 +
inclusion same
curve
of
lines,
(8.5).
if o n l y
F c Q(z,q),
t h a t by C o n d i t i o n
i Hd(exp~l(w))
of c o n d i t i o n
r (Bc)
The one
of
into z
and
has
! (Q-)-~n >- i+
e > 1 - ~ .
it s u f f i c e s
to show
(Y) Hn >-,,d exp n
proof
takes
connecting one
i.i I,
(w) Step e,
2.
every 0
Now,
to f i n i s h
0 < E < i, n ~ n22,
< q S 3d
n
and
there
the p r o o f
exists
for e v e r y for e v e r y
n22
= n22
z 6 Wn(W,Y,¼ subspace
(£,r,s,e) !
~n ) ,
F c T
that
such
for e v e r y
N n (w)
for e v e r y
that
for
q,
satisfying
(8.4)
93
one has t F (H(z, (i + ~)q) e
(8.6)
c B2n(ZF, (i + e)q)
and £ q) . (ZF' (i - e)q) c t Fn(H(z , (i -y) We will prove here only along the same lines. As z - ZF 6 F, G(z, (i +~)q)
(8.7)
The proof of (8.7) is exactly
(8.6).
then the set
= {x 6 T
~n(w)
N; x + (z
ZF ) 6 H(Z, (i +~)q) }
is such that tF(G(z, (i +~)q))
=
tF(H(z, (i +~)q)) .
Thus to prove
(8.6) it suffices to show that for
n22(Z,r,s,e)
one has
n ~ n22 =
(8.8)
tF(G(z, (l +2) q)) c B2(ZFn , (l+e)q) Let us remark that Let (a)
ZF 6 G(z,(l +y) q).
^ y 6 G(z,(l +~)q). y 6 E
Two possibilities occur
and
(b)
y ~ E
2%n(w)
. 2~n~w)
Let us examine separately these two cases. (a)
In this case
N Y - ~FIIn s pG(y,z F) ~ (i +~)q, where ~S denotes the distance induced by the Riemannian scalar product in T n(w)N on (b)
G(z,(l + ~ ) q ) .
Consequently
n(y) y = tF
B ~ ( z F , ( I + e)q).
The proof in this case is a bit longer but it uses only the
elementary trigonometry, max v~2(Vn,~A)
(8.4) and the inequality
lld~n(V) ii _< C n _< n .
(6.11) n, i.e.
94
Let
0 < 6 Z ~.
the u n i o n
of all
such that
for some
Let us d e n o t e
straight
lines
straight
C n (6)
by
(respectively
K c T n(w)N ,
line
L c E
passing
C'(6)) n
through
also p a s s i n g
ZF
through
2¢n (w) ZF
one has
Yn(K,L)
and
L
Yn(K,L))
measured
(respectively
with
measured
I Yn(K,L)
(respectively
!
(respectively K
~ 6
~ 6),
denotes
the a n g l e
respect
to the R i e m a n n i a n
with
respect
between
where
Yn(K,L)
the s t r a i g h t norm
to the L y a p u n o v
norm
lines
]J'l] n ! ll-ljn)
in
T~n(w)N" The i n e q u a l i t i e s every
6,
Cn
(2.9)
0 < 6 s [
6 ,seXp(ner,s)
2/2A
Now,
(2.16)
and
(2.16)
and
1 c C~(6)
(6.11) n imply
that
c C~(Nn).
Consequently,
one has that
for e v e r y
n t 1
Beginning angles
from this m o m e n t
norm
for every
n > 0
as
for every
n t 1
one has
~ = exp(-7er,s),
by
(8.9)
(8.10)
up to the end of the proof,
will be m e a s u r e d
in
and
(8.9)
c Cn(26-2 A r , s e X p ( - 6 n S r , s )) •
and lengths
Riemannian
that
c Cn (2/2 A £ exp(ner, )). r,s s
I
G(z, (i + [ ) q )
G(z, (i + [ ) q )
imply
one has
T
exclusively
with
all
respect
to the
N. cn(w)
If
y 6 G(z, (i +
in v i r t u e all
of
(8.10),
n t n22(~,r,s,s) :af
nd
Let one has
y 6 Cn(2/2
)q)
then
to prove
NY- z
(8.3)
n -
(y,z F) _
it s u f f i c e s
to p r o v e
~)q. that
Thus, for
one has
A£r,seXp(-6nSr,s )' Y ~ E
flY - ZFlln _< (i + ~E) q
n23 = n23(Z,r,s)
then
2¢n(w)
n IItw(Y) - z FIln_<
be a n u m b e r
such that
(8.11) (i + s)q.
for e v e r y
n ~ n23
95
2/~ A r,s £ exp(-6ngr,s) and that
(8.4)
< ~
is true
1
and
7[
~exp(ner,s
) < ~ .
for the R i e m a n n i a n
(8.12)
orthogonal
complement
to E 2$n(w) "
Let us c o n s i d e r the o r t h o g o n a l
a point
projection
of
y
as in
y
to the s u p s p a c e
(8.11).
Let us note
by
y'
E 2¢n(w)
As f o l l o w s through
y
and
from ZF
(8.12),
for
n ~ n23 ,
is n e v e r o r t h o g o n a l
the s t r a i g h t
to
E
.
line p a s s i n g
Moreover,
as
2¢n(w) y ~ E
,
the points
y, t~(y)
and
ZF
are not c o l i n e a r
(see
2¢n (w) Fig.
4) and w h e n
are never t~(y)
F # El , 2~ n (w)
in the same p l a n e
and
y'
except
are colinear.
Let us c o n s i d e r to a t r i a n g l e
with
ZF' y' t~(y)
the case w h e r e
When
F = El 2#n(w)
the t e t r a h e d r o n
the v e r t i c e s
of such a t e t r a h e d r o n
the p o i n t s
the points then
in Fig.
and
y' iF,
y' = t~(y).
w h i c h m a y be s o m e t i m e s
iF, y, t Fn(y)
is p r e s e n t e d
and
y'.
reduced
An e x a m p l e
4.
Y
t~(y)
Fig.
^~F
4
1 .£ exp(ner,s)
1 <- n , n I _< ~ - Z exp(nSr,s)
6 _< 2/2 AZr,seXp(-6n~r,s )
and
(cf.
fly -ZFIIn _< (i + ~)q.
(8.4)),
98 Now,
it is easy
the length
to see that
of the side AB of the
Let us denote
the length
Ilt~(Y)
- ZFIln
triangle
is never
ABC p r e s e n t e d
of the interval
[a,b],
bigger
in Fig.
by
than 5.
labl.
C
A
Fig.
5.
e = Z exp(ner, s) < 4 '
B = 2/2 AZr,seXp(-6e
n ) < ~, rts
IcBr = (l+~)q The s t r a i g h t f o r w a r d I
lAB I =
( ! + ~ )e q ( e o s
_< (i + ~e) q ( l
Thus
for
n ~ n23
I~(Y) But for
- ZFIIn
(8.11)
+ 2 ~) 8 _<
one has ~ (i + 5e) ( i
(8.13) and
that
Z Z exp(-nSr, s)) . + 4/2 A r,s
Z Z exp (-ner, s))q. + 4/2 A r,s
n t n24 = n24(Z,r,s,e)
Consequently,
shows
B + sin tg~ B~" <_ (i + 2 ) q ( l
Z (i +5) (I + 4/2 A r,s
one has
computation
thus
one has that
Z exp(-nSr,s))
implies
that
(8.6).
(8.13)
< ! + e.
for all
n ~ n22
def
max(n23,n24)
g7
Finally,
let us n o t e
substantially Let
the
fact
us d e n o t e
that
that
by
~
in the p r o o f
i exp
of
(8.7),
one u s e s
1 1 Z ) (n~r, s) ~ ql < ~ exp(nSr, s
the o r t h o g o n a l
projection
of
T
N
onto
~n(w) E
,
with
respect
to the R i e m a n n i a n
scalar
product
in
T
2~n(w)
PROPOSITION that
N. Cn(w)
8.2.
for e v e r y
There n > n3
(8.2.1)
for e v e r y
(8.2.2)
there such
and
a number
n 3 = n 3 ( i , r , s , ~ , Y)
3 z 6 ,n(W,y,~_~ ~ 6') n
7 0 < q -< ~ d n, i f
q,
exists
C1
mapping
It e a s i l y
, F = E± 2¢n(w)
follows
(8.9)
and
from
from
(8.5),
the
such
has
y ~ ~H(z,q),
then
p(z,y)
= q,
^ 7 ,~ a n ) ÷ E2@ n (w) : B2(~(z) n
~n
H ( z , ~7 dn ) c g r a p h
that
one
~ ~n c Q ( z , 3 d n) '
IId~zn(y) ]I -< 4 / 2 AZr,seXp(-6ner,s )
(8.2.3)
Proof.
exists
Lemma
inequality
for any
^ ^ 7 dn ) . y 6 B2n(Z,~
8.1 a p p l i e d
to
tge ~ 2a
valid
for
0 < a ~ . Let
us r e m a r k
that
in P r o p o s i t i o n
be r e p l a c e d
by any o t h e r
number
will
8.4.
n3
Let us d e n o t e
volume
on z
A(z,8)
It f o l l o w s All others
~n
from
P,~,
and
and
t,
and
(8.1.2) n4 h,
that
etc.,
0 < t < 3.
Hn
~,
i, r, in the
8.2 the
A(z,@) which
s,~,y,
7 ~
the m u l t i p l e In this
can
case
the
t. the
Wn(w,y,6~)
p
dimensional
Riemannian
respectively. 0 < Q < i,
let us d e n o t e (8.14)
<_ 2dn}.
c Wn(W,y,6~).
will
appear,
can d e p e n d
the d e p e n d e n c e
on t h e s e
among parameters
future.
aim of the p r e s e n t
Proposition
on
= {y ( ¢ n ( w ) ; 2dn(l - 8) _< ~ ( y , z )
n o t be m a r k e d The
by
6 W n (w,y,~3 6~)
numbers
on
number
depend
~(w,y,6~)
For
will
also
8.1,
section
following.
is to d e d u c e
from Lemma
8.1 and
98
Lemma
8.2.
every
There
i, 0 < 8 < ~
9,
n t n4
exists
and e v e r y
an a b s o l u t e
there exists
z 6 Wn(w,Y,¼
constant
C2
n 4 = n4(9)
6A)
such that
for
such that for e v e r V
one has
n
(A(z,8))
(8.15)
~ n ( Q ( Z , d n ) ) -< C2O. Proof.
S t e p i.
condition
Taking
(Bb) of Sec.
equivalent
the i n v e r s e
to the e x p o n e n t i a l
i.i I, one e a s i l y
sees that
m a p and a p p l y i n g
(8.15)
is
to
~n(A(z' 9)) C219
(8.16)
~n(Q(Z,dn) )
where
A(z,8)
= e X P n-i (w
be e a s i l y d e d u c e d ~(~(~),
) (A(z,8))
and
C21
a constant.
(8.16)
will
from
(8.17)
d n ( l - 8 ) 2) c ~(~(Z,dn) )
and (i(z,8))
c {x 6 E
; 2dn(l - 29) (1 - 8) ~ llx - ~(z)II 2#n(w)
2d
L e t us n o t e Lemma (8.17) for
that
(i + 8) def R(z,e)}.
7 2dn(l + 8) < ~ dn,
8.1 a p p l i e d
is true
for
to
F = E
n t n41(8).
because
± 2%n (w)
8
1 < ~.
immediately
N o w let us prove
implies
that
(8.18)
that is true
n > n42(9). First,
from
n
(8.18)
(8.14),
let us r e m a r k
that f r o m c o n d i t i o n
(Bc)
in the same w a y as in the p r o o f of
that there exists
n43 = n43(8)
such t h at
from Sec.
l.li,
(8.5), one d e d u c e s
for every
n t n43
one
has
I
A(z,@)
c
{y 6 W n ( W , Y , ~ n ) ; 2dn( 1 - 8) (i -e) 2 ~ P(y,z)
!
2dn(l + 8)2}.
(8.19)
99
Now,
the i n c l u s i o n Thus
n42(@)
it is e n o u g h
~ n43
Let
n
n3
c ~(Q(z,2dn(l-
~ max(n2,n3,n43)
which
exists
is evident.
a number
n42 =
n ~ n42
- 2 @ ) (I - @ ) )
8.1 and
2 d n ( l + @) )
to s h o w t h a t there
n42 = n42(@)
t h a t for
c ~2(z(~) n
such t h a t for
B 2( ( z (I z ) 'n 2 d n
Lemma
z(A(z,@))
comes
(8.20)
@ ) (I _ @ 2 ) ) ) .
(where
from Proposition
n2
comes
8.2)
from
be such a n u m b e r
~ n42
lld~ n(y)l I z @
for all
7 (~(z) ,~ d n)
y 6 B
L e t us fix there exists
H~(z)
(8.21)
n >_ n42
a point
(cf.
and s u p p o s e
I
be the
that
zn 6 S Q ( z , 2 d n ( l - @ )
- Z(Zn)ll n -< 2dn(l
Let
(8.23)) . (8.20)
is n o t true.
(1-@2))
s u c h that (8.22)
- 2 @ ) ( i - 8).
interval
in
Thus
E
connecting
z(z)
and
2%n(w) ~(z n) .
As
n42 ~ nl,
and to
e = @
then by L e m m a
one o b t a i n s
L e t us c o n s i d e r
by
(8.22)
8.1 a p p l i e d that
to
i F = E2%n(w)
I c ~(Q(z,2dn(l -2@))).
the c u r v e
F = {(~ n(V),V) ; v 6 I} c ~ ( z , 2 d n ( l - 2@)) ) connecting
z
and
z n.
l e n g t h d o e s not e x c e e d On the o t h e r
hand,
Now,
from
(8.21)
it is e a s y to see t h a t
its
2dn(l - 2%) ( 1 + O). it f o l l o w s
from L e m m a
8.1 t h a t for
n ~ n42
one has ~2(~(~) 2dn( 1 _ @)2(i _ @2)) n ' Thus,
for
n ~ n42
Consequently (i - @ 2 ) (1 -@) lent to (8.20) S t e R 2.
one obtains (1-@)2(1-@
< 1 - 28.
1 + @2 < @
c ~(Q(z,2d
For
(i - @) (i - 62))).
z(r) ~ 2 d n ( l - 8 ) 2 ( 1 - 8 2 ) .
2 ) < ( 1 - 2@) ( 1 + @), @ > 0,
and we o b t a i n
and c o n s e q u e n t l y
(8.18).
1 0 < @ < ~,
(8.21)
As
that
n
and
this
the c o n t r a d i c t i o n
(3.2)
i.e.
last i n e q u a l i t y
is e q u i v a -
which proves
imply n o w that t h e r e e x i s t s
100
a constant
C22
> 0
V c z ( Q^( z , ]7 an)),
such that
z ( Wn (w,y '43 6~) '
7 IVl < V n ( Q ( z , [ an) where
IVI
Riemannian
denotes scalar
for e v e r y m e a s u r a b l e
-1
n
one has
(8.23)
(v)) < - C 2 2 1 v I
the L e b e s g u e
product
subset
in
measure
T
of
V
induced
by the
N ~n(w)
Now,
(8.17),
ap - bp
implies
(8.18), that
(8.23)
for
and the s t a n d a r d
n t n3
n z-l(R(z,8))]
-i ^2 2 Pn(~ (Bn (z (z) ,dn (i-8) ))
~n(Q(Z,dn))
IR(z,@) I <- C22
C25[(I+8)
where
C22 - C25
(8.15)
is p r o v e d
If
with
respect does not
h.
denote
5 C218,
Thus
metric
is small of
W
(cf.
Q(z,h)
where
and c o n s e q u e n t l y •
(7.5)
the c l o s e d on
enough,
W, i.e.
and
(7.16)).
ball in
centered
at
0 < h < h z,
W, z
with and Q(z,h)
W.
A £
k,r,s,~,y
R U(x,q(2,C)),
(x, c)
=
(8.16)
for the sake of b r e v i t y
i (x,c) =
by
Riemannian
h > 0
touch the b o u n d a r y
Let us note
- (1-28)(1-8)]
to the s u b m a n i f o l d
we will When
dpn
n 4 = max(n2,n3,n41,n42).
to the i n d u c e d
of radius
(1-8) ] p
- [(1-2e)(I-8)]P}_ <
are some constants.
Let us return z c Int W,
[2dn(l+8) ]p-[2dn(l-2@)
25 dn)i < C23 IB 2 (~(z)'3-6
= C24{(I+8)P
8.5.
of
one has
pn[6(z,2dn(l+8))
ZnCACz,8))
factorization
(8.24)
~ k,r,s,e,7
the last symbol
(x,q(2,C)),
is d e f i n e d
in
(4.1).
101
Let
z E Int W
and
let
0 < h < h
.
Let us d e n o t e
D(z,h)
=
Z
A(x,C)
N Q(z,h).
D(z,h)
are
AS
also
L e t us r e m a r k for
n ~ 2,
because
~p,8
for
~n(y)
subsets although ~n
defined y c W
For
such
that
every
P
the p o i n t s
w i E AZ k,r,s,~,T one
every
A(x,C) use
E Int W 60,
n ~ 1
and
consequently
#n
on
is n o t d e f i n e d
D(z,h)
for all
for all
on
W
n ~ i,
n ~ i.
the n o t a t i o n
~n(y)
for the
set
and
every
B > 0
0 < 60 < ~P,8'
there
exists
N U(x,q(2
C)
there
every
exists
h,
n 5 = n5(n,P,8,d0,h)
,
1 s i s n 5,
such
and that
for
has
(8.3.1)
V ( w i) N W # @.
Let
Yi = V(wi)
us n o t e
(8.3.2)
is d e f i n e d on
for e v e r y
and
i
Q(z,h)
N.
in g e n e r a l
we w i l l
0 < h < hp
every
then
of
is d e f i n e d } .
8.3.(IV). > 0
that
is w e l l
Nevertheless,
LEMMA
is c o m p a c t ,
nevertheless
~n
~ n { y E Y;
W
compact
The
D W.
submanifold
c W n def
#n(D(P,h))
is w e l l
Wn(wi,Yi,~)
n5 ! U ~ n ( W i , Y i ,I 6n) i=l
c
defined
and
n
n5 def
Proof.
Step
implies
there
positive on
~,
and
r,
For y = V(w)
As
exists
the
c ~n(Q(P,h+
IWI ~ e £ (C), rtsr~,y a number
a strictly
s, ~, T, C
satisfying condition
i.
U Wn(Wi,Yi,~n) i=l
yZ (C) r,s,~,T
increasing
t(Y)
for
S Y
(2.9)
> 0
function
and d e f i n e d
inequality
then
and
5)).
applied
to
n = 0
a strictly
t = t(Y)
depending
only
0 < Y < Y£ (C), r,s,~,~ and
such
that
(8.24))
such
that
the
following
holds. every
w
E A(x,C)
N W E Int W
(cf.
one has
0 < Y < m i n ( q 0 ( w ' W ) ' Y ~ ,( s C, ~), 7) r ~0(w,y,t(y))
c Q(y,y).
that
for a n y (cf.
V(w)
N W # ~
and
Y,
(7.16)) (8.25)
102
Let us fix A(P) As
P E Int W
and d e f i n e
= inf{qc(w,W) ; w E i(x,C)
follows
from the d e f i n i t i o n
and
of
V(w)
qC(W,W),
N W E Q(P,h)}. A(P)
> 0.
Let us d e f i n e
~P,B
min(t[min(¼,
and fix n u m b e r s
(8.26)
YZ
n >_ i,
~ > 0,
and
60
such that
0 < 60 < dp, B . Step
2.
As
#n(D(P,h))
then one can c o n s i d e r the c o v e r i n g %n(w), open subsets of
0 < h < hp,
{Int W n ( W , y , }
@n ) }wEA(x,C).
F r o m the c o m p a c t n e s s covering
of that
type
Now
Wn
it r e m a i n s
(7.21)
defined
only
equivalent
!
If
i,
a finite
number,
say
that
c %n(Q(p,h
¢-n(y)
that
if
+ 8)).
(8 27)
!
y E W~n(Wi,Yi,@n),
E W0(wi,Yi,d0)
c W.
Thus
then (8.27)
}-n(y) is
] c Q(P,h+8)
(8.28)
1 _< i _< n 5.
(8.28)
1 _< i _< n5,
is not true,
such that
~-n [~n (wi' Yi ,dn)] '
NQ(P,h)
#~.
Now, q
z E ~Q(P,h+8)
there
N D(P,h)
as the set
exists,
then
exists
a point
q E ~-n [Wn(Wi,Yi,dn)] ~
But as
if a p o i n t
exists
possible
to
¢-n[~n(Wi,Yi,dn) for all
there
the m i n i m a l
to prove
it follows and
¢n(D(P,h)),
1 <_ i _< n 5 : n 5 ( n , P , 6 0 , B , h )
n5 ! U ~n(Wi,Yi,dn) i=l
=
From is well
of
containing
Wn (wi,Yi, ~i ~n' ) ,
sets:
of the set
by the f o l l o w i n g
there
# ~,
then
¢-n[Wn(Wi,Yi,@n)] exists
N #-n[Wn(Wi,Yi,dn)].
necessarily Thus
from
and
q
and some
i,
q ~ Q(P,h+B).
n }-n [Wn (wi ,y.,6')]l is connected,
N
then
a point (7.21),
(8.25)
and
(8.26)
103
one has z 6 8Q(P,h+6)
N %-n[Wn(Wi,Yi,6n)]
c (8.29)
c ~Q(P,h+B)
n W0(wi,Yi,60)
On the other hand, as (8.25) and (8.26) one has
c ~Q(P,h+6)
n Q(Yi,48-).
# @ ' D(P ,h) n #-n[w n (wi'Yi' 6')] n
I ~ # D(P,h) n ~-n[Wn(Wi,Y.,6')] 1 n
c D(P,h) N W0(wi,Yi,60)
by
(7.21),
c (8.30)
c D(P,h) n Q(Yi' 46-)" Finally, ~Q(P,h+~)
(8.29) and (8.30) give (see Fig. 6) that n Q(Yi' ¼) # ~ (8.31)
D(P,h) n Q(Yi' ¼) ¢ ~"
104
/
....."-. .
Fig.
6.
In g e n e r a l
Q[
,
8)
D(P,h)
~W
is a v e r y
complicated
set of C a n t o r
type.
But the o t h e r with
p(D(P,h) ,
~Q(P,h+~))
hand
Q(Yi'
diem
¼)
>_ B, <- ~
because
D(P,h)
and we o b t a i n
c Q(P,h) .
the
On
contradiction
(8.31). In the
8.6.
• future
we w i l l
L e t us c o n s i d e r
simply
n >_ 1
and
write i,
n 5 = n5(n).
1 ~< i _< n5(n).
We w i l l
use
the
^
following
notations.
If
!
z (Wn(wi,Yi,6n)
then
z = exp~ 1
(z), (w i )
t nwi = tEn
and
z = t nwi(z) .
For
Q(z,r)
c W n ( W i , Y i ,6 n' )
let
i% n (w i ) us n o t e
also
Q(z,r)
= e x P - n1
Q(z,r),
where
0 < r < 3d
(wi) z
may
where
f
belong
simultaneously
to
i # j,
nevertheless
these
.
Although
n Wn(Wi,Yi,~n) notations
and
will
W
I
11
never
(w~,y~,~n), J j lead
to
ambiguities. Let
{Ai}i( I
be a f a m i l y
of
subsets
of
set
X
and
let
y c X,
105
y c
U A.. i6I ±
{Ai}iE I
We w i l l
of
Y
say that the m u l t i p l i c i t y
is n o t b i g g e r
than
L
of the c o v e r i n g
if for any
y E Y, #{iEI;
yEA.}~L. 1 N o w we can p r o v e LEMMA
8.4(V).
For e v e r y
0 < d o < @0' there
exist
the f o l l o w i n g
L > 0,
60,
refinement
of L e m m a
0 < 60 < 6p, 8,
n 6 = n6(P,B,60,h)
n 7 = n7(n,P,~,60,h)
8.2.
there exist
d 0,
such that for e v e r y
and the p o i n t s
n t n6 c ~n(w)
{Zj}l~j~n7
such that:
(8.4.1)
for e v e r y
j,
1 <_ j -< n 7, t h e r e e x i s t s
such that
Q(zj,2dn)
Wn
=
1 6') ~ n
lj ~ n ( W i , Y i i=l
n7 U Q(z.,d ) c j=l 3 n
c
n7 c
(8.4.3)
n5
J ~ i Q ( z j ' 2 d n ) c Wn
The m u l t i p l i c i t y Q(zj,2dn) ,
Proof.
Let
of p o i n t s 1 ~ i,
n ~ i.
j s n7
-
is compact,
n
prove
(8.4.3).
fied in v i r t u e
such that
p(z,zj)
and its c a r d i n a l i t y
of P r o p o s i t i o n
follows
from
(8.4.1).
Thus,
~n
by the b a l l s
than
L.
t h e n we can find a f i n i t e
~ (zi,zj)
> dn
for all
z E Wn
there
exists
on the c h o i c e
{Zj}l~jSn7
set
i ¢ j, some
j,
Now,
of points.
satisfies
let us r e m a r k
8.1.
satisfied
of
Let us n o t e t h a t such set is not
may depend
F i r s t of all
are e v i d e n t l y
Let us r e m a r k
< d n.
t h a t the set
(8.4.2)
i=lUW n ( W i , Y i , n6')"
is not b i g g e r
and that for any p o i n t
such that
We w i l l (8.4.1)
W
=
of the c o v e r i n q
1 ~ j < n7
{ Zj}l~j~n 7 c Wn
1 ~ j s n7 unique
As
1 < i <_ n 5,
c W n ( w i , Y ,6')1 n
n5 (8.4.2)
i,
that
conditions
(8.4.1)
the f i r s t two i n c l u s i o n s
and the last i n c l u s i o n of
it r e m a i n s
is s a t i s -
o n l y to p r o v e
t h a t b y the d e f i n i t i o n
of p o i n t s
(8.4.2)
(8.4.3). {Zj}l~jsn7,
of
106
for e v e r y
i # j,
1 ~ i,
j ~ n7
{ dn) (dn 1
Q
zi' - T
N Q
zj, -]-
=
one has
@.
(8.32)
For the sake of b r e v i t y we w i l l d e n o t e We w a n t to p r o v e n6(P,B,60,h)
that t h e r e
Qj = Q ( z j , d n ) ,
exist a number
such that for e v e r y
n >_ n 6
K
1 _< j _< n 7.
and
and e v e r y
n6 =
j,
1 _< j _< n 7,
one has #{1 _< a _< n7; Qa n Qj # ~} def K(n,j)
(8.33)
implies
For some zj
E Wn(Wi,Yi,
immediately j,
1 ~ 6~)
Qak N Qj # @, 1 ~ k ~ s.
Now,
exist a number and e v e r y
j,
to p r o v e K'
In v i r t u e of
maximal in
exists
(8.32), a number
(8 4.1)
Thus
•
(8.33)
n6
such that
then
it is s u f f i c i e n t
Let us d e n o t e
Qa k c Q(zj,3dn) ,
to p r o v e
that there
such that for e v e r y
n ~ n6
one has
to p r o v e K" > 0 disjoint
for some
and of P r o p o s i t i o n
d Q(z, ~ )
exp -in
d i e m Qj ~ 2dn,
n o t e d by K' (n,j),
Let us c o n s i d e r
of
1 i i ~ n 5.
Qf n Q(zj,3dn) } def YJ ~ K'
n u m b e r of m u t u a l l y
Q(zi,3dn),
As
Q(zj,dn),
all the i n d i c e s
and a n u m b e r
1 s j ~ n7,
#{i s f ~ n7;
that there
> 0
L = K + i.
i = i(j),
< as < _ n7
1 s k ~ s.
with
(8.33)
let us c o n s i d e r
for some
1 s a I < a 2 < .....
by
(8.4.3)
1 S j ! n7,
_< K-
z,
(8.34)
it is s u f f i c i e n t
that for open balls
n
and
d Q(z,~)
and
Q(z,
j
K".
c Q(zj,3dn).
d ~)
Q(z
to p r o v e
as above, the d n --3- c o n t a i n e d
of r a d i u s
does not e x c e e d
8.1 one o b t a i n s
is w e l l d e f i n e d
(8.34)
,
In v i r t u e
d -~ ) c Wn(wi,Y i
c Q(zj,3dn)
c
[~n(Wi,Yi,6n)] " (w i ) As
Q(z,
d ~)
from Proposition
c Q(zj,3dn) 8.1.1.
and as
one o b t a i n s
zj
that
( ~n(Wi,Yi,{ z ( Wn(wi,Yi,
6~),
then
3 , ~ ~n ) .
Now,
n
107
d from Lemma
8.1 applied to
respectively,
d ~2(~,n T )
n > n6
is defined
immediately
_<
)
and to
Q(zj,3d n)
n 6 = n6(P,B,~0,h)
9d ¢ B2n(Z j, ~ )
c t nw. (~(zj~'3dn)) 1
at the beginning
of Sec.
(8.35)
8.3.
implies that
9
K'(n,j)
Q(z,
one has
d c tnw. (Q(z, ~ ) ) 1
B2(v r) ^ n ' (8.35)
and to
one sees that there exists a number
such that for every
where
E = ~
Vol (BP( dn) ) d
=
27P
def K
Vol where
BP(r)
denotes
and Vol its volume.
9.
COMPARISON
9.1.
the
p-dimensional
Thus
(8.33)
the TLSM
V(w)
closed.
N U(x,q(2,C)) (cf.
Naturally
for TLSM V(w)
(8.24)),
and satisfying
element of the family A(x,C)\V(w)
that the set
continuity
two submanifolds in
W1
U(x,q(2,C))
all the conditions
from Sec.
and
W2
is not trivial
7.4.
transversal 1 IWII < ~, -
4.1. we will consider
V(w)
at the points
2
~2
where
yl
and 1
y ,y
2
Let us fix two numbers
y2
is defined by
~2
i.e.
~i
and
intersect 1 y = V(w) N ~i,
~ Int U(x,q(2,C)). 6i0,
i = 1,2,
0 < 6i0 < ~1 min (qC(w,wl) , qC(W,W2)) qC(W,W)
and
respectively,
IW 2 I < 1 -
of Poincar4
the open submanifolds ~i
to the
In order to avoid
related to the domain of the definition
Let us suppose that the submanifolds
where
question becomes
such that
~2.
n
such that
i.e.
as in Sec.
= V(w)
N U(x,q(2,C))
is not an isolated
map,
y
r
is proved.
which does not satisfy the above condition.
Si (x) k,r,s,~,y
the problems
w 6 AZ k,r,s,~,7
the absolute
Let us consider family
(8.4.3)
OF THE VOLUMES
Let us consider a point
of TLSM A(x,C)
Euclidean ball of radius
and consequently
(7.16).
such that
~ qC(w,wl,w2), def
(9.1)
108
N o w we described
can a p p l y
in Sec.
to the m a n i f o l d s
7.5 and o b t a i n
for
W1 ~
and
~2
i = 1,2,
the c o n s t r u c t i o n
and
n > 0
the
manifolds
•
n ~i = ~ n ( W , y z , 6 1 n )
=
<eXP%n(w)
(T' )-l(~l(v) ,v) ; %n(w) (9.2) (V n , n J
@[mn = 6i0(~') n ,
where {~}jt0
defined
Let
z I = zl(n)
z l ( }-n(~ Consequently can a s s u m e some
point
by
n
i = 1,2.
(9.2)
the e s t i m a t i o n s
# yl
be a p o i n t
) n i(x,c)
z I (V(w) that
zI
which
Let us n o t e
that
(6.6)
such
for the
-
(6.11)
functions
hold.
that
c ~w io n A(x,C). for
some
is so c l o s e
we d e n o t e
by
w to
z
2
( i ki, r , s , ~ , y yl
(cf.
that Fig.
n U(x,q(2,C)).
V(w)
meets
~2 W0
We at
7).
i
/
f
W
/ f
;
z 2
X
U(x,q(2,C))
Fig.
7
j/VV(w)
109
PROPOSITION
9.1.
p(¢k(z2),
i.e.
For any
~k(w))
n t 0
and
k,
0 ~ k s n
one has
< (Q(s,~,y)~2bk) 2
~(z 2) c V k ( W )
(cf.
(3.9)i).
Proof.
Using
(7.1.i) I,
(6.11 n' c o n d i t i o n
(7.14),
(6.1)
and i n e q u a l i t y
(Bb)
0 < C z i,
from Sec.
i.i I,
(2.9),
we have
f
)(¢k(z2),¢k(w))
~ p(%k(z2
,¢k(~))
+
+ p(¢k(zl
,~k(yl))
p(¢k(~),%k(zl))
+
+ p(%k(yl),~k(w))
3 • 200/2 Air,s(~r,s )k d i a m U(x,q(2,C))
+ diam W
i
v
(w,yl,@ik)
+ 2/2 q 6 0 ( ~ ' ) k ( l
_< 2 4 0 0 / 2
+C)
q A Ir,s(
<_ 2400/~
+ qq(l,C) ($') k < (2A Z
+
q AZr,s(
+l)qq(1,C)
(<') k
<-
r,S
< @Z -
The last i n e q u a l i t y 4b ~ < 9.2.
According
to
)k
(Q(Z,e,¥)
2bk)2
r,s,~,y(~'
follows
(5.1)
from
and
(6.1),
(5.2)
(6.7) I and
we will
denote
(6.5) I b e c a u s e
for
i = 1,2
and
1 _< k _< n t" i
PO
=
' (T w o exPwl)(yl)
Pk i
=
(T'% k (w) o e x p -I (w) ) (¢k(yl))
=
i i (Uk,V k) ,
and by P r o p o s i t i o n
9.1.
=
i i (Uo,Vo)
=
we can also d e f i n e
j=O
+
~,]
(po) =
(9.3)
110
f --i P0
=
' (T w o exPwl ) (z I)
Pk
--i --i (u0,v0)
=
( i>0
(T k
o exp -I ) (~k(zl)) ~ (w)
(w)
=
~'. ]
'=
(p)
(9.4)
--i --i (Uk,V k) .
= Let us notice
that these
points
actually
depend
on
n.
w(k) ~ Pk
(u ,v k
p2
(u2,v 2)
Pk=(Uk,Vk) M
Fig.
8.
F w(k)
=
w(k)
=
2 Mk -i ) (¢k(w)), (T' o exp k #k (w) ~ (w) -i ) (~k (Q)) (T' o exp k Ck (w) ¢ (w)
i = Mk (T' o exp -I ) (Wk) , i = 1,2, ~k (w) ~ (w)
LEMMA
9.1.
For every
n8(i,r,s,e,y,t)
> 0
C > 0 such that
zl ( #-n(~ )i n A(x,C) (cf.
(2.2)
and
(2.11)).
__if
and
t > 0
for every
II~In - Vnllnl _< ~ n
there
0 <_ k < n.
exists
n >_ n8(t) then
n8(t)
=
and --2Vn 6 B 2n(v2, (C+t)~n)
111
Proof.
Using
(5.4)
as well
as
(2.11)
we have
f
r~n Vn~l + HVn~ vn~r + rv~ V~nr< V~n Vnr~< ~ ~n
2 2 n + ]](u0,v 0) II) (
<
9.3.
< ~
r,s
volumes
which
At this T
the Lemma
We p r o c e e d
~n(w)
N.
~i n
=
play a crucial
stage
F i(y) ]
role
in the proof
to work
a few extra
the c o m p a r i s o n
of T h e o r e m
in the tangent
notations.
of the
Namely,
4.1.
spaces
let
-i (W~) , i = 1,2 eXP~n(w ) y 6 ¢-n(~i)
and for
•
concerning
it is c o n v e n i e n t
requires
+ Const(
is proved.
n o w to the lemmas
will
This
u~ v~l
+ ]I(u , n I[ + II( n' n N <
--2 --2 --I --i 1 1 + /~ 200([] (u0,v 0) ]] + [[(u0,v 0) I] + II(u0,v 0) [I +
<_ ~ n
Since
n~V~l
~ $ n + /2rl (u ,~ ~ )El + ]I(Un,V ~ -~n)
_<
=
T
and
-i
j = 0,i," .... ,n
Cj (W~) ( (Y))
exp
and
yj
let
-i (¢j (y)) = exp j (w)
.
CJ (w) We will
also use the n o t a t i o n
Beginning
from Sec.
from this m o m e n t
we will
2.6. assume
for c o n v e n i e n c e
that
6 1 0 , ~ 2 0 ~ C. LEMMA n >_ 1
9.2. (VI). and every
There
exists
C 3 = C3(Z,r,s,e, Y)
-n ~i a ~ % (Wn),
Ide(n-l) w (an)
b 6
¢-n
such that
for
2 (~)
Fln (a) -
1
~
C3C
d#(n_l) w(bn ) F2n (b)
Proof.
Step I.
particular
cases:
Clearly,
it is enough
to prove
the Lemma
for two
112
(i)
two transversal manifolds
(ii) a,b 6 V(w),
-
d%(n-l)w(bn)I F2(b )
y
a
=
yl ,
b
=
y
2.
-n
(
i,
d~(n_l) w Yn) iFI(yl )
)
n
d~(n_l)w(Yn) -n 2 F2( n y 2,~ i
coincide,
d¢(n_l)w(an)IFl(a) =
n
and ~2
i.e. in notations of Sec. 7.5.
d¢(n_l)w(an)l Fl(a ) For, since
~i
those two cases imply
n id~(nn-l)w(bn) F2(b) I
the general one via the following general inequality: for A,B,C > 0 l~c-
(9.5)
iI ~ IA-IJBc + IB- 1 c + Ic- it.
Step 2. We proceed to the proof in case (i). We have
d~(n-l)w(an) F l(a)
= n d~(~-l)w(aj) F ~ ( a ) <
Ln (a,b) = d~(n-l)w(bn) IFl(b )
~ld~(~-l)w(bj)
F~(b)
(9.6)
_< ~ <
i+ d@(~-l)w(aj)F~(a)-
d@(~-l)w(bj)IF~(b)1}
We will estimate the numerator and the denominator of the last expression.
We o b t a i n
using
Lerama 3 . 2 .
f
d#-~j-l)w(aj) F~(a) Cl<
sup
y~Aj (w)
- d#~-l)w(bj)F~(b)1
~
lld,~_llw
[I,d¢(~_l)w(aj) - d~(~_l)w(bj)II + F(F~(a),F~(b))]. From Mean Value Theorem and (3.3.6)I one has
(9.7)
113
lld<~_l)w(a j) -d~(jll)w(bj)II ~< ( sup lld2#-%j_l)w (u) II) -< u6A.(w) 3
-< IIaJ-bJll~J(w) _< diam(wj)l
(9.8)
c8Z3 b 4b(j-l) y
Step 3. We will now estimate separately aperture F (F~(a) ,F~(b)) .
diam(W~)
and the
From condition (Bb) from Sac. 2.1i, from (2.9) and from (6.11) n we have diam(W~) < q/2 .diam' (W~) _<
rld (vl JI12 I /2 q[26 i + ( sup v6B 2 (vj, 63 ) = /2 q 2610(~')J(i +Cq n)
=
19 91
4/2 q~10(~')J
Substituting (9.9) into (9.8) one obtains _
Ild¢~_l)w(a j) - d¢(~_l)w(bj)I I ~ 4/2 q c813
610 ~4b ~, j b (-~6)
(9.10)
In order to estimate the aperture we first estimate it with respect to the Lyapunov norm using Proposition 3.4 and (6.11). We use (3.4) tO pass from aperture F to aperture F' computed with respect to the Lyapunov norm. F(F~(a) ,F~(b)) <_ /2 A ~r,sexp(3Sr,s ) " F' (F~(a),F~(b)) < < 4/2 A Z seXp(j r, Er,s)
-
sup lld~ (v) II -< v6B2(vj,~) (9.11)
<_ 4/2 CArZ,sqJexp(3er, s) = 4/2 CA r,s Z exp(-6Jer, s) '
=
114
where
~i is defined in (9.2). 3 Finally, substituting (9.10) (3.3.5) I one obtains r I d ¢ ~ - l ) w ( a j ) F~(a)
C7Z~ b . a b3
<- C 1
-
(4/2 q c813
from
(6.5) I,
Step 4.
(9.11)
into
~
~, j + (--4b) a
(9.12)
s C31(610 +C)exp(-6Jer,s),
~' < a4bexp(-6JCr,s).
It remains now only to estimate
From estimate
(9.7), and applying
d#~-l)w(bj)F~(b)
610a4b b y
+ 4/2 CA~,seXp(-6Jer,s))
because
and
the denominator
in (9.6).
(3.3.2) I one obtain that
_<
lld~(j_l)w(bj_l) IIp _<
(9.13)
c5 £ I p" b (j -i) Thus finally, for
0 =
b
substituting
1
< 1
(9.12) and
(cf.
(9.13)
into
(9.6) we have
(2.8))
exp(6er, s )
n
Ln(a,b)
where
~ j=l ~
(i + C32 (610 + C)~ j-l) <- 1 + C33(610+C) '
C33 = C33(9~,r,s,d,y).
easy fact.
For any
e,
such that for every
a,
Indeed this follows
0 < 0 < i,
(9.14)
from the following
there exists a number
K = K(e)
oo
0 < a < 2C32,
one has
~ (l+a@ i) < 1 + Ka. i=0
115 Since
a
and
b
appear
symmetrically
in all our c o n s i d e r a t i o n s
we also have 1 = Ln(a,b ) _< 1 + C33(~I0
Ln(b,a)
and thus [Ln(a,b)
Ln(a,b)
- i[ ~ C33(610
Step write
~ i - C33(610
5.
Proof
the a n a l o g u e
only d i f f e r e n c e i.e.
(9.6)
appears
the c o u n t e r p a r t us to use
inequality
4~j (w)
exactly
610 5 C.
the same
line,
namely
we
in the same way as above.
the e s t i m a t e
that
yl, y2
and P r o p o s i t i o n
Thus
/2(IJY~II'
one obtains
of
The
1 2 Ilyj-yjll j(w ) ,
(9.9).
IIY~ - Y~II j(w )
<-
because
and p r o c e e d
we note
(5.3)
(2.9).
follows
in p r o d u c i n g
of
For that purpose allows
(ii)
Finally
+ C).
+ C) 5 2C33C,
of of
+ C).
for every _<
y
7.2 along
j ~ 1
- y
6 V(w)
n U(x,q(2,C)) with
.
our usual
This tool
-
we have
j (w)
+ fly II j (w) ) ~< 200/2(Kr, s)
=
200/2(
+
lleXPw- I (y2)
-i
1 (Y)
-1 - eXPw (w) Hw
=
J(,,yllw +
+
_ exp~l(w) llw , -< (9.15)
200/2 K(
+
ileXPx I (y2)
_<
800/2 K(Xr,s)Jq(2,C)
+
_ exPxl (w) IIx) _<
_< C34(Z,r,s,e,y)
(Xr,s)JC.
Here we use the i n e q u a l i t y
q(2,C) which
~q-I ~ 1600FK
follows
from
C (6.1)
and
(7.13).
Indeed,
(6.1)
implies
that
116
q(l,C) of
<
~q-I
(9.14),
9.4. on
Inequality
L n) r (c because
K
Let us denote ~i
by t h e
n
i
Riemannian
The following
B,
9.3(VII).
n >_ n 9,
the measure
n
(p-dimensional
volume)
induced
structure.
as the third main lemma
4.1. C 4 = C4(i,r,s,~,y)
there exists
A 1 c ~i,
the analogy •
There exists
0 < 8 < i,
us to produce
statement may be considered
in the proof of Theorem LEMMA
allows
< ~'.
r,s
by
(9.15)
A 2 c ~2,
such that for any
n 9 = n9(Z,r,s,~,7,C,6) 2n (A2)
> 0
such that if
and
1 (A I) 1
2 2 ~n ( A )
(9.16)
< B
then ~(~-n(Al)) 2
v0(¢
Proof.
-n
i
Let
structure
_< C4(8 +C).
(A2))
in
p~
be the measure
T n(w)N.
on
Obviously
~in
induced by the Riemannian yi = exp -in
for
(Ai),
i = 1,2,
¢ (w) we can write n ~l(yl)
pln (,yl.)
~in(A I)
~2n(A 2)
P2n(Y2)
~I(AI)
m2(A2)n
~n2 (Y2)
AS follows
from condition
last multiplier
in
(9.17)
with a speed depending
W 2.
if
Thus
n
(Bc) of Sec.
go to 1 as
A 2 c w n z,-
is big enough,
(9.17)
on i.e.
2.2.
n ÷ ~, C
the first and the on A1 c ~1 Wn,
uniformly
but independent n >_ n91
of
(Z,r,s,e,y,C,8)
W1
and by
(9.5) 1 1 Pn ( Y ) ~(Y2)n
1
-< 2B.
(9.18)
117
From Proposition 3.1 and Mean Value Theorem we have I
i -n ~0(~ (n-l)w (yi))
=
Iy
d%(n_l)w Tm¢~i d i(m)
=
l =
for some points Therefore,
m I 6 yl
(9.18),
(9.19)
1 -n (yl)) P0(} (n-l)w 2 -n (y2)) ~0(# (n-l)w
- 1
(9.19)
d~(n_l)w Tmi~i
t
m2 ~ y2
-
a n d Lemma 9 . 2 .
_< C 3 C ( i
• ~i(yi), i = 1,2
+ 2S)
imply
+
that
(9.20)
28.
Similarly to (9.17) we can write
~)i(~-n (AI))
=
~ (¢-n (A2))
~)i0(%-n (AI))
1
-n (yl)) 0 (~ (n-l)w ~02 (~-n (n_l) w(Y2) )
1 ~i (~-n (n_l)w(Y))
p20( -n 2 ~ (n-l) w (Y)) 20(¢-n(A2))
AS above, taking into account condition (Bc) of Sec. 2.2. and also Proposition 7.4, we see that the first and last multiplier go to
1
as
n ÷ ~,
uniformly on
A 1 c W~,
depending on C but independent of follows from (9.20) and (9.5). 10.
W1
A 2 c W~, and
W 2,
with a speed so that the Lemma
THE PROOF OF THE ABSOLUTE CONTINUITY THEOREM
i0.i. We will apply now the results of Sec. 8 to the transversal submanifold W I. Thus we will consider the submanifolds n(Wi,Yi,l ~n )
(cf. Sec. 8.1) , the sets
7 In
(cf. (8.3.1.)) ,
(cf. (8.4.2)), Q(zj,dn) c ~in(Wi,Y il.6n ) . (cf . Sec . (8.5)) any additional explanations. Let n >_ n 6 (cf. Lemma 8.4). Let us consider j, 1 Let i, 1 _< i _< n 5, be such that Q(zj,d n) N wl(wi,Y ~ , n _
~in
etc.
without
1 _< j _< n 7. 1 , 2 6n) #
J
and
Q(zj,dn) c wl(wi'yl'6n)'n Let us consider the covering of the ball
~2(~jn , (2-28)d n) c
118 tE
by the closed
p-dimensional
cubes
2~n(wi ) 1 ~ m ~ Nj,
of diameter
~d n
mj
c
E
c
2~n(wi )'
(with respect to the norm
If.If n
) (w)
with disjoint interiors. Let by
i
be the length of an edge of the cube
(Dmj)h
the concentric cube with edge
By Vol we will now denote the (measure)
in
mj"
Let us denote
Z + h.
p-dimensional
Riemannian volume
E 2¢n(w)
Let us remark that if for
~n = ~ n
0d 0
,
then for every
n > 0
and
one has
IVOI ((~mj) ~n)
10.2.
0 < ~ <
I
2P
(i0.i)
Let us define D mj 1
=
)-l{(~(v),v);
exp n (w i) °(Tin(wi)
v ~ ~' ^ }, @n(wi) (Dmj)
62 mj
=
exp n )-l{(P2n(V) ,v); v 6 T' ^ } ¢ (wi) °(T'~n(wi) #n(wi) (Dmj)~ n '
D1 mj (e)
=
(olj) , exp@ 1 (wi)
52 mj (e)
=
-i (D2 ") eXPcn (wi) ]
(i0.2)
I ~r is the mapping which defines the submanifold wr(wi,Y[,~ n) in n (cf. (7.20) and Sec. 9.1), r = 1,2. To be sure that ~r is well in ' ((Dmj) ), we will suppose beginning from now defined on v ~ T n(wi) ~n
where
that
0 < ~ < ~i qc (w,wl ,W 2)
Dm2 3. c ~2 2 Wn(wi'Yi'
(cf.
(9.1)).
~1 qc(W'WI'w2)(~.)n).
This implies that
Let us note also that the
intersection of the interior of any two sets empty, when
m I # m 2.
D1 . m13
and
D1 . m23
is
119
PROPOSITION and all
i0.i.
j,
1 0 < 8 < [.
Let
1 s j ~ n7,
For all
(cf. L e m m a
n t n4(8)
8.2)
one has
N
Q(zj,dn)
Proof
(10.3)
(i0.3)
c Q(zj,2dn).
By a p p l i c a t i o n
•
that
3 D1 c m=iU mj
-i eXPn(wi)
of
is e q u i v a l e n t
to
(10.3)
one
immediately
sees
to
N.
t n (Q(zj,dn)) wi
where,
3 U m=l
c
let us recall,
F r o m Lemma
~i mj
tn (Q(zj,2dn)), wi
c
Q(zj,r)
8.1 a p p l i e d
(10.4)
= exp ~ ( w i ) Q ( z j , r ) .
to
F = E
we have
that
l~n (w) tnw. (Q(zj'dn))
c ~2(~jn , (i + e ) d n )
for all
n >_ n 4 = n4(@)
>_ n2(e).
Thus
l
to p r o v e
the first
inclusion
of
(10.4)
it s u f f i c e s
to p r o v e
that
for
N
~j all
n > n4
one has
~2(
-
By the d e f i n i t i o n 1 0 < 0 < ~,
But as first
inclusion
"
of
{D j}
of
of
one d e d u c e s
that
) c n
3 ~I U mj" m=l
N. 3 DI B2(zj (2 - 2 8 ) d ) c U m j" n ' n m=l
one has
~2 (zj, ^ ^ (i + @)d n) c ~2 (zj,(2 - 28)d n ) (10.4)
N o w let us p r o c e e d The d i a m e t e r
(i + @)d
n
~imj
and thus
the
is proved.
to the proof
is equal
to
of the
8dn
second
and as
inclusion
of
(10.4).
~imj N ~2(~jn ' (2-28)dn) # @
N.
follows
m=iU
mj c
from the first
n(Zj,(2 - @ ) d n ) .
inclusion
and g = 8. n By P r o p o s i t i o n i0.i
in L e m m a
Thus,
the second
8.1 a p p l i e d
to
inclusion
F = E
s
lCn (w)
r = 2d
n7
~i c n
and L e m m a
8.4 we o b t a i n
that
Nj
u
u
j=l
m=l
D I. c ~i.
Moreover,
from
such that
for every
m3
(i0.5)
n
(8.4.3)
it f o l l o w s
n > 1, -
that
there
the covering of
exists
some n u m b e r
~1 by t h e s e t s n
L > 0
120 I
{D~j}l!j~n 7
is of m u l t i p l i c i t y
at m o s t
L.
We will
denote
this
I~mSN. 3 covering 10.2.
by
A.
We will
describe
has m u l t i p l i c i t y
one,
n o w h o w one can c h o o s e
except
it we p r o c e e d
consecutively
Q(Zj+l,2dn) ,
j = 1,2, ..... ,n 7 - 1
all sets
D mj+l 1
the c o v e r i n g
of
~i n
the m u l t i p l i c i t y
N DI
U i=l i
=
~l
c
n
For
The
10.1
every
e,
N
D m1 j + l
Q ( z j , 2 d n)
and in
(j+l)th
j Nk c k=l U m=l U Dmk"
f o r m e d by all r e m a i n i n g
of c o v e r i n g
N
{ D Ni } l1! i S
wh$ch
To o b t a i n
to the b a l l step we e l i m i n a t e 1 {Di}12i2 N
Let
elements
of
is not b i g g e r
N.
be
Clearly,
than
L
and
(10.6)
NU D I c ~1 . i=l 1 n
(10.7)
we w i l l n o t e
following
LEMMA
of
n 7 Nj D 1 U U mj ' j=l m = l
D~l = D mj
the p r e v i o u s
f r o m the b a l l
such that
a subcover
a set of a v e r y small m e a s u r e .
Lemma
lemmas
(VIII).
6 i = nmj' ^
~2I = ~2mj
etc.
is the last of our e i g h t b a s i c
it is e s s e n t i a l l y There
0 < 8 < m i n ( ~1,
exists iC2)
(see
(10.2)).
lemmas.
Unlike
global.
C 5 = C 5 ( Z , k , r , s , ~ , Y)
(cf. L e m m a
8.2),
such that
and for e v e r y
for
n ~ n4(%)
1
i~iVO (¢-n (DI)) N
- 1
comes
f r o m Sec.
(10.8)
~ C5(0+C),
I #-n v0( ( U D )) i=l
where
C
Proof.
Let us d i v i d e
G
B
and
(G=good,
7.4.
the set of
B=bad)
{i .... ,N}
into two d i s j o i n t
subsets
121
l i 6 G
iff
i 6 B
D i1 E Q(zj,2(l-0)d n)
j
otherwise.
It is clear that if Moreover, (where
for some
n4(@)
i E G,
]
I
then
Int(D~ N D~) = ~ for every i # j. I 3 it follows from Proposition 8.2 that for n ~ n4(@) ~ n 3
arise from Lemma 8.2) and for
1 5 i ~ N
one has
I
diam D~ ~ 28d . 1 n such n
Thus,
from Proposition
i0.i one deduces that for
n7 U D~ c U {z E Q(zj,2dn) ;~(z,~Q(zj,2dn)) i6B 1 j=l where we note by metric
p
on
~
the metric induced on
~n(~l)
(10.9)
by the Riemannian
N.
We will denote Aj(@)
~ 4@dn},
A(zj,28)
defined by
(8.14) by
= {z 6 Q(zj,2dn) ; ~(z,~Q(zj,2d n) ~ 4@dn},
Since the multiplicity
of covering
{Di}l~i~ N
Aj(8),
1 ~ j s n 7.
does not exceed
one has
f i~190Ni ( _ n ( D ! ) )
=
~ i6G
i(_n(Dl))
+
[ vl(_n(Dl)) i6B
_<
i G 90(~-n(D )) + L~I(i~BU ~-n(D )) =
= v0(
( U D )) + L~l(~-n( U D )) -< iEG i6B
l~-n N 1 1 1 _< ~0 ( ( U D )) + L~ (~-n( U D )). i=! iEB
Thus consequently
so that
L,
122
N
[ vl(
i=l 0
v~(¢-n( _< 1 + L
1 VO(¢-n(
N D1 U
))
V~(¢-n(
i=l i
Clearly, l(¢-n( v0
(10.8)
U DI i)) i(B
is e q u i v a l e n t
U i~B
D1))
N U D.]-)) i=l m
to
(iO.lO)
C5(8+C).
N
l(}-n( U DI)) ~0 i=l
To p r o v e
(10.10)
we w i l l
use the f o l l o w i n g
evident
remark.
a.
If
al, .... ,a N > 0 and b I, .... ,b N • 0 and
if b~i - h (i0.ii)
for all
i,
al+ ..... +a N < 1 _ < i _ < N, t h e n bl + .... .+bN _ h.
Let us pass n o w to the p r o o f of (10.9)
{Di}!~i~ N
1
(i0.i0).
and of t h e fact t h a t the m u l t i p l i c i t y is not b i g g e r
1 (c-n( U D )) iEB (~-n( U D )) i=l
than
L,
l(~-n( V0 _<
In v i r t u e of
(10.3),
of the c o v e r i n g
one has
n7 U A. (8))) j=l 3
n7 1 ¢-n ( U A . C e ) ) ) ~0 ( j=l 3
--< L
l(¢-n( U Q(z ,dn))) ~0 j=l J
[ 1 -n j=l ~0(¢ (Q (zj'dn)))
n7 ~ l ( # - n ( A j (8))) i=l
< L
n7 j=l ~ l(¢-n(Q
Thus prove
in v i r t u e of
(i0.Ii),
t h a t for some n u m b e r
1 ~ j ~ n7
one has for all
(zj,dn)))
to p r o v e
(i0.i0)
C 5 = C5(Z,r,s,~,y) n ~ n 9
it is s u f f i c i e n t and for all
j,
to
123
l(}-n(Aj (8))) _
~0
-< C5(8 + C ) .
(10.12)
~01 (¢-n(Q(zj ,dn) ) )
Let us fix
j,
A 2 = Q(zj,dn).
1 ~ j ~ n 7 , and d e n o t e
From Lemma
8.2
one o b t a i n s
A 1 = Aj(8) that
for all
U Q(zj,dn) , n t n4(8)
one has
Vn(A
)
-
1 < ~)nl(A2)
9 (Aj (8)) -
In o t h e r words, vl (AI)
1
+
c2e.
n4(8)
_< C28.
n >_ n4(8)
1
n >-
-< 1
1
by a p p l i c a t i o n
for all
1
9 (Q(zj'dn))
for all
_
Thus,
+
of L e m m a
9.3 to
S = C28
< 1
>_ m a x ( n 2 ( 8 ) , n 3 , n 4 ( 8 ) , n 9 ( 8 ) )
one d e d u c e s
,
that
one has
-n
v0(¢
(Aj(8)). -< C 4(c28
1
+ C).
v0 (~-n (Q (zj ,d n) ) This
proves
10.3.
Now,
i0.i)
that
(10.12)
and c o n s e q u e n t l y
it follows for all
p(D 1 n A(x,C))
from L e m m a
i,
(10.8).
9.1 a p p l i e d
1 ~ i } N,
(p
to
denote
t = ~
(cf. Sec.
the P o i n c a r 4
c ~2
map) (10.13)
1
if only Lemmas
n ~ nl0 = n10(~,8) 8.3,
D = D(P,h)
~n(p(D)
8.4 and c ~i
(10.7)
one o b t a i n s
(cf. Sec.
= p(#n(D))
de__~fm a x ( n 4 ( 8 ) , n 8 ( w ) )
8.5)
c p
that
for
from now,
C,
i=UlD
8,
Consequently,
n > nl0
by
and for
one has
=
U
p(D
)
i=l Beginnlng
"
610
and
c
U
i"
(10.14)
i=l 620
will
be a s s u m e d
so
124
small
that all our p r e c e d i n g
those
enumerated
we will
suppose
assumptions
in c o n s e c u t i v e
lemmas,
about
them,
in p a r t i c u l a r
are satisfied.
In a d d i t i o n
that
8d 0 0 < 03 < - ,
PROPOSITION nll(e)
10.2.
such that
2 --2 ~n (Di) 1 1 ~n(Di )
Proof.
(10.15)
There
exists
for any
a constant
e,03,n ~ nll
03
1
Using
and
and
Vn(Di)2 --2
i, 1 ~ i ~ N,
(I0.16)
introduced
Vol((Di)03
in
(10.2)
one can w r i t e
V°l((Di)03n)
)
V o l ( D i)
n V o I ( D i)
(10.17) MI(DI(e))
1 1 Mn (Di (e))
product. exists
now e s t i m a t e
From
condition
nll I = n l l l ( @ )
1 - e -<
2 ~ IJn (D~i (e))
Let us e s t i m a t e from
(3.2)
such that
1 _<
it follows for
one has
W
~n2(D~ (e))
pn2(D~(e))
We will
nll=
_< c6(e + e-~o)(1 + e-~o).
the n o t a t i o n s
~(~)
C 6 = C6(P)
I(DI
separately (Bc)
each of the m u l t i p l e s
from Sec.
such that
for
)
2.2 one o b t a i n s n ~ nll I
that there
one has
-< 1 + e.
(10.18)
the second m u l t i p l e . immediately
n ~ nll 2
in this
From Proposition
that there
exists
8.2 and
nll 2 = nll2(e)
one has
~ n ( D (e)) ^ _< 1 + 2Pe. Vol((Di) n)
(10.19)
125
By
(10.15),
fourth
and the
second
and
the
third multiple
fifth multiples
first one
is e s t i m a t e d
are e s t i m a t e d
respectively.
in
(I0.i).
in t h e s a m e
The
way
as t h e
Indeed,
V o l (D i ) 1 - 2P8 _<
_< i,
(10.20)
< _ 1 + 8.
(10.21)
1 (D 1 (e)) ~n 1 (D 1 (e)) 1 - 8 _ < ~n i vl(D 1 ) n
Finally, for
n t nll(8),
From the
(i0.18)
five multiple
-
(i0.21)
imply
(10.16)
that
is true,
now on we will
following
together
there with
suppose
with
exists
the c o u n t e r p a r t
nll(@)
a constant
that
C,
C6
8, d 0
depending
only on
and
~
satisfy
also
condition
the right
can apply
to
that
n t n12
for
(9.5)
for
(10.22)
c6(e + e-~o)(l + e-~o) < 1 so t h a t
of
such that
hand
s i d e of
A 1 = D~ 1 =
and
(10.16)
A 2 = D} 1
is s m a l l e r
the L e m m a
than
i.
Thus,
one
9.3 a n d o n e o b t a i n s
(max n9,nll)
2(¢-n(~)) v0
1 (~-n (D~)) v0
- 1
_< C 7
[C +
where
C 7 = C7(£,k,r,s,~,7).
10.4.
Now,
we can pass
the notations
o f Sec.
L e t us r e m a r k
p : W
^~
of
note
as t h e P o i n c a r e
~_/_) (i + 8d 0 ],
(4.1.2). now
is a h o m e o m o r p h i s m ,
then of
p(D), ~2.
where
Thus,
(10.24)
In t h e
~i = ~
'' W1
spirit
of
i = 1,2.
map
(x,c) ÷ ~ 2 n ~k , r , s , ~ , 7
a measurable
subset
to the p r o o f
4.1 w e w i l l
that
fl A k , r , s , ~ ,
(8 + ~ u)
(x'C)
D = D(p,h)
in v i r t u e
of
(cf.
(10.14)
Sec.
8.5)
one obtains
is
P.
126
t h a t for all
n t nl0
N
[ ~2 (¢-n(D~)) • i=l
Consequently,
from
(10.24)
n t n13 = m a x ( n 4 , n l 0 , n l 2 )
and L e m m a
i0.i
it f o l l o w s
t h a t for
one has
r
V2 (p (D)) _< N (8 +8-~0) (i+8--~0)]}i=i [ Vl(~-n(D
-< {i + C 7 [ C +
)) <(10.25)
W
-< {i + C 7 [ C +
W
(8 + 8 ~ 0 ) (i+8-~0)]}
•
N
{i + C 5 ( @ + C ) }
L e t us take > 0,
dO > 0
1 0 < C -< 2'
m = and
• ~l(}-n(
U D~)). i=l
(6d0)2.
When
8
n
m a y be c h o s e n
one d e d u c e s
from
(10.25)
increases
indefinitely,
as small as we like.
Since
that
~2(P(D))
I where
<
(i + C 8 C ) v I
C 8 = Cs(Z,r,s,e,y) From Lemmas
(<) ~-n
N
U i=l
(10.26) D'1 1
'
= C 5 + C 7 + C 5 C 7.
8.3 a n d 8.4 o n e has that for e v e r y
n5 U D. c U w l ( w . , y l 6') i=l 1 i=l n 1 ]' n N
Consequently,
~2(P(D))
from
(10.26)
c ~n(Q(p h+~)) '
one o b t a i n s
_< ( I + C 8 C ) ~ I ( Q ( P , h +
~)).
that
"
n ~ 1
127
As
~ > 0
may be chosen as small as we like, this implies that
~2(P(D))
~
(10.27)
(i + C8C)~I(Q(P,h))
i.e. that ~2(p(A(x,C) But as
N Q(P,h))
Q(P,h)
5
(i + C8C)~I(Q(P,h)).
is an a r b i t r a r y s u f f i c i e n t l y small ball in
the a b s o l u t e c o n t i n u i t y of the m a p p i n g is c o m p l e t e l y proved,
q(2,C)
and
and
e(C)
er,s,~, Y
of the sets
(4.1.1)
VZ r,s,e,7
Z w 6 Rk,r,s,~, Y
Thus,
(4.1.2)
e(C),
are a r b i t r a r y pairs of numbers satisfying
all our p r e c e d i n g conditions, Assertion
is proved.
if one sets
qr,s,e,y = q(2,C)
where
p
~i,
i.e.
(7.12)
follows now from
-
(7.14).
(6.1.1),
and from the proof of
and let
V(w)
from the d e f i n i t i o n
(4.1.2).
Z 6 Sk,r,s,~, Y (x).
Let
Indeed,
let
w i = W i n V(w),
i = 1,2. From (6.1.1) and from the d e f i n i t i o n of the sets V k,r,s,~,X ~ n 1 > 0 such that one deduces that there exists
~nl(w)
6 A ik,r,s,~,y
and that the s u f f i c i e n t l y small pieces
1 Cn
~i
of
1 (W I)
c o n t a i n i n g the points
a s s u m p t i o n s of
~n
(wi) ' i = 1,2,
(4.1.2) with respect to the point
satisfies all the 1 x I = ~n (w). Now
it remains only to prove the absolute c o n t i n u i t y of the m a p p i n g P--I --2, where this P o i n c a r 4 map is defined with respect to the W ,W snl family W i t h few u n e s s e n t i a l m o d i f i c a t i o n s , (Rk,r,s,e, Y) of TLSM. the proof is exactly the same as the proof of
(4.1.2).
Let us u n d e r l i n e that the above proof r e q u i r e s the use of the iterations 10.5.
~n
w i t h an a r b i t r a r y large
n.
Up to now in the proof of a b s o l u t e c o n t i n u i t y we never needed
the a s s u m p t i o n that essential
~(Ak,r,s,~,y)
in the proof of
(4.1.3).
> 0.
This a s s u m p t i o n becomes
I n c i d e n t a l l y the i n v a r i a n c e of
never d i r e c t l y appear in the proof of T h e o r e m 4.1. 10.6.
We pass now to the proof of
(4.1.3)
i.e. to the e s t i m a t i o n
•
128
of
IJ(p) (y) - 1 I . To this end we w i l l
to all our p r e c e d i n g
0 < C < ~
Let
i
transversal
As Fubini
C
so small
that
in a d d i t i o n
.
(10.28)
x 6 A£ k,r,s,a,7
For a n y
W6 =
consider only
conditions
be a d e n s i t y
6, 0 < 6 < q(2,C) to the f a m i l y
p o i n t of the set
let us d e f i n e
SZk,r,s,e,y(x) ,
A£ k,r,s,a,¥"
the s u b m a n i f o l d
W~,
by the f o r m u l a
! (eXPx o (Tx) -l){(6,v) ; v 6 B 2 ( q ( 2 , 0 ) ) }.
x
is a d e n s i t y
Theorem
p-dimensional.
p o i n t of the set
we can find Riemannian
and consequently
A kZ, r , s , ~ , y ' then by the Z 0 < ~ < e such that the rwsi~,~
6,
volume
V w ~ ( W 6 N A(x,C))
V w d ( W 6 N A(x,C))
Let us c o n s i d e r
the m a p p i n g s
> 0
(cf.
(8.24)). and
P~I,~
is p o s i t i v e
p~
~2"
Clearly
W~,W
P = P~I,~2 = ~ 6 ' ~ 2 As
VW6
° P.~iw,w 6~
(W~ N A(x,C))
> 0
and as m a p p i n g s
p~ ~ i = W~,W
and
P~i ~ are a b s o l u t e l y c o n t i n u o u s in v i r t u e W ,W 6 one o b t a i n s t h a t v. (~i N A(x,C)) > 0, i = 1,2.
of
( ' P~Z,~
)-i
(4.1.2) , then
1
Let us n o t e b y that
w
T
the set of all p o i n t s
is a p o i n t of d e n s i t y
to the m e a s u r e the set
~i
~2 n ~(x,C)
~l-almost
all p o i n t s
with of
absolutely
continuous,
belong
p(T).
to
L e t us c o n s i d e r the p o i n t s of d e n s i t y s u c h t h a t for e v e r y
of the set
a n d that the p o i n t respect
y 6 T.
are
v2- almost It f o l l o w s
t h a t for e v e r y h,
~ i N A(x,C)
p(w)
0 < h < h(6),
of
v2.
density
all p o i n t s
of
p o i n t of
As and as
there exists
o ne has
p-i
is
~2 N A(x,C)
from the d e f i n i t i o n
e > 0
such
with respect
is a d e n s i t y
to the m e a s u r e
~ i N A(x,C) then
w E ~ i N A(x,C)
of
h(e)
> 0
129
~l(Q(y,h))
Now,
_< (I + e ) ~ I ( T
(10.27)
and
v2(P(T N Q(y,h)) i.e.
(10.29)
NQ(y,h)).
(10.29)
imply
that
if
0 < h < h(e),
one has
_<(l+e) (I + C s C ) ~ I ( T N Q(y,h))
that ~2 (p(T N Q(y,h) ) ~I(TNQ(y,h)) _<(l+e) (i + C8C).
As
~ > 0
Lebesgue
can be c h o s e n
- Vitali
Theorem
(10.30)
as small
(cf. Sec.
as we like,
3.1)
(10.30)
implies
that
(10.31)
J(p) (y) _< 1 + C8C. As
y E T,
~2 N ~(x,C). symmetrical
Now, roles
j(p-l) (p(y))
But (cf.
then
p(y)
is a d e n s i t y
in our c o n s i d e r a t i o n s
p o i n t of the set p
and
p-i
play c o m p l e t e l y
and h e n c e S 1 + C8C.
J(p) (y) =
(10.28)),
by
then
(10.32)
1 j(p-l) (p(y)) from
(10.31)
(cf. and
Sec.
3.1)
(10.32)
and
C8C <
one d e d u c e s
1
that
IJ(p) (y) - iI -< 2C8C. Let us fix q ( 2 , C 0)
and
(7.12)
- (7.14)
C0:
e(C 0) and
0 < CO ~ 1
which
is so small
such that all our p r o c e d i n g (10.28),
are satisfied.
Then
that one can
conditions,
Z q r , s , ~ , y (B)
2-~)
for
0 < B _< 2C8C 0
)
for
B > 2C8C 0
=
L
q(2,C
i.e.
it is s u f f i c i e n t
to take f Iq(2,
find
130
for
0 < B ~ 2C8C 0
for
B > 2C8C 0 .
Sr,s,~,y(B)
(c o )
This
finishes
the p r o o f
ii.
ABSOLUTE
CONTINUITY
ii.i.
The
(Theorem of
aim of
ii.i)
which
is due
a n d of F u b i n i
asserts
that
11.2.
Let
the
in fact
on
(X,~,o)
sigma-field
Roughly
are
subsets
of
Sinai
speaking,
space, X
an
important
(cf.
this
where
on w h i c h
5
Continuity
theorem
on local
continuous
theorem
Lecture
of the A b s o l u t e
induced
absolutely
be a L e b e s g u e of
is to p r o v e
to Ja.G.
measures
N,
MEASURES
consequence
Theorem.
the c o n d i t i o n a l measure
section
is an e a s y
Theorem
4.1.
OF C O N D I T I O N A L
the p r e s e n t
[Ano] 2) and w h i c h
by a s m o o t h
of T h e o r e m
stable
manifolds
on them.
we n o t e
as u s u a l l y
the p r o b a b i l i t y
by
measure
is d e f i n e d . A partition
B
X/B
is a L e b e s g u e
into
a family
of
X
is c a l l e d
space.
Then
of c o n d i t i o n a l
measurable
the m e a s u r e
measures
Oc'
if the
~
factor-space
can be d e c o m p o s e d
c E 6,
so that
= F o . Jx/B e These detailed
measures
discussion
conditional For
E X
definition PROPOSITION measurable
ii.i.
absolutely
CB(x)
Let
~
almost
measurable
zero.
partition
the e l e m e n t
of
B
is a s t r a i g h t f o r w a r d
(X,~,~)
of
~- m e a s u r e
For
and
containing
corollary
x.
of the
measures.
X.
all
with
be a L e b e s g u e
Let
continuous
continuous
spaces,
denote
proposition
partition
for
up to a set of
[ROC]l_ 3.
of c o n d i t i o n a l
absolutely
Then
see
let
following
unique
on L e b e s g u e
measures
x
The
oIn_n M,
are
~
with x
respect
E X,
respect
space
be a n o t h e r
to
t__oo ~
the m e a s u r e aCB(x )
and
let
probabilitz
and
so t h a t gCB(x )
B
be a
measure d ~ do _ f. is
131
d~c 8(x)
f Cs(x)
~C~(x)
C
(x) fd~c B(x)
8 11.3.
Beginning
assumptions measure
of
f r o m n o w we w i l l
(4.1.3)
i n d u c e d on
X c N,
normalized
q rZ, s , ~ , y ( B )
If compact
0 < ~(X)
measure
6
~ ~(X)
submanifolds, measure
its m e a s u r a b l e
Let us n o t e by
z (B(q(B)), on
@
and
then by
vx
~ P.
denotes
the
For the sake of
i Sr,s,~, Y (B) = e(B). we w i l l d e n o t e
the
X.
~,
of
X
x ( X,
f o r m e d by the smooth, we w i l l d e n o t e
i n d u c e d by the R i e m a n n i a n B(q(B))
: BI(q(B))
the
metric
x ~2(q(B))
P.
c m m
and
~ = {y x ~ 2 ( q ( S ) ) } y 6 ~ l ( ~ ( B ) ) . the n o r m a l i z e d
Lebesgue
measure
measure
with the normalized
let us c o n s i d e r
one d e d u c e s
I
= q(B)
t h a t the c o n d i t i o n a l
coincides
def ~.
the m e a s u r a b l e
From Proposition
measure
a l m o s t all
¢
D
metric
on
B(q(B)).
pC~(z ) ,
p-dimensional
Lebesgue
C~ (z). n
U(x,q(B))
v
C6(x)
implies
the c o n d i t i o n a l for
As b e f o r e ,
partition
the set
partition
Fubini Theorem
of
on
then by
on
Let us c o n s i d e r
Now,
< +~,
is a m e a s u r a b l e
normalized
measure
fulfilled.
by the R i e m a n n i a n
b r e v i t y we w i l l n o t e If
are
N
s u p p o s e o n c e and for all t h a t the
v~ (y)c
y ( U.
depends
o n l y on
ii.i
q =
it f o l l o w s
is e q u i v a l e n t
Moreover,
that there exists
partition
directly
to the m e a s u r e
from condition
a positive
l (exp x o (Tx)
function
%
(Bc)
-1)~
that ~N Y
f r o m Sec i.i I
such that
i, r, s, ~, y,
(ii.i) is d e f i n e d
for
0 < t _< i,
lim $(t) t÷0
= 0
and that
d~)~n (y) 1
< @ (B) .
(11.2)
132 11.3.
Let us The
recall
following
that
A(x,C)
Proposition
is d e f i n e d
is a d i r e c t
by
(8.24).
consequence
of C o r o l l a r y
7.11 •
PROPOSITION
11.2.
If
y ( ~. ( X , T
(T'x o exp~l) (V(y)
n U(x,q(B))
where
~y: BI(q(B))
11.4.
Let
A c C
÷ B2(q(B))
(x) n A (x , ~ )
be a m e a s u r a b l e equivalently •
subset
then
= {(U,¢y(U)) ; u ( B I ( q ( B ) ) is a
C1
},
mapping.
)
of
(11.3)
C
of p o s i t i v e
Let us note
(B)),
(x)
of p o s i t i v e
(x) - m e a s u r e
~C
or
v ~ - measure. x
X = {z (A(x, q(S)) ; V(z)
n A ¢ @}
and ^
[A] =
U (V(z)
n U(x,q(B)).
z~X Let us c o n s i d e r {V(z)
the p a r t i t i o n
R U ( x , q ( B ) ) } z ( ~.
Theorem ~([A])
and > 0
this
and that
~
from now,
last a s s u m p t i o n
Under
ii.i.
For
is e q u i v a l e n t function C~ (a)
~
Continuity
Theorem,
is a m e a s u r a b l e
partition that
suppose
Fubini
[A] is m e a s u r a b l e ,
we will
~ -almost
to the m e a s u r e satisfying
1
[A] into the sets
in
of
[A].
(4.1.3),
any r e s t r i c t i o n
that
q(B)
= e(B) ;
of g e n e r a l i t y .
we will n o w p r o v e every v ~. a
a ( [A], Moreover,
(ii. i) such that
one has
dv [A] C___i(a) I d~a
of
sees that
does not p r e s e n t
this a s s u m p t i o n
THEOREM
From Absolute
(7.1.3) I one e a s i l y
Beginning
~
_< ~ (B).
the m e a s u r e there
exists
~ a -almost
~)[A] C~(~) a positive
everywhere
on
133
Proof.
Step
1
For
"
a 6 A£ k,r,s,~,y
and
n U(x,q(B)
K c V(a),
we d e n o t e
K(D)
=
U C z6K
(z)
and
~a(K)
= ~[A] (K(n)).
(11.4)
Y
K(T]) /
~
Fig.
Let ( A c C will
use
write
s ( [A]. n
(x) ; y
In v i r t u e every
y
and
(y,s)
and s
s = Cq(y) s
are
of
the P o i n c a r 4
n
n C~(s),
uniquely of
where
determined s 6 [A]
y 6 V(x) by
s.
and
Thus
and we w i l l
we
sometimes
s.
of P r o p o s i t i o n
A = Cq(x)
9
as c o o r d i n a t e s
instead
y 6 C~(x)
Pxy:
Then
I
" Cn ~ (Y) )
11.2 map
[A] -~ Cr](y)
N
and Pxy
[A]
(11.3) def =
PC
it f o l l o w s (x)C
(y)
that
for
134 is w e l l d e f i n e d . L e t us d e n o t e
sets
{Cq(y)
Let
Q
by
~A
the m e a s u r a b l e
[A]}yE[A].
y 6 C~(x)
and
L e t us d e n o t e
s (Cq(y)
N
partition
also
[A].
of
[A] into the
vqA = i ~q. y vq (C n (y) N [A] ) Y Y
We d e f i n e
d~ A] n(Y) - - ( s ) dv nA Y
=
ty(S),
(11.5)
dP x (y)
=
h(y)
(ii.6)
dv x~ (cf. of
(ll.4). y
and
The functions s
and of
Let us n o t e foliation
q,
y
Pxs'q • C~(x)
i.e.
2
p ~(y)
is also a s m o o t h m a p p i n g .
dv ~ x d(v~o rl )) Px~ The f u n c t i o n
T
Moreover, (with
~ = g)
Ity(S)
for
for
for
-
ii
y (C~(x)
ITs(Y)
(y)
=
and
h
are
÷ C~(s)
functions
= s.
As
q
is a s m o o t h
i n d u c e d by the
foliation,
(11.7)
is a m e a s u r a b l e there exists
Pxs
T h u s one can d e f i n e
T~(y) .
f u n c t i o n of
a positive
s
function
and g
y. satisfying
(ii.i)
such t h a t
and
s (Cq(y)
-< g(B)
(11.8)
N
[A],
that
(11.9)
and t h a t
- iI -< g(B)
y (C~(x)
measurable
the P o i n c a r 4 m a p
- 1 I < g(B)
y 6 C~(x)
lh(y)
t
respectively.
and
s 6 A.
(ii.io)
135
Indeed, tion Ii.i.
(11.8) (11.9)
follows i m m e d i a t e l y from follows from the fact that
and from the C o n d i t i o n same as above,
(11.2) and from P r o p o s i -
(Bc) of Sec.
i.i I.
~
is a smooth f o l i a t i o n
(11.10)
follows from the
together with the T h e o r e m 7.1.3 I.
As one admits that C o n t i n u i t y Theorem,
q(B)
for any
= e(B),
then in virtue of A b s o l u t e
y E C~(x)
and
s 6 A
one can define
d(, DA Vy OPxy) (s) = H (s). d~ A Y x The f u n c t i o n (4.5)
H
(ii.ii)
is a m e a s u r a b l e
function of
y
and
s
and from
one deduces that
IHy(S) where
L
(11.12)
- 1 I ~ L(B), is a function s a t i s f y i n g
F i n a l l y let us defin~ formula:
if
D
(ii.i)
(with
~ = L).
the p r o b a b i l i t y m e a s u r e
is a m e a s u r a b l e
subset of
A,
~
on
A
by the
then
(D) = ~[A] ([D]) , where
[D]
is d e f i n e d like
implies that the m e a s u r e
~
[A].
A b s o l u t e C o n t i n u i t y T h e o r e m easily
is e q u i v a l e n t to the m e a s u r e
~A. x
Let
us define dgn A x
(~)
=
B(s).
(11.13)
d~ From
(4.5) and c o n d i t i o n
6 A,
the function
18(s)
where
f
Step 2.
- II
B
(Bc) of Sec.
5 f(B),
is a function satisfying Let
1.11 one deduces that for
satisfies the i n e q u a l i t y
Q c [A]
(ii.i)
(with
be an a r b i t r a r y m e a s u r a b l e
~ = f). subset.
It follows
from the u n i q u e n e s s of the set of c o n d i t i o n a l m e a s u r e s that to prove the T h e o r e m ii.i it is s u f f i c i e n t to see that v[A] ( Q ) =
I
[[ [A]
where
1
IQ R C~(a) (r)Ga(r)d~a(r)]d~[A] (a)'
(11.14)
C~ (a)
denote the c h a r a c t e r i s t i c Z the f u n c t i o n G a is such that
function of the set
Z
and where
136
IGa(r) - II < ~(B), for
~a
almost all
(11.15)
r 6 [A] N V(a).
From the definition of conditional measures
[A] ) } , from t~Ch(y
11.4)-(11.7) , (ii.ii), (11.13) and from the Fubini Theorem, one obtains that f
[A] (Q) =
II
IQ n C n (a)(s)d~[A] (s)~d~ [A] (a) ~ ~a)
[A]
C (a)
[A]
C (y)IQ n C (y)
c~ tY)
[A]
Cn (Y)IQ n C D(y)
q~YJ
Fr
IQ
x
(s) ty(S) d~y A (s)~ h(y) d~x (y)
[A]IQ (y,s)tY (S) h (y) dgyA (S) d~x (y) = (11.16).
[A]IQ(y'S)ty(Pxy(S))HY(~)h(y)dm~xA(s)d~x(Y)
=
(s) IQRC~(s) (y' s) ty (Pxy (~)) Hy (s) h (y) dm~x (y)~ dV~xA(s) =
(s)IQ n C~ (s) (y,S) ty(Pxy(S))Hy(S)h(y) • Tg (y) d ~ (s)IQ n C~ (s) (Y'S) [(y's)d~(y~d~Dx A(s) ([) IQDC~ (s) (y' ~) i (y,s) 8 (s) dv~ (y~ d~ [A] (y,s) , wi~h L(t,s) = ty(Pxy(S)) Hy(3) h (y) T~(y) . The fact that the last integral is equal to ~[A] (Q) is nothing else but (11.14) written in a slightly different manner. (11.15) follows now from (11.8)-(11.10), (11.12), (11.13) and from the counterpart of inequality (9.5) for five multiples. •
137
11.4.
From
COROLLARY
(ii.16)
ll.l.
one
For
immediately
~
almost
deduces
the
all p o i n t s
following
y 6 A
and e v e r y
point
X ^
z ~ A~ k,r,s,e,y
N U(x,q(B))
Ve~(z) (C~(z)) 11.5.
If
induced
= ~C~ (z) (V(z)
z 6 A£ k,r,s,e,y
we w i l l
Secs.
by
metric
suppose
11 and
that
V(z)
N U(x,q(B)) then
by t h e R i e m a n n i a n
Now, (cf.
such
that
v p
is e q u i v a l e n t
U
AZ
one
has
= i. we w i l l
z
on local
the
4)
N A = y
denote
stable
~-invariant
the m e a s u r e
manifold
V(z).
probability
to the m e a s u r e
9.
measure
Let us r e c a l l
that =
Ak'r's'e
I>2
k,r,s,e,7"
y>0 Theorem COROLLARY
ii.i
one
the
following
I_~f ~ ( A k , r , s , e)
11.2.
z E AZ k,r,s,e,y
implies
where
£
> 0
is big
then
enough
for
and
~
almost
y > 0
every
is small
point
enough,
has ^
~z(V(z)
Proof.
N Ak,r,s, e N U(z,q(B))
Let
G = M\H. measure
H
denote
As the ~,
For
~
then
the
also
G1 =
measure
= ~(G)
z ~ AZ k,r,s,e,7
us d e f i n e
set of r e g u l a r
invariant ~(G)
= 9z(V(z)
N U(z,q(B)).
points
~
(cf.
(11.17)
Sec.
is e q u i v a l e n t
2.2 I) and
to the
= 0.
let us d e f i n e
A
z
= {y ~ V(z) ; y ~ G}.
U {z 6 A Z - ~z(Az) Z>2 k,r,s,~,y'
> 0}.
G1
Let
is
y>0 measurable. Let such
us
that
First
we w i l l
suppose
that
prove ~(G I)
~(G 1 N A £k,r,s,~,7)
(with r e s p e c t As we
to
~)
suppose
one
sees
that
one
has
Vz(V(z)
z,
9z(V(z)
of the
that
on the
> 0.
> 0. set
v(G I)
set of
that
Thus Let
> 0,
> 0
x
one
can
find
be a p o i n t
£
and
y
of d e n s i t y
G1 N A~ k,r,s,~,y" then
z E U(x,q(B))
N G 1 N U(x,q(B)))
N G N U(z,q(B)))
~(G I) = 0.
> 0.
in v i r t u e
of T h e o r e m
of p o s i t i v e This
implies
and c o n s e q u e n t l y
ii.i
~
measure
that
for s u c h
from T h e o r e m
ii.i
138
one obtains Now, and
from
that
~(G)
l e t us n o t e
INFINITE
12.1.
1-3 of
infinite
and
[Rue] 3) a n d
b y M. B r i n
and
4.1.
ik,r,s,
indicate
From
is t r u e
that
v ( G !) = 0.
(7.2.1) I,
for all
\G 2 c H.
how, our
counterpart
The last
Z. N i t e c k i
•
be very
using
some
previous
of T h e o r e m s
result
(see
a n d P. C o l l e t
We will
~ ( G 2) = 0.
(11.17)
can modify
dimensional
F. L e d r a p p i e r subject.
one
proves
CASE
now quickly
the
that
because
[Rue]3,
contradiction
G 2 = G @ GI;
DIMENSIONAL
We will
Secs.
This
(7.2.2) I it f o l l o w s
z ( ik,r,s,~\G2,
12.
> 0.
has
[Bri]2).
been We
to o b t a i n
6.11
[Man] 2
thank
and we leave
from
arguments (cf.
proved
for t h e d i s c u s s i o n s
sketchy
results
independently
sincerely
M.
we had about the d e t a i l s
Brin,
the
to t h e
reader. 12.2.
Before
recalling
the existence emphasize
that
the a p p r o a c h approach
of LSM even
of
(cf.
in f i n i t e
21 a n d
purpose
only
of P a r t
for t h e
Thus
the m a p p i n g s
Indeed,
invertibility
12.3.
E
be a separable,
be an
open
Let us suppose
that
K.
that at least
The well one
the
u p to now,
Indeed,
K c U
known
such measure
of P a r t s
to -~.
the
of
metric
of
and, whose
for
the
}: N ÷ %(N) of
in w h a t
%
is u s e d
follows,
existence
is
~.
Let
dimensional
¢: U ÷ E
is a c o m p a c t
n0 > 0 be a
I-III we can
Moreover,
invertibility
metric
infinite
subset.
#(K) c K, a n d t h a t for s o m e no d¢ (x) is c o m p a c t . Let ~ on
equal
of L y a p u n o v
the
measure
allows
let us
to the P e s i n
2).
of i n v e r t i b i l i t y I,
a k i n d of L y a p u n o v
of
Let
LCE
in P a r t
construction
construct
U c E
compared
for t h e p u r p o s e s
with
independent
and
space,
considered
from Appendix
I, the a s s u m p t i o n
c a n be d r o p p e d .
we will
case
advantageous
(T.I)
[Rue] 3) w h i c h
Hilbert
+~
is in f a c t u n n e c e s s a r y . also
(cf.
(1.2) I)
I log+fld~l[ld~< consider
framework dimensional
dimensional
[Rue] 3 is m o r e
(cf. Sec.
condition
Ruelle's
in i n f i n i t e
and
subset
every
%-invariant
Krylov-Bogoliubov exists.
Hilbert
be a
C1 such
x 6 K, Borel
space
mapping. that the o p e r a t o r
probability
theorem
guarantees
139
L e t us s u p p o s e
I
that
log+lld~xlld~j(x)
(12 .i)
< +~.
K
For
every
x (x,u)
The
x ( K
=
1-3 of case
in A p p e n d i x
2.
in a s o m e w h a t
theorem,
the O s e l e d e c Let
us n o t e
different
the LCE w i l l
let us d e f i n e
the LCE
X(x,u)
all
need.
the n e c e s s a r y
It g e n e r a l i z e s
Multiplicative
that
manner
in
unlike
the r e s t of
are
formulated
formulated
the book. way;
proved
infinite
Theorem
facts
in d e c r e a s i n g
facts
to the
Ergodic
[Rue] 3 t h e s e
be e n u m e r a t e d
by
(12.2)
summarizes
[Rue] 3 that we
dimensional
section,
u 6 E
1 logIId~n (u)If-
lira sup n++~
following
in Secs.
and
In this
i.e.
Xi(X)
>
Xi+l (x)THEOREM #(A)
12.1.
c A,
(12.1.1)
There
p(A)
= 1
The
exact
and
for e v e r y
u
6 E.
When
u
varies
number
of d i s t i n c t
Xl(X) where
> X2(X)
s(x) = + ~
determined of
E, of
exists
limit
in
that
in
subset
A
for e v e r y
(12.2),
E\{0},
> .....
is not
finite
such
values
sequence
a Borel
x
perhaps
k(x,u)
{Xi(X)}
of
takes
includinq
K
such
that
E A:
equal
too
at m o s t
-~,
exists
a countable
-~,
> Xs(x) (x) = -~,
excluded.
of c l o s e d
codimension
Moreover,
linear if
there
subspaces
exists
a uniquely
{Li(X)}l~iss(x)
i < +~, def
E = Ll(X) such
that
X(X,U) (12.1.2) closed
V
~ L2(x)
for e v e r y
L e t us linear
fix
and
for e v e r y
c A,
T < 0.
subspace
= {u 6 E;
¢(A)
i
~ Ls(x) (x)
~ Ls(x)+l(X)
{0}
u (Li(x)\Li+l(X)
o n e has
= Xi(X).
n As
~ .....
this
lim m÷+~ exact
For
every
V n = Vn(X)
of
n ~ 0 E
let us d e f i n e
by
ull = X ( ~ n ( x ) ,u) ! logIId¢ m m ~n(x ) limit
exists
the
in v i r t u e
of
-< T}
(12.1.1).
(12.3)
140
Then
for e v e r y
has
d~n(L) x
(12.1.3) we
linear
~ V
For
n
subspace
every
a > 0
A
such that
defined
on
IIdCmn
vll ~
L c E
satisfying
L ~ V0(x)
one
= E,
(x) = E. there
exists
for e v e r y
a positive n , m ~ 0"
and
measurable every
v
function E Vn(X)
IlvlI~ (x) e x p (m T) e x p ( ( n + m ) e).
(x)
(12.1.4)
If
L
is a l i n e a r
subspace
of
E
such
that
L N V0(x)
= {0},
then lim 1 log y(d~(L)
where
the angle
y(-,-)
L e t us n o t e Vn(X) ,
n ~ 0,
for e v e r y
,Vn(X))
that are
n ~ 0,
was
=
defined
(12.1.1)
L e t us n o t e
all o f the
12.4. Sec.
also
f r o m Sec.
We will
(2.15).
(12.1.2)
same
finite
imply
that
codimension
the
subspaces
in
E.
Moreover
2.2
from
is a n i s o m o r p h i s m .
that
(12.1.1)
follows
are directly
3.1 o f
now describe
from Corollary
related
to the c o n d i t i o n s
[Rue] 3 r e s p e c t i v e l y . in o u r
framework
the c o u n t e r p a r t
of
2.3 I
L e t us
fix
T < 0.
A T = {x E A; X £ ( x ) (x)
Obviously the
by
X
(12.1.2)-(12.1.4)
(S.2)-(S.4)
and
de n : L + d~n(L) X
[Rue]3;
0,
AT
introduced For
I
f(x)
(x)
x
L e t us d e f i n e
> T >_ X z ( x ) + l ( X )
is m e a s u r a b l e notation
6 AT
and
Vn(X)
let us d e f i n e
= exp XZ(x)+l(X),
e x p X £ ( x ) (x).
the
¢(AT)
set for
c AT .
= Lz(x)+l(~n(x))
some
Z(x)
L e t us n o t e
>_ i}.
that
for x ~ A T.
in
141
Clearly
l(x)
Let
r
us d e f i n e 0 < e ~ £
< 1. and
s
£r,s
be
positive
exactly
as
in Sec.
let us d e f i n e
r,s
the
l(x) A
This
> 0.
r,s,e
last
reader
A
=
case
r,s
Now,
l'(x)
For
in
positive for
x
8e
Now, n , m ~ 0;
in
the E
now
has
in
leave
it to the
of the
it f o l l o w s
dimensional subspace
for e v e r y
(12.6)
subspace
V0(x)
m,n ~ 0
W0(x)
(cf.
(12.3)).
and e v e r y
(12.7)
,
function
that
there
AT,
such
defined exists that
on Ar,s, e. a measurable
(cf.
(2.1.3) I)
(12.8)
~ Y(Wn(X),Vn(X)),
< +~.
"
Now,
For
every
of the
the
on
~' (x) = exp(-£).
(2.1.i) I)
measurable
d~(W0(x))
E
Moreover,
c Ar,s,s.
n t 0
let us r e m a r k indeed
(12.5)
for x ( Ar,s,e:
finite
that
(cf.
defined
every
d i m Wn(X) ; p(x)
codimension
one
(12.1.4)
W n ( X ) def
and we
~' (x) = ~ ( x ) e x p ( - 3 e ) ,
positive
and
here
~(hr.~.s)
IIVlI~s(x)rmexp(s(n+m))
function E AT
e,
hr,s, £
it f o l l o w s
(Be(x))-lexp(-en) where
some
complement
(12.1.3)
from
Let
S r < s S ~(x)}.
and
let us d e f i n e
2£ is some
Moreover,
=
to
E Vn(X) , x ( hr,s,c,
where
I"
r < s. For every
< r < s < ~(x)}
(2.6)i-(2.8) I we d e f i n e
x E AT,
lld%~nxVll _<
l(x)
is m e a s u r a b l e
as the o r t h o g o n a l
v
S l(x)
be not c o n s i d e r e d
= l(x)exp(3s),
From
and
(2.5)
= 0, let us d e f i n e
= {x E AT;
belongs
like
by
exercise.
hr,s, e
x E AT
l(x)
will
as an e a s y
Obviously each
For
2.31
r < 1
set
Ar,s, e = {x E AT; r exp(-s) when
numbers,
subspace
that
proof
the
given
let us d e f i n e n ~ 0, V
n
p(x)
p(x)
= d i m W O(x)
is also
=
the
(x).
inequality in Sec.
(2.1.2) I r e m a i n s
1.3 of
[Pes] 1 w o r k s
true
for
also
in
142
our f r a m e w o r k . function v ~ W
~
Thus
for e v e r y
defined
on
AT
e > 0 there
exists
such t h a t for all
a measurable n,m ~ 0
positive
and e v e r y
(x)
n
1 lld¢~n vll >-Nvll y - ~ [ ~ ( x )
]mexp(-s(n+m)) .
Let us fix once and for all a n a t u r a l Ap =
{x ~ AT; p(x)
number
(12.9)
p _~ Z. Let us d e n o t e
= p}.
It is c l e a r
that ~(A ) c A . P P Let x E A . L e t us d e n o t e by ll(X) ~= 12(x) ~= ... all L y a p u n o v P c h a r a c t e r i s t i c e x p o n e n t s at x , w h e r e a ny e x p o n e n t Xi(X) is r e p e a t e d k.l(x)
times
following base
; i.e.
k2(x)
the first kl(X)
with
fl(x) ..... fp(X)
i,l _-< i _-< p l i m n÷~
1 n
9 x
log
every
Ep
vectors
1 -~ and
{fi(x) }iEp
¥(d~n(Ep),
(12.12)
n ~ 0 P Z i=l
Xl(X) , the
an a r b i t r a r y
such that for e v e r y
= U(Ap)
subsets
implies
C
a ~ invariant
such that for e v e r y x E Z and for
p,Q
c {1,2 .... ,p } we have:
1 (x)
exp
c that for
al,...,ap
(12.11
subspaces
of W 0 (x)
spanned
by the
respectively.
there exists
n
d~x(Eo))~= -
mapping
d,n(EO) ) = 0
{fi (x) }i~Q
e > 0
to the m e a s u r a b l e
that for p ~_ 2 there e x i s t s
on Z such t h a t for e v e r y
easily
let us c o n s i d e r
2 applied
the l i n e a r
and
and for e v e r y
laiI
disjoint
EQdenote
> 1 defined
Now,
u(Z)
log y ( d , n ( E p ) ,
T h u s for e v e r y C
, implies
Z c Ap,
two n o n - e m p t y lim n÷~
where
subset
Now, W0(x),
with
n IId~x(fi(x))II = li(x)
of A p p e n d i x
> d*xiw0(x )
measurable
etc.
of s u b s p a c e
The T h e o r e m T.I. Ap
X2(x),
of t h e m c o i n c i d e s
a measurable x E Z
positive
and e v e r y
(-
n)
u-almost
every
function
n ~ 0 (12.12)
x ( Z, for e v e r y
E IR
n n ~ IId~ (fi(x))II =< H e (x) exp(en) II d ~ x ( i = 1
a .1f . 1 (x)) II (12.13)
143
where
qe(x)
implies
=
(4C
(x)) p.
t h a t for e v e r y
1 y ( g l U , e 2 v) -~ ~,
Indeed,
two v e c t o r
81 = ±i,
the s i m p l e
geometrical
u, u ~ E,
u ~ 0, v ~ 0, such t h a t
g2 = -+i, one has
consideration
IIul[ + IIvl[ _-<4qHu+v[].
For p ~_ l, let us d e n o t e A
= A
p,r,s,e
p
FI A
~5(Ap,r,s,e)
Clearly, For
r,s,e"
c A p , r , s , e.
~ > i, let us d e f i n e
p,r,s,e
the set
p,r,s,e (12.14) Be(x)
Obviously, =
A
if
£i > £2
~ Z, y
(x) ~ Z, n
~2 ~£2 p,r,s,g~ p,r,s,g
then
and
(x) ~ Z}
U />0
p,r,s~
p,r,s, Now,
12.5.
let us d e n o t e ~i = A~ 6' w h e r e p,r,s p,r,s,
We pass n o w to the h e a r t of the m a t t e r ,
Theorem
2.2i;
product) THEOREM
i.e.,
to the c o n s t r u c t i o n
a l o n g the semi o r b i t 12.2•
For e v e r y
a scalar product
<.,.>' x,n
a measurable
and p o s i t i v e
way,
for
and e v e r y
on the space numbers
the c o u n t e r p a r t
of the L y a p u n o v
{~n(x) }n~0'
x 6 ~l p,r,s
~ = --~Ds.
E,
norm
of (scalar
x 6 ~Z p,r,s n ~ 0,
there exists
which depends
A = A(/,p,r,s),
on
x
in
B = B(£,p,r,s)
such t h a t (12.2.1) II'll
on
The associated E
norm
by the f o l l o w i n g ng
B
exp(-
i ll-llx, n
is r e l a t e d
inequalities ng
• rrS, Pr's)ll.II ~ ll.llx,n ~ a e x p t ~ ) I I ' l l !
to the H i l b e r t
norm
144
lld%%n(x) Vn (x) II'
~ l'(x),
(12.2.2)
(l![d~%n(x) where
IIAII' denotes
between two spaces generated ~2.2.3)
the usual operator norm of an operator
(or their subspaces)
by the scalar products The subspaces
to the scalar product
and 12.2.
First,
required.
Secondly,
substantial
Let
<-,.>' Cn(x) ,0 instead of
12.2.
1.5.1 in
1
The proof
[Pes] I. Vn(X)
are orthogonal
with respect
between theorems of
%
is not
coincide, even if one has
~nx = x.
nE B exp(- ~-)
[PeS]l,
exp(x(~n(x),u))
inequality,
First we construct
V
space
n
Wn(X)
(x)
independently
scalar
= d~(W0(x)).
and
<. ' .>'x,n'
W
n
(x)
are othogonal with
we obtain
the scalar product
E.
and let
n ~ 0.
the following
For
u,v ( V (x) n
let us define
scalar product
S l(x)
and
one easily obtains
exp(x(#n(x),v))
xrn
(12.]5)
~ l(x),
that the series in
We will now verify that for some constant II'll'
The
is not a
'x,n = m>_0[(~' (x))-2m < d ~ n x ( U ) ' d#mn%x(V)>"
AS
2.2 I
follows the same idea as the proof
and on
as the subspaces
x E ~l p,r,s
following
(x)
one.
one whole
x,n
n
12.2 the invertibility
respect to the scalar product <-,->'
acting
in Theorem 12.2 we do not claim that the scalar
and
product we need on Afterwards,
W
that two main differences
(12.2.1)
Proof of T h e o r e m of T h e r o e m
A
If'l;' y,n
<.,->' y,n"
and
for Theorem
<.,.>' x,n
fact that in
V (x) n
endowed with norms
<.,.>' x,n"
Let us underline
product
Wn(X)]-l!I') -I t ~' (x),
then from Schwarz
(12.15)
A > 0
converges.
the norm
verifies
II'II ~ II'II~,x ~ A exp( n6)rl-[i
(12.16~
145
and the first Indeed, obtains
that
inequality
of
from
(12.15),
for
u 6 V
n
(12.2.2). from Schwarz
inequality
and from
(12.7)
one
(x)
r IiUjix,n
= /
[ (~' (x)) m>_0
-2mltd~ ~
(u) II2 ' nx
_<
-< ~$(x) ilullexp(n~)/[ (I' (x))-2m(r m >_ 0
exp
6) 2m~
=
= A(x)exp(n~)Iiuil, because As
from
(12.5),
r exp(26)
(cf.
A(x)
= e6(x)/
[ (l' (x))-2m(r m~0
(12.16)
is proved.
and thus
Let us prove
(12.14))
< l' (x) = l(x)exp(36)
x ~ Hi p,r,s
~ r exp(36).
one obtains
now the first
exp 6)2m' ~ I i -ex ~ (-6)' def =
inequality
of
(12.2.2).
For
u 6 Vn(X)
one has
Ild~¢nxUilx'n+l
=
/m~0 [ (I' (x))-2mild~m+l~n(x) (u) II2'
=
/ I' (x) ¢ [ (I' (x)) m>0
-2(m+l)ild~m~l (u) ll2' (x)
l' ( x ) / ~ (I' (x))-2mIld~m
m~0
We pass on subspace define
now to the c o n s t r u c t i o n
of the
Wn(X).
u 6 Wn(X),
For any vector
!
(u) ll2' = I' (x) IiUIix,n-
~n(x)
scalar
product
u # 0,
<',->'x , n
let us
146
llUlIx'n
=
i (~' (x))m
~
m>_0 lld#m
(u) II
<~n (x)
Let us recall
(12.]7)
that
lld¢m (u) ll # 0. #n(x)
Moreover,
as exp X(X,U)
)m ~-~(x),
the series
[ (~' (x) m>_0 lld~m (u) II
converges
and thus
]l~Nx, n > 0
~n(x)
is well defined,
ll~-llx,n is not a norm on
homogeneous,
for
i.e.
Let us note that
7 ~ IR,
y ~ 0,
Wn(X),
but
}!7U!Ix,n = 171
llTllx,n
is
NUllx,n.
IIUIIx,n -< llulJ.
l~ow, let us fix once at for all a measurable family of bases fl(x), .... ,fp(X) of subspaces W0(x), such that (12.10) is satisfied. Then,
f
for every
= n,i
n ~ 0,
dCnf x i l ~ x , n'
form a basis of
the vectors
1 < i _< p,
Wn(X).
We will define now the scalar
follows:
if
U = i=-~laifn'i
'
=
xrn
basis of
Wn(X) ;
the norm
First,
Wn(X)
as
then
product,
in particular
{fn,i}l~i~p llfn,illx,n
that on subspace
Wn(X),
is an ortho-
lifn,ilIx,n
1.
for some constant
' verifies ll'llx,n
exp(-n6)ll'll-< and the second
v = i=l~bif~..,i'
on
(12.18)
to this scalar
Now, we shall verify > 0,
<. t .>'xtn
~ aibi , i=l
i.e. with respect normal
and
product
!
II'Nx,n
inequality
of
II'il
(12.2.2).
let us prove the left side of the inequality
(12.19)
147
12.19.
From
vector
fn,i'
[
(12.9),
(12.i'4)
and
(12.17),
one o b t a i n s
that for e v e r y
1 < i _< p,
(N' (x))m
_< Hd#nfillx,n y6(x)exp(6n)
m>-0 [Id~nx(fn, i) II
[ exp(-2~m) m>_0
]Id~nfill
_<
exp(6n)
l-exp (-2~)
~
-<
,
i.e.
llfn,illx, n,
> l-exp(-26)£
Consequently,
for e v e r y
exp(-6n) IIfn,ill
v (Wn(X),
(12.19)
is true w i t h
(12.19)
is evident,
because
llfn,ill
and b e c a u s e
the i n e q u a l i t y
Let us prove v (Wn(X).
the left side
~ _ l-exp(-26)
for e v e r y
now the second
The second
The r i g h t
i,
(12.13) of
IIfn r illx't n
is true w i t h of
(12.2.2)
of
side i n e q u a l i t y
1 _< i _< p,
inequality
inequality
inequality
of
-<
~6(x) -< Z.
(12.2.2).
Let
is e q u i v a l e n t
to the
inequality. !
>
lld~ n(x ) (v)llx,n+l
Like
in the proof
one e a s i l y follow
!
of the first
verifies
(12.20)
for
inequality
in
<.,.>' x,n+l
(cf
Wn+l(X)
(12.2.1)
(12.18))
"
To c o m p l e t e
with
n t 0
respect
(12.2.2),
1 <- i -< p.
using Now
{d# n x ( f n , i ) } l ~ i ~ p to the scalar
(12.17),
(12.207 are
product
"
the proof
for every
of
v = fn,i'
from the fact that the v e c t o r s
orthogonal
(12.20)
!
_ ~ (x)llVlIx, n
of T h e o r e m
12.2,
and every v e c t o r
we have only
to p r o v e
w 6 E = Vn(X) 9 Wn(X),
i.e. , that B exp(-2@n)llwH
< llWlJx,n -< A e x p ( 2 ~ n ) H w N .
(12.21)
148
Let
w ~ E,
one i m m e d i a t e l y inequality
of
deduces (12.16)
from
obtains
that
B ~ 1
one obtains
As Z > l,
and thus from the left side
that
for e v e r y
u E Vn(X) :
(12.22)
as the s u b s p a c e s
with respect
to the s c a l a r
the left side of
(12.21)
then
from
(12.8)
and
Vn(X) product
W
n
<'t°>Xl
(x) n ,
are
finally
one
w i t h B = -2-
Let us go n o w to the r i g h t x 6 ~l p,r,s'
v 6 Wn(X).
llull-< IIUIIx,n.
(12.19),
orthogonal
u ~ Vn(X),
!
exp(-~n)
Now
w = u + v,
side i n e q u a l i t y
and
of
(12.21).
As
(12.10),
1 Y ( V n ( X ), W n ( X )) >_ ~ e x p ( - 6 n ) . The s i m p l e g e o m e t r i c and
v 6 W
n
considerations
imply
that for every
u 6 Vn(X)
(x)
I]u]l + ]]vll -< 4Z e x p ( ~ n ) llu+vll. consequently,
from
(12.16)
and
(12.19)
one o b t a i n s
llu+VIlx,n _< IIUlIx,n + IIVllx,n _ < E exp(dn) llull + Z e x p ( n ~ )
NvH <
_< (A+Z) exp(~n) (Ilull+llvN) < 41(A+Z) exp(2~n)llu+vll,
i.e. one o b t a i n s As u s u a l square
I
A = 41(A+Z).
by
£2
se q u e n c e s .
in Sec.
measurable
with
let us d e n o t e
summing
Like
(12.21)
2.3i,
the H i l b e r t
~Z
for
x ( Ap,r, s and
f a m i l y of i n v e r t i b l e
linear
n a 0,
'
: V
n
(x)
÷
£2,
T'
2x,n
: W
n
(x)
we can d e f i n e
transformations
' : E + Z 2 x ]Rp T' = T' x T' Yxtn r x,n lx,n 2x,n'
Tix,n
space of all real
÷
]R p
'
{T~,n},
a
149
such
T x'
that
standard 12.6.
scalar
L e t us
~: U ÷ E i.e.
transforms product
suppose
be a
for
some
C2
in
bounded
derivative
Now (cf.
this
remains ~m-k
5i),
the
~
¢
singularities exponential
of
text
Moreover uniformly
the
•
All
second
12.2
proofs
derivatives,
remain
the p r o o f s
Thus
consider
~
sets valid
Manifold
in Sec. the
Theorem
2 of
[Pes] 1
spaces
~k
~
results
and
with
are
even
it can
and
y
~l p,r,s'
~. 3w
only
one
simpler.
and
I, up to considers changes.
as the
Indeed,
identified
forgetand
the
we are
some m i n o r
space
n o w be
etc.
among
2.2i,
of P a r t
A1 k,r,s,a,¥
completely of
of T h e o r e m
of a l i n e a r
and
we can
instead
sets
with
to the reader.
Invariant
replaces
mapping
but we do not
the d e t a i l s
of all
on a s u b s e t
we drop
C1
condition,
instead
of the
is u n n e c e s s a r y
on the
is a
given
if o n e
counterpart
absent,
satisfied,
sup
and
assumptions,
spaces.
mapping.
singularities,
the
6 U
~
Abstract proof
Theorem
is d e f i n e d are
x
that
leaving
his
if i n s t e a d
mapping
identity
further
for
~l . p,r,s as
preceding
first
every
the H 6 1 d e r
change
Banach
to p r o v e
In fact
the
any
using
sets
into
(12.23)
Pesin's
with
Consequently,
7.1.31
bounded
to s u p p o s e
that
true without
the T h e o r e m
all our
and
refinement,
together
by a r b i t r a r y
able
with
satisfying
easy
let us n o t e
Sec.
<.,.>' xtn
lld2~xll ~ L.
it is s u f f i c i e n t
into
product
x ]Rp.
L > 0
In fact
enter
12
mapping
and
scalar
that besides
constant
lld~xlI ~ L
the
the with
in the
Moreover
in a b s e n c e
indices.
the e s t i m a t i o n
(6.1.5)
is
I
because
G (x) A 2 (x)
< ~.
x~ 1 p,r,s In the o t h e r TLSM 12.7. such
words,
are u n i f o r m l y N o w we w i l l that
A1 k,r,s,e,y
for
the m a p p i n g s
@x
which
defines
Lispchitz. define
~(AP'r's\/>0U one
x 6 ~l p,r,s'
the m e a s u r a b l e
A P1 ' r ' s ) = 0
considers
the
sets
and
subsets that
A1 p,r,s'
A1 p,r,s
if i n s t e a d
c ~l p,r,s of the
the C o r o l l a r y
7.11
sets
remains
150
valid. The space of all space
is separable if
p
dimensional
p < ~.
subspaces of a separable H i l b e r t
The space of all scalar products on
finite d i m e n s i o n a l vector space is also separable. theorem
(see e x e r c i s e II. 7.3
du calcul des probabilit4s, measurable that
W0(x)
it depends c o n t i n u o u s l y on
As
Neveu, Bases m a t h e m a t i q u e s 1964)
such that
one can find a
~(~lp,r,s \A~ ~,r ,s ) ~ ~'
and the scalar product x 6 AZ p,r,s
<''" >'x,0
on
and that the function
is c o n t i n u o u s on A Z p,r,s. V0(x)
=
[W0(x)]±,
c o n t i n u o u s l y on and from on
J.
Paris, Masson ed.,
A1 p,r,s c ~lp,r,s
subset
the subspace
= l(x)
from
Thus from Luzin's
Moreover,
V0(x)
depends
from the d e f i n i t i o n
(12.15)
it is easy to see that the scalar p r o d u c t
depends c o n t i n u o u s l y on
scalar product Now,
x 6 A1 . p,r,s
(12.1.3)
V0(x)
then the subspaces
<.,.>' x,0
x 6 A1 . p,r,s
C o n s e q u e n t l y the
depends c o n t i n u o u s l y on
from the above p r o p e r t i e s of the sets
with the last remark of Sec.
<-,'>' x,0
x 6 A1 . p,r,s A1 p,r,s
together
12.6 one easily follows that on the sets
A1 C o r o l l a r y 7 11 remains valid. p,r,s' 12.8. Let us suppose now that $: U - - - - - - ~
E is an injective mapping.
The A b s o l u t e C o n t i n u i t y T h e o r e m 4.1 also remains valid in our infinite d i m e n s i o n a l point,
framework.
As for the proof,
there is only one
the proof of Lemma 9.2 w h i c h needs m o r e substantial changes.
The remaining parts of proof need only minor,
inessential changes.
The point is that in the proof of Lemma 9.2 one uses d2~ -I
w h i c h are p r e s e n t l y not defined.
the m a p p i n g
d~-n(an )
etc.,
de -I
and
Let us note by the way that
used in the f o r m u l a t i o n of
F~(a) Lemma 9.2 are n e v e r t h e l e s s well defined in our new f r a m e w o r k as the l(a) (cf. remark inverse of the i n v e r t i b l e m a p p i n g d~n(a) : F0l(a) ÷ F n following
(12.1.4)).
We will now d e s c r i b e in detail how to adapt the proof of Lemma 9.2
to our infinite d i m e n s i o n a l
framework.
The unique point in the proof of Lemma 9.2 d25 -I
are used is the estimate of the q u a n t i t y
where
d~ -I
Ln(a,b)
In fact, one proves that when the transversal
submanifolds
~2
b E ~i
coincide
(case
(i)) and w h e n
a E V(w),
then
and
from ~i
(9.6) and
151
l ld¢-l(aj )
F~(a)
Rj (a,b) def
I - Id¢-l(bj) 1 II Fl(b) <
d# -l(bj) F~(b) I
(12.24)
C(l,p,r,s)@ ]
for appropriate 0 = e(£,p,r,s), 0 < e < 1. The analogous estimations in the case (ii) when a,b 6 V(w) have similar proof and we leave it to the reader. We will give now a proof of (12.24), which assuming (12.1), avoids completely the use of de -I and d2~ -I. It is clear that Id¢-l(aj) F~(a) I =
Id¢(aj_ I) Fj_l(a) 1 I-l' (12.25)
~Id~-l(bj) F~(b) I =
Id~(bj_l
F 1j_l (b) l
--!.
Consequently, Id~-l(aj) F~
I - Id~-l(bj) (a)
F~(b) II
Id¢(aj-l) F 1 I-I _ Id#(bJ -I) F~_l(b ) l j_l (a) (a) I - Id~(bj 1 )
Id~(aj-l) F 1
-
i ~'_i
Id~(aj-l~
<
1
Fj_I (a)
1
--1
=
(12.2 6)
1
IF 4 l(b)
I • la~Ib 1~i 1 3-
,Fj_l(b)
I
Substituting (12.26) into (12.24) one obtains
R.(a,b)
3
=
Id#(aj_1 ) F~_ l(a) I - Id¢(bj_ I) IF1 (b) I j-i _
(12.27)
d~(aj_l ) F~_l(a)I
NOW, uslng Lemma 3.2, the Mean Value Theorem and (12.20) we
152
estimate the last numerator (cf. (12.23)) I]d~(aj-1) F~_ l(a)
_<
[
-
Id~(bj-1) F~_ l(b)'I
cILP[IId#(aj_ I) -d~(bj_l)II + F(F~_I{a) ,Flj_l(b))]
<
{12.28) _< cILP[LIlaj_l-bj_lll + F (Fj_ I I (a),F~_l(b)) ] _<
_< Cl(max(L,l))P+l[diam(W~_l) + F(F~_l(a) ,Flj_l(b))]. The estimation of
diam(Wj_ I)
and of
F (Fjl_l(a) 'F 1j-l(b))
remains unchanged (cf. (9.10) and (9.11) respectively).
Thus, as
in the proof of Lemma 9.2 from (12.28) one obtains that
I Id~(a1) j-
'
F 1 (a) I -Id{bj_ I) F1j_l(b) II ~ j-i
(12.2 9)
C(£,r,s,L) (610 +C)exp(-5Jer,s ) . We will now estimate the denominator of (12.26). Like in 12.28), but using the fact that a 6 V(w), applying (6.1.1) I and the inequality K r,s < ~ one obtains
<
cILP[IId~(aj_I) -d~(~J-l(w))II + F(F~_I(a), Wj_l(W))] -< Cl(max(L,l))P+l[iIaj_l - ~J-l(w) ll + F (FI3- !
(a)
, Wj -i (w))
]
_<
Cl(l,p,r,s,L) (Na-wIl(
153
where recall, one obtains
q(2,C)
is defined by (7.12)-(7.14).
I
Id~ (aJ -I) F~_I (a)
I > Id~(¢J-l(w))
Wj_l(W )
Consequently,
I + (12.30)
- C2 (/,p,r, s,L) q (2, C) exp (-5j Sr, s) .
To estimate from below
Id}(}J-l(w)
Wj_l(W) I we proceed
as follows. From (12.1.1) and (12.2.2) one immediately obtains that for any vector h ( W~_l(W)] 2j (12.31) IId¢(~J-l(w)) (h~ II ~- C3(i,P,r,s) exp( pr,S ) llhll. Now,
from (2.16) and (12.31) one deduces easily that Id~(~J-l(w))
Wj_l(W) I >- C3(/,p,r,s)exp(-:4JSr,s)-
This, together with (12.30) implies that
Id# (aJ -I) F~_I (a)
I t C3(/,P,r,s)exp(-%JSr,s)
+ (12.32)
-C2(/,p,r,s,L)q(2,C)exp(-5Jer,
s) •
Thus if, C~ (/,p,r,s) q(2,C) ~ 2L~(/'P'r's'L)~2
'
(12.33)
154
then
(12.32)
implies
that
Id% (aj-l)IF 1 j_l (a) Finally, that if
q(2,C)
Rj(a,b) i.e.
putting
for all
I >- C3(Z'P'r's) 2
together
satisfies
together
Theorem 101 of
13.
[Str]
the c o n t e n t of
We w i l l
formulate
M,N,~,% 1 I.
is an
the a n n o u n c e m e n t
of A b s o l u t e
convex.
This
of d ~ x for x ( U (and not
[Str]),implies
t h a t %: U
)E is an
of Sec.
of local
families, 7 of
stable manifolds
one can r e w r i t e
[Pes] 3.
The
same is true
r e l i e s o n l y on the facts
satisfy
conditions
(A),
and the
almost without for the
f r o m Sec.
ii-4 I.
in our f r a m e w o r k .
(B) and
(i.i)
-
(1.3)
Let
2.2i). following
[Pes] 3.
[Pes] 3"
9E
case w a s p u b l i s h e d .
n o w t h e s e two sets of r e s u l t s
{x E A; Xi(x)
from
[Str], w h e r e
of i n j e c t i v i t y
104 of
of t h e i r
[Kat] 2, w h i c h
Let
(cf. Sec.
that %: U
the set ~. has to be s u p p o s e d
the e x i s t e n c e
continuity
f r o m Sec.
unchanged.
REMARKS
content
The
t h o s e of
on p a g e
changes
13.2.
one o b t a i n s
jSr,s ) ,
in i n f i n i t e d i m e n s i o n a l
Once proved
absolute
(12.34
mapping.
FINAL
13.1.
remains
w i t h the a s s u m p t i o n
x E K as w r i t t e n injective
and
then
n o w the a s s u m p t i o n
mapping with
On p a g e
(12.29)
(12. 34)
is p r o v e d .
L e t us c o m p a r e
injective
(12.27),
(12.33)
The r e s t of the p r o o f
Continuity
exp (- 4~ r' s ) .
_< C ( l , p , r , s ) (~i0 + C ) e x p ( -
(12.24)
12.9.
j t 1
# 0
for e v e r y
Let us s u p p o s e theorem
Corollary
i,l ~ i ~ s(x)} de_~f A'
that
~(A')
> 0.
r e p e a t s w o r d by w o r d T h e o r e m s
12.1 b e l o w c o r r e s p o n d s
7.2 and 7.9
to C o r o l l a r y
7.2 f r o m
155
THEOREM
13.1
There
•
exist
subsets
A
c A'
i = 0 i 1,2
t
s u c h that
t - - . -
1
00
(13.1.1)
A i N Aj = @
if
i ¢ j
and
U A i = A', i=0
(13.1.2)
~(A 0) = 0,
u(A i) > 0
(13.1.3)
¢(A i) = A i
for
(13.1.4)
for e v e r y
respect (13.1.5)
for e v e r y
A. =
~(A
)
n.
1
j=l
=
6 9'7.+. ,
there
1
Aj+l.
for
is a
K
1
1
exist a decomposition
,
A3.1 n A3.2 = @
1
_<
j
_< n .
is e r g o d i c w i t h
¢ Ai
U,
.
U1 A j,
1
the m a p p i n g
i ~ 1
n,
i > 0,
i >_ 0
i > 0,
to the m e a s u r e
for
if
Jl # J2'
1
-
1
f (A~ i ) =
1,
A~
such that
and
1
the
mapping
~ IAi
has a
n.
l
COROLLARY
13.1.
continuous Ai
is a
authomorphism.
If for some
spectrum K
i > 0,
(see Sec.
2 of
the m a p p i n g
[Roc] 3 ) , then
$ restricted
to
authomorphism.
In fact t h e r e
is no d o u b t
t h a t the m a p p i n g
~ni A~
is B e r n o u l l i
l
(cf. T h e o r e m
8.1 f r o m
n e e d an e s t i m a t i o n given
in T h e o r e m
13.3. from
[Pes]3).
of J a c o b i a n
J(p),
t h a t to p r o v e
stronger
this r e s u l t we
t h a t the e s t i m a t i o n
4.3.
Let us f o r m u l a t e [Kat] 2
It seems
(Theorems
now the c o u n t e r p a r t
(4..1)-(4.3))
of the p r i n c i p a l
w h i c h do not i n v o l v e
results
topological
entropy.
Sec.
Let
M,N,~,#
1 I.
Let us d e n o t e
of p e r i o d THEOREM Lyapunov
13.2.
satisfy by
conditions Pn(~)
(A),
(B) and
(1.1)-(1.3)
the n u m b e r of p e r i o d i c
points
n. Let us s u p p o s e
exponents.
Then
t h a t the m e a s u r e
U
has n o n z e r o
from of
156
(13.2.1)
periodic
points
(13.2.2)
max
(13.2.3)
if in addition
of
~
are dense in the support of measure
P, O, lim sup n+m
log Pn(~) ) h (~) n > '
the measure
concentrated
on a single trajectory,
points o_~f ~
having a transversal
support of measure
p.
p
is ergodic and not
then the periodic
homoclinic
hyperbolic
point are dense
i__nnthe
PART III THE E S T I M A T I O N OF E N T R O P Y F R O M B E L O W THROUGH L Y A P U N O V C H A R A C T E R I S T I C E X P O N E N T S * F. L e d r a p p i e r and J.-M. S t r e l c y n
i.
I N T R O D U C T I O N AND F O r m U L A T I O N OF THE RESULTS
i.i.
The c e l e b r a t e d Pesin entropy formula asserts that when
compact R i e m a n n i a n manifold, when an a b s o l u t e l y c o n t i n u o u s the m e a s u r e on
M
~ ( Diffl+e (M)
and when
is a is
(i.e. a b s o l u t e l y continuous w i t h respect to
induced by the R i e m a n n i a n metric)
p r o b a b i l i t y measure,
M ~
invariant
then
r =
(
h~ (~)
where
]M
Xl(X)
< X2(x)
exponents of
#
respectively. system
[ ki (x)xi (x)) d~ (x) , Xi(X)>0
at
<...< Xs(x) (x) x,
{k
(x)}
1 2.2 I)
(Cf. Sec.
(i.i)
denote the L y a p u n o v c h a r a c t e r i s t i c
l~i~s (x) and h (#)
their m u l t i p l i c i t i e s the metric entropy of the
(M,~, ~) .
The aim of this and of the next part is to give a proof of formula (i.i) w h i c h remains v a l i d if one considers maps w i t h s i n g u l a r i t i e s as well as some class of m e a s u r e s
somewhat larger than the class of
a b s o l u t e l y continuous m e a s u r e s
(see Sec.
1.3 below).
The fact that the Pesin entropy formula holds for the class of m e a s u r e s c o n s i d e r e d here was well known to Pesin himself, not clearly stated.
Furthermore,
R. Man~
[Man] 1
although
r e c e n t l y gave a
very ingenious simple proof of the e s t i m a t i o n of the entropy from b e l o w in formula
(i.i).
It seems that Man~'s proof can also be
e x t e n d e d to the more general framework we are considering. less, the original proof given in G. Sinai
(see
[Sin] 1 ) is also beautiful,
natural and,
only the Pesin theory of invariant manifolds, rather simple. the system, entropy
if one admits
this proof is in fact
It allows us also to identify the Pinsker algebra of
that is the
h(~,yA)
Neverthe-
[Pes] 2 whose idea goes back to Ja.
o - a l g e b r a of sets
of the p a r t i t i o n
given here follows Pesin. by a large readership.
One o f o u r
A
such that the m e a n
~A = {A,M\A}
is zero.
The proof
aims is to m a k e it u n d e r s t a n d a b l e
Except for the k n o w l e d g e of results stated in
* With minor m o d i f i c a t i o n s , this part reproduces the paper A proof of the e s t i m a t i o n from b e l o w in Pesin's entropy formula. Ergod. Th. & Dynam. Sys. (1982), ~, 203-219.
158
Part
I
(Propositions
theory
1.2.
(see
The 0
subset
~
let us
+
X
V
A c M
3.3 below)
part
and
for e l e m e n t a r y
entropy
is s e l f - c o n t a i n e d .
is d e f i n e d
in Sec.
2 I.
For
x
( A,
( TxN,
denote (x,v)
by lim 1 iogIid n(v)II x n ÷~ n
=
the L y a p u n o v
characteristic
Like inSec,
i
l.l~and
[Roh] 3) this
ll.4iI
=
r,s,~
U
exponent
of v e c t o r
v.
let us d e f i n e
£
Ak'r's'a'Y
£>2
y>O O~ksm (cf.
Secs.
(6.5) I,
3.5 I)
(2.8)ii
We call
where and
i+
thus
the
A+
=
Let
us d e n o t e
=
0 < e < i, of P a r t s
results
following
{x 6 A ; X + ( x , v )
A+ r,s,e
e = e(r,s), all
> 0
subset for
of
some
satisfies
I and
(2.5) I,
II are
applicable.
A: 0 ~ v
( T N}. x
also
A+ N A
and
P+
=
U r
r,s,~
The
local
unstable
for
%-i)
passing
manifold
of
through
x E P+
~
(i.e. will
the
A+ r,s,e
local
be d e n o t e d
stable here
manifold by
Vloc(X) .
L e t us d e n o t e
u~n(Vloc(~-nx))
for
x E P+,
for
x
oo
v(x) {x} We
call
us a l s o
V(x) note
Theorem in
the g l o b a l that
i.i
[Pes] 3 or in
unstable
U(A +) = ~(P+)
is p r o v e d [Rue] 2.
in o u r
elsewhere.
manifold and
that
framework
passing P+
is
exactly
through ~
x.
Let
invariant.
in the
same
way
as
159
+ THEOREM
i.i.
(l.l.1)
if
y ~ V(x)
then
V(x)
N V(y)
(1.1.2)
if
y E V(x)
then
V(x)
= V(y) ;
if
K(x)
(i.i.3)
Let
K(x)
Then
open
if
y 6 V(x)then
Pv(z)
submanifold
denotes
us n o t e
obtained
can
also
ing e q u i v a l e n c e
the
(see T h e o r e m call
denote
1.3. and
the
~
that
the
we
(mod 0)
connected on
is e a s y
components
N.
if d e r i v e d
from
the
6.1 I. of
N
as the
the p o i n t s
-n(y))
shall (see
into
global
classes
x
and
unstable
of the
follow-
y
< 0
has
C
subsets
The global
We
manifolds.) and we w i l l
of
disintegrated
~ ~
of
~
~
M.
For
which
M
x 6 M,
contains
measures
means
By
foliation But we m a y subsets
(A ~ considez
by the
partitions
~.
that
we
defined
If for
we d e n o t e
denote
the p o i n t
p p
by
~
on
is a n o t h e r almost
the
x
every
o-algebra
of all
M.
of the m e a s u r a b l e manifolds
of m e a s u r a b l e
[Roh]3).
of the p a r t i t i o n
then
unstable
also
the n o t i o n s
partition
(x) c C~(x).
partition.
shall
use
set of c o n d i t i o n a l
{C~(X)}x~ M
partition,
We
(1.1.5)
of the p a r t i t i o n
x E M
unstable
is a
= 0;
metric
of T h e o r e m
between
measures
measurable
sisting
on the
the p a r t i t i o n
be d e f i n e d
be a m e a s u r a b l e
elements
measurable
induced
Riemannian
of
in the p r o o f also
follows,
{~C~(x)}x{M
measurable
V(x)
is g e n e r a l l y
V.
the e l e m e n t
one
which
and the v e r y d e f i n i t i o n of u n s t a b l e I p a r t i t i o n the g l o b a l u n s t a b l e f o l i a t i o n
conditional
and
then
N
(%-nx,~'nY)
distance
the p r o o f
relation
this
In w h a t
C~(x)
<_ 0}
6.1
it by
Let
of
V (~-nx)
by the
lim sup ~1 log p (~-n(x), n÷~
will
X+(x,v)
lira p
V(z)
that
us r e m a r k
manifolds
true:
= ~;
submanifold
n+~
of the
Let
are
= V(~ (x)) ;
where
estimates
following
connected;
--
Let
the
= dim N - dim{v 6TxN,
(V(x))
(1.1.5)
6 P
dimensional
not (1.1.4)
x,y
global
iff
defined
in ~1.2
consider of
M
A { M
measures unstable
the
which and
which
are
is g e n e r a l l y
o-algebra are
the u n i o n
x E A
implies
in some
foliation.
~
For
sense if
W
not a con-
of V(x) c A). well is a
160
m a n i f o l d imbedded in
N, W
inherits from
N
a R i e m a n n i a n structure
and hence a R i e m a n n i a n m e a s u r e that we shall call the induced m e a s u r e on
W.
We denote
D e f i n i t i o n i.
~x
the induced m e a s u r e on
A measure
~
on
M
V(x) o
will be called a b s o l u t e l y continuous
with respect to the global u n s t a b l e f o l i a t i o n of measurable partition ~x(C~(x)) ZC~(x)
> 0
for
~ ~
of
M
~
iff
such t_hat C~(x) c V(x)
almost every
x ( P+,
for any and
the c o n d i t i o n a l m e a s u r e s
are a b s o l u t e l y continuous w i t h respect to the induced m e a s u r e
~X" This p r o p e r t y could also be called q u a s i - i n v a r i a n c e by the global unstable
foliation
(think of the foliation by the orbits of a flow,
for instance). A short name ought to be Sinai measures. the i m p o r t a n c e of that p r o p e r t y in the e x i s t e n c e of such a m e a s u r e 'stochastic' [Pes]2,3,
[Sin] 1 and in a lot of examples,
is the key p r o p e r t y for e x p l a i n i n g
p r o p e r t i e s of the system
[Rue]4).
In fact Sinai did stress
(see
One is led to believe
[AnO]l, 2, (see
a b s o l u t e l y continuous with respect to the u n s t a b l e same role and have the same importance a b s o l u t e l y continuous
[BOW]l, 2,
[Bun] 5,
[Rue] 5) that m e a s u r e s foliation play the
for m u l t i d i m e n s i o n a l
systems as
invariant m e a s u r e s do in the theory of maps of
an interval. The a b s o l u t e c o n t i n u i t y t h e o r e m p r o v e d for d i f f e o m o r p h i s m s of compact m a n i f o l d s Sac. N,
in
[Pes] 1 and for the m a p p i n g s w i t h s i n g u l a r i t i e s
llii asserts that if then
~
~
in
is an a b s o l u t e l y c o n t i n u o u s m e a s u r e on
is a b s o l u t e l y c o n t i n u o u s w i t h respect to both global
unstable and global stable foliation of
%.
It is i n t e r e s t i n g to note that the m e a s u r e s w h i c h are a b s o l u t e l y c o n t i n u o u s with respect to global u n s t a b l e
(or stable)
foliation of
but w h i c h are not a b s o l u t e l y continuous occur very f r e q u e n t l y in A n o s o v and related d y n a m i c a l systems Let
M, N,
¢
and
~
(see
[Sin] 3 and
satisfy conditions
(A),
[Bow] I.
(B) and
(1.1)-(1-3)
from Sac. i.I ILet us now recall C o n d i t i o n s %-invariant m e a s u r e
C o n d i t i o n l.ii.
(1.2) I c o n c e r n i n g the
There exist p o s i t i v e constants
for every p o s i t i v e (U s(A))
(i.i) I and
Z.
-< C1 ~a.
s,
C1
and
a
such that
161
Here
A
denotes
the
singular
set of m a p p i n g
~
and
its
U
s
neighborhood.
Condition
Let
these
1.2.
the a b o v e respect
h
If the
to the
(i.e.
=
automatically
(1.2)
satisfied The
We w i l l
~,
~
satisfying
continuous
with
then (1.2)
the
A
in the P i n s k e r
system
coincides
(mod 0) w i t h
a-algebra
there
is a
= 0.
conditions
in the
(C) and
inequality
characterization in
(1.2)
prove
here
only
the
is c o m p l e t e l y
independent.
i.i I,
Let
i.e.
is an a b s o l u t e l y
This Theorem
has
M, N,
the
is a d i f f e o m o r p h i c
one
are I -an e q u a l i t y .
has
Sec.
o-algebra
estimation
estimate
I.i.
one
from
1.1
was
given
by
[Pes] 3.
exponents
COROLLARY
(1.4)
of P i n s k e r
characteristic
~
conditions
continuous
of the
f r o m below.
entropy
Indeed,
It is p r o v e d
and
~
satisfy
(A)-(C),
imbedding.
Moreover
measure
on
in the n e x t
all
the
N.
suppose
Then
part.
conditions
(2.1)i-(2.4)i let us
through
the a b o v e
in the
from
and
~ :N ÷ ~(N)
that
the
inequality
an e q u a l i t y .
corollary
follows
ll.iIiwhich
immediately
implies
from above
(by a p p l i c a t i o n
to
Theorem
~-i)
1.2 a n d
that
~
from
is a
measure.
Before exchange
proving
the r o l e
replacing
the
Theorem of
in e v e r y
characterization and
of
of
Lyapunov
Sinai
measure
i ss a b s o l u t e l y
foliation
compact
ki (x)Xi(x)) d ~(x )
subset
the
then
above
B. P e s i n
(1.2)
of a s m o o t h
x,0).
satisfied.
probability
and
unstable
~ (BkA)
If m o r e o v e r
Sec.
log + x = m a x ( l o g
where
with (A\B)
Ya.
are
o-algebra
for any
< +~
is a d i f f e o m o r p h i s m
}-invariant
~lobal
the P i n s k e r
( ~
~
(1.1) I
[ > ( [ _ . IM Xi(x)>0
(~)
Iid~(x)
when
conditions
M B
that
conditions
and V
M log+lid
us n o t e
manifold
THEOREM
1.21 .
%
place
we give
coefficient
1.2,
and
we w a n t
%-i
unstable is m o r e
of e x p a n s i o n
to e m p h a s i z e
in all
that
by s t a b l e 'natural'
that
is w r i t t e n
and v i c e because
of the v o l u m e .
you
can
here,
versa.
it r e l a t e s
thus
The entropy
162 If
A
is a linear o p e r a t o r between two E u c l i d e a n spaces of the
same finite d i m e n s i o n of
A.
m,
we denote
~k
the
k'th exterior power
Let us denote m
IIAAII
=
1
[ IrA K=I
+
With this notation,
Akl I
•
it follows
from
[Rue] 1 (see also A p p e n d i x
2) that
almost e v e r y w h e r e [ ki(x)xi(x) Xi(x)>0
=
(1.3)
lim ~n l°gN (d~)All n÷~
and i ( x~ ki(x)Xi(x))d~(x) M X. ( )>0 1 Thus,
=
lira 1 i l°glI(d@n)AIId~(x)" n÷~ n M
the Pesin entropy formula may be also w r i t t e n as follows:
h
1
:
r
(1.4)
lira n I l°gtIId l lld (x)" n÷~
M
In their i m p o r t a n t work
[Led]6, F. L e d r a p p i e r and L. S. Young
prove that for the d i f f e o m o r p h i s m s of class manifolds,
the p r o p e r t y
C2
of smooth c o m p a c t
(1.4), or e q u i v a l e n t l y the e q u a l i t y in
is a c h a r a c t e r i s t i c p r o p e r t y of Sinai measures.
(1.2)
Earlier L e d r a p p i e r
(Led]4,5 proved this fact for m e a s u r e s w i t h n o n - z e r o L y a p u n o v exponents. It seems that the proofs from maps w i t h s i n g u l a r i t i e s
[Led]4, 5 can be adapted to the case of
studied in this book.
Let us note that i n e q u a l i t y mappings
formed by all finite c o m p o s i t i o n
s a t i s f y i n g the c o n d i t i o n s Secs.
2. 2.1.
(1.2) remains true for the class of
1.21 and 4.2i) .
(A),
(B) and
~K o...o ~i
(2.1)-(2.3)
of m a p p i n g s
from Sec.
1 I.
(See
We leave the details to the reader.
PRELIMINARIES If
H
is a finite d i m e n s i o n a l E u c l i d e a n space, we denote
the volume on dimension, mapping.
H.
E1 c E
Let
E
=
F
VOlEl(U)
Vol
H
be two E u c l i d e a n spaces of the same
be a linear subspace of
Let us define VOlFI(A(U))
IAIEll
and
E
and
A :E ÷ F
a linear
163
where
U
arbitrary A(U)
is an a r b i t r a r y
open
linear
of
c F I.
We
subspace
also
and b o u n d e d F
of the
subset
same
of
E1 , F1
dimension
as
is an
E1
and
denote
Idet A I = IAIE 1 . Let
X
dimension
Y
JX
induced
be d e n o t e d
2.1.
~ x formula.
If
(M,M,~)
a measurable
and
~
by the R i e m a n n i a n
metrics)
respectively.
We r e c a l l
y
f E Ll(y,Vy),
be a m e a s u r e
measure
on
X
the
then
r Iy fd~y.
=
(f o T) Idet dT1d~ x
Let
known
the m e a s u r e s
of v a r i a b l e s
PROPOSITION
2.2.
be two R i e m a n n i a n m a n i f o l d s of the same f i n i t e 1 T :X + Y be a C diffeomorphism. Riemannian
(i.e. will
change
Y
and
measures and
and
preserving
space map.
of We
finite shall
measure
use
the
and
T :M~M
following
well-
result.
PROPOSITION
2.2.
defined
M
on
Let
such
g
be a p o s i t i v e
finite
measurable
function
that
log - ~ o T E LI(M,~) g
,
where
log - a
=
min(log
a,0).
Then lim ~ log g(Tnx) n
f
=
log g ° T d ~ g
Proof.
Let
immediate to the
us
and
=
first (2.1)
0
~-almost
everywhere,
(2.1)
(2.2)
0.
note
that when
follows
from
log g E L I ( M , ~ ) ~
the B i r k h o f f
ergodic
(2.2)
is
theorem
applied
function
log ~ ° T. g Let when h
us also
applied
E LI(M,~),
note
that
to a f u n c t i o n but
the B i r k h o f f
ergodic
h, h = h+ - h_,
in g e n e r a l
the
limit
theorem
with
is still
h+ ~ 0, h_
can be i n f i n i t e .
As
true
~ 0
and
164
log - 9. ° T 6 L 1 (M ,~ ) , g this
shows
that
n-i lim ~ [ n+= n i=0
the
following
Ti+l log g o g o Ti
limit
exists
~
almost
everywhere
1 Tn l i m -- log g o dsf K n+~ n g
=
and moreover I
Kd~
f J
=
M
where
log ° ~ g T
d~,
M
both
sides may be equal
+~.
As 1 -- l o g n we have K
g
÷
almost
0
everywhere
therefore =
l i m ! l o g ( g o T n) n
On the other
hand,
almost
w e know,
as
everywhere.
almost
0 < g <
everywhere,
that 1 l o g (g o T n) n converges Thus
to
there
0
in m e a s u r e ,
is a s e q u e n c e
lim 1 log(g o T i+~ n.1 This
implies
K(x)
because
n i ~ +~
ni)
=
= 0
~
0
T
such
~
preserves
~.
that
almost
almost
the measure
everywhere.
everywhere
and proves
(2.1)
and
(2.2).
[]
Let the
M1
space
o f all
measurable and
2.3.
call
Let
be
sub
M1
partition,
a function
~
f
o-algebra
space
of
M.
measurable
functions
we denote
M
the
n-measurable
b e an i n v e r t i b l e
on a p r o b a b i l i t y theory.
some
We
in
f
denote
L2(M,~).
G-algebra
iff
measurable
(M,M,~) .
We will
is
If
generated
M
and measure
recall
L2(M,MI,~) a
is a by
measurable.
preserving
two r e s u l t s
map
of e n t r o p y
165
PROPOSITION ~ -I. h
2.3.
(¢) t h ( ¢ - l , a )
Proof.
of
PROPOSITION -i
and
((~)
that
2.4.
the
(see Sec.
Let
H(q~-l~lct)"
of the
partition
From
12.1
of
M
such
that
at the p o i n t
a measurable
{#ne}n( Z
generate
M.
(mod 0)
with
is o n l y of
definition
of
M
such
that
a reformulation
Then
the P i n s k e r
A M n(~ ~n
in t e r m s
of
o-algebra
[Roh]3.
will
theorem
that
x ( A, T N x
[]
be a m a p w i t h
we want
l e t us r e c a l l For
partition
coincides
~ :M ÷ M
x.
from the very
[Roh]3).
< +co
12.3
of the
x ~ N
=
and
now on
statement
T N x
2.4
immediately
7.1 of
be
system
Proposition
For
e
the p a r t i t i o n s
of theorems
2.4.
be a measurable
2.3 f o l l o w s
¢-i
=
G-algebra
Proof
e
t H(¢-I~I~) .
Proposition
of e n t r o p y
h
Let
Then
singularities
as
in
to
N
to p r o v e .
T N is t h e t a n g e n t x d e c o m p o s e s in
space
Eu @ E0 @ E s x X x
where
E u, E 0, and Es are linear subspaces corresponding respectively x x x to p o s i t i v e , zero a n d n e g a t i v e L y a p u n o v c h a r a c t e r i s t i c e x p o n e n t s o f at
x.
u d%x(Ex)
This
=
decomposition
is i n v a r i a n t
u x0 E%(x) ,d~x(E )
0 E#(x)
=
in t h e s e n s e
and
d~ x
(E s)
that
=
s E~(x) •
u L e t us n o t e
that
for
x ( A+
,
is a n o n - t r i v i a l
E X
L e t us c o n s i d e r
TU(x)
(see
2.1)
= [d~xl ul" E
PROPOSITION
Sec.
2.5.
x
log Tu
(LI(M,~)
and
subspace
of
T N. x
166
JM(xi(~x)>0r k i ( x ) X i ( x ) ) d u (x)
Proof. in
The p r o o f
=
is b a s e d on the f o l l o w i n g
lOse] b u t e x p l i c i t l y
in
(2.3)
fM log T U ( x ) d U (x).
fact i m p l i c i t l y
[Rue] 1 (compare w i t h
(1.3)).
contained
If
x ( A
then
Ix
k. (x)X, (x)
X. ( )>0 1
1
=
1
lim 1 l o g IdSxn I ul "
n÷~ n
(2.4)
E
x
F r o m the H a d a m a r d
inequality
1
we h a v e
1
-1
< Ild¢¢(x)N m - ]d$$ x) I u
I
-<
ldCxl
1 EUx
=
TU(x)
< lld#xlIm,
E¢ (x) where
m = d i m N.
measure (2.4)
U,
Thus c o n d i t i o n
implies
that
and the B i r k h o f f
(2.2)
and the
log T u ~ L I ( M , u ) .
ergodic
theorem,
~
Now
invariance (2.3)
of the
follows
from
because
n-i
l°gld nl ul
=
i=0Z ruI ilx)l •
°
x 2.5.
Let us now f o r m u l a t e
[Ano] 2 T h e o r e m
4.4).
a classical
r e m a r k due to E. Hopf
We w r i t e d o w n the p r o o f h e r e
(see
for the sake of
completeness.
PROPOSITION
2.6.
subsets
of
M.
Proof.
We shall p r o v e an e q u i v a l e n t
L
2
Let
MI
Then
(M,MI,U)
c
L2
f-(x)
exists
lim 1 n÷~ n
on a set
of the
invariant
inclusion
(M,Mv,U) . a dense
By the B i r k h o f f =
o-algebra
M I c ~.
We f i r s t c o n s i d e r L2(M,M,~) .
be the
n [
set
ergodic
F
of c o n t i n u o u s theorem,
f ($-i(x))
i=O
Mf,Mf
( M I, D(Mf)
= i.
for any
functions f 6 F,
in the l i m i t
167
Let
P
By m e a n
be the o r t h o g o n a l
ergodic
f ( F
theorem,
is d e n s e V
in
which
are
with
x, y ( V ( z )
f-(x)
= f-(y).
the g l o b a l (mod 0)
2.6.
Suppose prove
such
by
Therefore
an
But
i.e.
we have
such
any
V
now
u ( P +)
Theorem
1.2
for a n y
restricted
= I.
The
on set of
two p o i n t s
there
L 2 (M,MI,~).
exists
Pf
where
x, y ( M f+ some
z ( P
l i m p ( @ - n x , @ - n y ) = 0 and thus n+~ in { P f ; f 6 F} is c o n s t a n t a l o n g
function
measurable
L2(M,~)
Pf.
that
(1.1.5)
manifolds
M
of
represents
L2(M,MI,P).
unstable
shall P+
f-
equivalent,
with
projection
to
Mf
and
so c o i n c i d e s
function,
Using
a
Propositions
by c o n s t r u c t i n g
2.3,
a measurable
2.4 a n d
2.5 we
partition
q
of
that
¢-I , n(A ~ M @n q = M
~ <
v
.
1 (2.5)
n
{~ n}n( Z
generate
M (with respect
to
la)
and r log T U (x)dp (x) . H(~-l~In) = ]M
This
will
formula
at o n c e
(1.2).
by a p p l y i n g Let if
The
=
almost
0.
=
+
= 1
shows
also
q
there
is w h a t
3.
CONSTRUCTION
3.1.
In this
(2.5)
for w h i c h
partition
is n o t h i n g
o-algebra
o-algebra
to p r o v e
is thus
and
hence
comes
q.
M
also coincides (mod 0) v + 0 < ~(P ) < i, T h e o r e m 1.2
and
in T h e o r e m
and as with
V(x)
1.2 = x
M.
follows
clearly
as
~2 (P+)
to f i n i s h
the
rest
OF THE
section we
from below
of the P i n s k e r
to the
M
satisfying
this
of e n t r o p y
(I-~(P+))~2'
that
and
to p r o v e
that
be w r i t t e n
~(P+)~I
a partition
2.4
also
when
can
pl (P+) This
characterization
everywhere,
~
estimation
The Pinsker
In g e n e r a l , because
the
Proposition
us r e m a r k
p(P+)
with
prove
(2.6)
we
= 0. the p r o o f
(2.5)
and
of the p a p e r
PARTITION construct
compute
the
we
(2.6)
have
only
in the
is d e v o t e d
to c o n s t r u c t
case
when
u(P +)
= i,
to.
q a measurable
entropy
in
~4.
partition More
satisfying
precisely
we w a n t
168 PROPOSITION
3.1.
Let
~
that
= i.
Then
there
~ (P+)
such
be a
$-invariant
exists
probability
a measurable
measure
partition
n
such of
N
that
(3.1.1
n s %-in; for
(3.1.2
~- almost
V(x)
(3.1.3
U %nc n
every
neighborhood (#-nx)
= V(x)
(3.1.5
for any B o r e l
subset
is m e a s u r a b l e
and
us
emphasize
continuity
does
not
We w a n t
also
that
~ being
equivalence The
~
with role
Cn(x)
c V(x)
everywhere
iff
it
and
function
Proposition respect
in this
continuous
n M n ~n
x
the
to the g l o b a l
(mod V
(C
(x) nB)
absolute
unstable
(3.1.3);
we h a v e
with
respect
to the
for
such
that
(3.1.3)
that
is h y p e r f i n i t e
(3.1.3)
= U
of
z
from
= M
section.
exists
easily
~(x)
3.1 t h a t
there
in
a
finite.
~ is q u a s i - i n v a r i a n t
(M,~;V)
property
the
that
follows
contains
M;
everywhere
out one meaning
absolutely means
relation
second
B c M
proving
any
to p o i n t
V :x ~ y
framework,
Note
( N,
generate
~ -almost
before
play
foliation
relation that
{~nn}n( z
of the m e a s u r e
foliation
said
x and
~ -almost
the p a r t i t i o n s
unstable
x
n
(3.1.4
Let
point
of
the
x, y
already global
equivalence
E V(z).
In
the m e a s u r e d
(see
[Sch]).
is a r e f o r m a t i o n
of the
first
one.
that nC
(~-nx)
=
c
%n
(x).
Therefore U
C
n
~n
implies
3.2.
We [Led]
=
clearly
set w h i c h
of
(x)
that
is in all
first 3
v(x)
prove
Mv c M M
~n
for all
%n
belongs
a general
to
lemma
n
and
that
any m e a s u r a b l e
M v.
from measure
theory
(see a l s o
3.1
).
PROPOSITION
3.2.
measure
~,
on
Let
r0 > 0
concentrated
and o_nn
~
be a f i n i t e
[0,r0],
non-ne[ative
0 < a < i.
Then
Borel
the L e b e s ~ u e
0);
169
measure of the set
La , os
{r;0 _
La
is equal to Proof.
[ v ([r-ak,r+a k]) < +~] k=0
r 0.
Let us define v([0,r0])
Na, k
=
{r;0 ~ r s r0,u([r-ak,r+ak])
> k
Let
'bad interval'
denote an interval of length
in a point of
Na, k.
bad intervals
Ci,k,1 ! i s S(k),
two bad intervals.
It is easy to see that
k
Na, k
with its centre can be covered by
so that any point meets at most
~(Ci, k) -< 2~([0,r0]) i=l
and
INa,kJ
set
W. N
2a k
S(k) <
of
}.
We have
S(k)u ([0,r0])
lemma,
2
< 2S(K)a k,
We get
where
JW[
JNa,k j, -< 4akk 2
Lebesgue
almost any point
denotes
the Lebesgue measure of the
and therefore, r
belongs
by the Borel-Cantelli
only to a finite number
and thus the series
a,k co
~ ([r-ak,r+a k]) k=0 converges.
3.3.
D
Let us recall
denote
some facts from Secs.
the closed ball
define the distance
in
=
I
+
The distance PROPOSITION
3.3.
and
C
such that
x
and radius
B(x,r) r.
Let us
if
x,y
(V(w)
for some
w 6 P+,
otherwise.
Pv(X,y)
closed subsets of
with centre
Let
pV:
P~ (w) (x,y)
pv (x,y)
M
51 and 6 I.
does not depend on the choice of the point
There exists an increasin@
P+, ~(P+\~A~) = 0
sequence
and p o s i t i v e
{Az}zt I
numbers
w.
of
r , A£, B~
170 (3.3.1)
for each
y 6 A Z,
(3.3.2)
for e a c h
x ( A£
B(y,3r£) there
c N;
exists
e (£),
such
0 < e (£) < i,
that for any y ( A£ N B(x,s (£)r) , 0 < r _< r£, the local u n s t a b l e B(x,r)
( 3 . 3 . 3)
manifold
Vloc(Y)
the m a p
y + Vloc(Y)
N B(x,r£)
for each
(3.3.5)
and
if
y ( i£ Vloc(Y) PV
radius
z (Vloc(Y), -n
pV($
(3.3.6)
for any
A£
=
contains
Secs.
61 a n d 7 I.
where
rl
to
follows
Vloc(Y)
leaf
zI
A£
z2
belong
+3
i.e.
Vloc(Y)
N B(x,r)
then
for
pv(Zl,Z2)
> 2r.
(and f r o m the proofs) of
the set
1 1 1 r = ~'3 s = 1 + 7' ~ = ~J' Y = ~
in
A£ k,r,s,e,y. 3.4.
We will now prove Proposition
choose
£
0 < r ~ r£. p(A£) U of
on M
> 0 A Z.
and
x ( AZ
This
is p o s s i b l e
and t h e n F o r any
defined
such that
x r,
3.1 in a p a r t i c u l a r ~(S(x,r))
by c h o o s i n g
> 0
first
case.
for all £
We
r,
such t h a t
in the s u p p o r t of the t r a c e of the m e a s u r e 0 < r 5 r£,
by all the sets
to
N B(x,r)
A£ N B ( x , g ( £ ) r ) ,
from results
and
,~j t-V 3
~) '
~j = ~ , £
n t 0
the two p o i n t s
If one t a k e s as
m-i £+i Aj U U k=O j=2 k , ~ , l +
and
pV(y,z);
U y ( A £ N B (x,e (£)r)
belonging
This proposition
B ( x , r Z)
-nC£
) ~ B£e
and are not in the same l o c a l y
from
of
the c l o s e d ball of c e n t r e
i__nn V(y)
r,0 < r ~ r£,
S(x,r)
some
N
topology);
then for e v e r y
-n z
y,$
is c o n t i n u o u s
into the s p a c e of s u b s e t s
(endowed w i t h the H a u s d o r f f
y
Vloc(Y)
is c o n n e c t e d ;
B(x,g (£)r£) N i£
(3.3.4)
is such that
we c o n s i d e r
the p a r t i t i o n
~r
171
Vloc(Y) for
Y
D B(x,r)
E A Z D B(x,s(£)r)
follows
clearly
that
and the set
~r n
We d e f i n e a partition
0r = n = 0 ~r
M\S(x,r).
is a m e a s u r a b l e ~r"
for some
The p a r t i t i o n
r,
From
partition
0 < r ~ rZ
n
(3.3.3)
of
it
M.
of L e m m a
that we c h o o s e
3.1 w i l l be later.
Let
us d e f i n e S
=
U nt0
r
%ns(x,r).
We n o w p r o v e p r o p e r t i e s
(3.1.1)
This property
(3.1.2)
It is c l e a r
C
(3.1.1)-(3.1.5)
is c l e a r
~(S
f r o m the d e f i n i t i o n
that for
(z) c C n V l o c ( ~ - n z )
when
z ( S
r
r
) = i.
of
a n d for some
H r.
n > 0.
c V(z) .
~r On the o t h e r ~r'
8r ~ 0,
hand,
we c l a i m t h a t there
such that
y (V(z)
pV(y,z)
exists
a function
} Sr(Z)
implies
y 6 C
'
The p r o o f of choosing
r
consists
in c o n s t r u c t i n g
such that
We d e f i n e
E(z)
(3.1.2)
=
8r
(z). ~r
8 > 0 ~ r o n l y on U A~.
such a
B
r
a n d then
almost everywhere. For
z ~ U A Z, Z
put
inf{£' ;z E A z , }
and nC£ Br(Z)
=
inf{A£ 1 n>_0 (z) ' 2Bz(z)
p (¢-nz,3B (x,r)) e
L e t us f i r s t p r o v e our claim. pV(y,z)
S 8r(Z).
C ~r(# -n y )
=
We h a v e to c h e c k
that
y
6 Vloc(Z)
z 6 U AZ,
that for any
y
i }r . ' BZ(z)
E V(z)
and
n ~ 0
C~r(~-nz)
F i r s t we k n o w by
any
Let
(z)
(3.1)
(3.3.4)
and that
as
y
(3.3.5)
6 V(z)
and
applies.
pV(y,z)
~ AE(z)
T h e r e f o r e we h a v e
for
n ~ 0 -nCz pv(~-ny,~-nz)
< Bz(z)e
(Z)Pv(Y,Z)
i < ~ P(#-nz, ~B(x,r))
(3.2)
172
and -n pv(~
We h a v e (i)
-n
by
z) pV(y,z)
z) < B£ (z)e
four c a s e s
If
(ii)
-nC~
-n y,~
y
to c o n s i d e r . ~ -n
and
(3.3.6)
and
If n e i t h e r
z
both belong
If
versa,
~-n y
~ -n z
nor
~-ny
belongs
-n
y,~ (x,r))
which would proves
we h a v e
(3.1)
belong
of
~r"
to
S(x,r)
to
S(x,r),
but not
we have
~-nz,
or v i c e
< pv(~
contradict
-n
y,{,
(3.2).
z)
Thus only
(i) and
(ii) occur,
which
the claim.
We w i l l everywhere. of
S(x,r)
we s h o u l d h a v e
-n
pv(¢
to
(3.3).
(3.1) by the d e f i n i t i o n (iii)-(iv)
(3.3)
_< r.
r, 0 < r <_ r£
now choose We w i l l
even s h o w that
r, 0 < r ~ rz 8r > 0 Let
x ( M.
[0,rz]
defined
v(A)
=
and let
p
applied
to
Let
~
~
almost
8r > 0
such that
for L e b e s g u e
almost
~
almost
every
choice
everywhere.
be the f i n i t e n o n - n e g a t i v e
measure
on
by
p({y (M;p(x,y)
6A}).
be an i n t e g e r , -Cp a = e , that
p >_ 1. IKpl
We g e t by P r o p o s i t i o n
= r,
3.2,
where -kC
K
=
{r;0 _< r < r£,
[ ~({y (M;Ip(x,y)-r I <e k=0
P As
}
preserves
the m e a s u r e
~,
P]) < + ~ } .
we have also -kC
K
=
{r;O _
P
Note
p({y 6 M ; I p ( x , ~ - k y ) - r l
<e
P]
<+co}.
'k 0 t h a t f r o m the u n i f o r m i t y
(see P r o p o s i t i o n p(z,
=[
3.3.1)
~B(X,rl))
< T
implies Ip(x,z)
- rll
< DT
there
of the R i e m a n n i a n
is a c o n s t a n t
D > 0
m e t r i c on such that
B ( x , r Z)
173
for
r]
and
Thus
T
we have
such
that
for
r
0 < T s r]
in
s r~.
K P -kC
co
~({y(
M;p(~-ky,~B(x,r))_<
e
P})
k=O
This y
< +co
D
implies
there
by the B o r e l - C a n t e l l i
exist
only
a finite
lemma
number
of
that k
for
~
almost
every
with
-kC p(~-ky,~B(x,r)
) < e
P
-
The
set T
=
{r;~(U #n(~B(x,r)))> n
is at m o s t
countable
8
almost
r
D
> 0 Fix
etc.,
~
and
for
0}
r
everywhere.
r 6
( N Kp)\T ptl except S . r
in This
and o m i t
( N K )\T we h a v e c l e a r l y ptl P c o m p l e t e s the p r o o f of (3.12).
the
subscript
(3.13) F i r s t l y it is c l e a r t h a t for all n are c o n t a i n e d in V(z). S r , ~ C~(~-nz) On the o t h e r lim n÷~ The
hand,
@V(~-ny,%-nz)
invariance
let =
of
~
y
be
in
n
V(z).
r
and
By
in
z
Br'
~r'
Yr
in
(1.1.5)
we h a v e
0.
implies
n
lim 1 [ n n i=l and hence
8(~-iz)
there
8(#-nz)
will
> pV(%
By the p r o o f
-n
of
> 0
be
y,~
~
some -n
almost
n
everywhere
as l a r g e
as we w a n t
y (V(z) y 6 Cnc
there (¢-nz) •
n (3.1.3)
is p r o v e d .
that
z) .
(3.1.2),
we have
%-ny
( C
(~-nz). n
any
such
exists
some
n
such
that
Therefore
for
174
(3.1.4). show
To
that
measure
show
any
in
that
the p a r t i t i o n s
two d i f f e r e n t
S
are
points
separated
{%~}n(Z
y
and
by some
generate
z
M,
of a set of
partition
~-nq,
we w i l l
full
n t
i.
Suppose
r
y
belongs -n
%nz
to i n f i n i t e l y
Cq(z)
and
a n d by
~ -n C
=
#ny
are
many
(z)
%-ns(x,r)
for all
infinitely
and
that
n.
often
in the
same
local
unstable
manifold
(3.3.5) -nO
pV(y,z)
~
Therefore
(3.1.5)
2Bzre
pv(y,z)
Let
B
= 0
and
be a B o r e l
they
coincide.
subset
of
M.
By
(3.3.3)
the
function
y ÷ Z y ( C ~ (y) N B)
is m e a s u r a b l e
and
finite
on the
set
S(x,r),
possibly
not
in the
complement. It f o l l o w s f n (Y)
that
the
function
(y) D B) ~ y (Cn - 1 V ~i~ i=0
=
is m e a s u r a b l e not on
clearly
and
finite
the c o m p Z e m e n t .
~y(Cq (y) N B)
=
on the
n-i U ~-iS(x,r), though i=0 Therefore fn > f n+l"
set
Moreover
possibly
lim fn (y) n-~oo
is m e a s u r a b l e
and
completes
the
proof
3.5.
proof
The
particular
if
Proposition
finite of
~
almost
Proposition
of P r o p o s i t i o n p
is e r g o d i c .
2.6 any
invariant
everywhere 3.1
on
S
r
on
3.1
is c o m p l e t e d
If
p(S r)
set
A
S
and
r
this
.
< i,
if
~(S r) = i
let us r e m a r k
is a u n i o n
of g l o b a l
and that
in by
unstable
manifolds. So if support take
the
~(A)
< i,
let us c o n s i d e r
of the r e s t r i c t i o n trace
in S e c . 3 . 4
on
starting
M\A
for
of the m e a s u r e
of the p a r t i t i o n
from a small
some ~
i to
a point (M\A)
constructed
neighborhood
of
x,
x
N A Z.
in the
in the If w e
same
it c l e a r l y
w a y as
175
satisfies
the p r o p e r t i e s
of
of p o s i t i v e
M~A,
Therefore
w e can c o n s t r u c t
(3.1.1)-(3.1.5) countable
almost
COMPUTATION
4.1.
OF
=
Let us d e f i n e
subset
a partition
the p r o o f
satisfying
As this w i l l be d o n e in a
all m e a s u r a b i l i t y of P r o p o s i t i o n
properties
are p r e s e r v e d
3.1.
[]
THE ENTROPY
We h a v e o n l y to p r o v e
H(~-I~I~)
on a n e w i n v a r i a n t
inductively
everywhere.
n u m b e r of steps,
and we thus c o m p l e t e
4.
(3.1.i)-(3.1.5)
measure.
(2.6),
i.e.
iM log T U ( x ) d ~ ( x ) .
the B o r e l m e a s u r e
(2.6)
~
on
M
by
t v(K)
=
J
~y(C M
for all B o r e l measure
on
subsets
(y)}yEM
(K)
time)
Subsets
(y) (K)
g
K
of
i n s t e a d of
unstable we h a v e
We call
M.
By
t h a t by d e f i n i t i o n
As the m e a s u r e global
of
(3.1.5)
~
foliation
that
~
on
M.
M.
d-finite
of t h e c o n d i t i o n a l
PROPOSITION
measures
(4.2)
~C
(y) (Cq(y) nK) . continuous
4.1.
continuous
derivative
The next result
For
~
almost everywhere
on
with respect
(we n o w use this a s s u m p t i o n
is a b s o l u t e l y
almost
d]JCT] (y) d~y C
(y).
with
d~/d~;
is a w e l l - k n o w n
statement.
~y
is a
In f u t u r e we w i l l use the s h o r t
is a b s o l u t e l y
the R a d o n - N i k o d y m
everywhere
g
~
JM ~C (y) (C~(y) NK)d~ (y)
=
~C
K
we h a v e
for all B o r e l form
(4.1)
S.
L e t us r e c a l l {~C
(y) n K ) d ~ ( y ) D
every
y E M
to the
for the f i r s t respect
g(r)
~ 0
measure
to ~
~. almost
theoretic
176
Proof.
Let
A
ARB gdv
=
J from
(4.1)
( M , B ( M n
sets.
As
]AnBdu,
and
IAOB gd~
be two arbitrary
(4.2)
=
one obtains
that
I(IrA BnCq(y) rg(z)d~ I (z))dp(Y) y
= (4.3)
= ]anBdp = ]a ~c~ (y)(B)d~(~)" The measure [Roh]3) .
Let
the standard d(C,D) Fix
j,
space
(M,M,u)
{Bi}j= 1 c M
is separable
be a dense
subset
as a Lebesgue in
M
space
(see
(with respect
to
metric =
u(C\D)
i ~ j < ~,
+ p(D\C)) . and apply
(4.3)
to an arbitrary
set
A c M
and
to B = B.. As A is arbitrary, (4.3) implies that there exists a 3 measurable subset Z i of M, p(zj) = 1 such that for every y ( Zj one has f g(z)dpy(Z)
=
UC
B. nC (y) 3 q This implies
(y) (Bj). q
our assertion.
We may therefore
write
for any Borel
subset
f g(z)dp BnC n (Y) almost
I
(z)
=
~C
Y
everywhere.
4.2.
everywhere
We now compute I (<~-lq lq) (y)
By
(4.4)
as
=
M (4.4)
(y)(B)
In particular
g(z)d~q(Z)
almost
of
q
=
(4.5)
1
C q (Y)
p
B
. the entropy. -log
C _lq(Y ) c C
~Cq
(y)
(y),
We denote (C~-i n
(y)).
we have
177 I(~-lql~) (y)
Using now that
=
-log [ JC
g(z)dUy(Z).
~-1 n (Y)
C _i (y) = ~-I(c
lc-1 (y) g(z)d~y(~)
J -1 ~
JrC
=
The following proposition will prove it in Sec. 4.3.
L(z)
4.2.
=
q
2.1, we have
g(z)d~y(Z) =
[ c (~y) ] -i
(%) Y
g(#
i z)
Tu(¢-iz)
d~
%(Y)
(z)
"
is the key point in our computation.
We
The function
@ (z)TU(¢-iz)
g(~ is
and Proposition
t
=
n
PROPOSITION
(~y))
-i
z)
measurable.
We have now I(¢-I~ID) (y)
t -log J n(#y ) L(z) g(z) d~ ,(y) (z). C
=
But by Proposition
4.2, the function
elements of the partition L(#y)
for
U
almost every
Consequently by i
~.
L
is constant on
Therefore on
C (#y)
~
almost all
we have
L(z) =
y.
(4.5)
g(z) (z) C (¢Y) L(z) dU~(y) n
=
1 g(z) d~¢ (y) (z) L(~y--~ JC (~y) q
=
1 L(~y~
and finally
I(¢-IDID) (y)
Now
I(¢-lql~)
quently
~ 0
=
log L(¢y)
log TU(y) + log g ( ~ .
and by Proposition
(4.6) implies
log-gO~ g
=
6 L I(M I-[).
2.5
log ju 6 LI(M,~).
(4.6)
Conse-
178
But the entropy H ( ~ - I ID)
H(}-IDI~) =
is given by
I(}-i nI~) (y) d~ (Y)
[ JM
and
(2.6) follows immediately
4.3.
from
(4.6) and from Proposition
We give now the proof of Proposition Let
Z(y)
Z(y)
=
denote the
M
(y) (C}-I
(y)) "
~C
~-i n
2.2.
4.2.
measurable
function defined by
n We shall write the family of conditional measures with respect to -i ~ in two ways. Firstly as %-lq t q, we can write for any Borel subset K of ~C ~C
(y)
(K)
(y)) C~-I~ _ (C -i (Y)) ~Cn(Y) n
=
(y) (KN
~
~-in
M
1 ~ Z(Y)Jc
g(z)d~ #-i D
(y)NK
(z). Y (4.7)
Secondly by invariance of of
~
we also have for any Borel subset
K
M ~C
(y) (K)
=
~-i n
UC
(~y) (#(K)) .
(4.8)
n
Therefore we get from
(4.7) and
(4.8) for any Borel subset
K
of
that 1 f Z(y) J (y)NK g(z)d~y(z) C~-i n
=
r g (z) d~# (z) = JC (~y)N~(K) (Y) n
r = J g ( z ) d ~ ( y ) (z) = J g(~z)TU(z)d~y(Z). ~(C-i ~ (y)NK) C _ ~ (y) NK The last equality follows from Proposition Consequently we get for 1 g(z) Z(y) for
~y LO~
=
1
-Z
2.1.
almost every
y ( M
g(#z)TU(z)
almost every =
~
z
in
C
~-i
(y).
Thus the function
M
179
is
~-i
measurable
proof of T h e o r e m
4.4.
This
achieves
the []
p o i n t out that this
4.2
(bis).
Let
properties
from P r o p o s i t i o n
measurable
function
Tu
h°~ h
n
measurable.
This systems
4.2.
last P r o p o s i t i o n
4.2 can be
as follows:
PROPOSITION
is
Proposition
1.2.
Let us finally
written
and proves
h
turns out that
be a m e a s u r a b l e Then
such that
result was p r o v e d by A. N. L i v s i c
N 3.1.
G. Sinai
in a more
with
the positive,
function
in a c o m p l e t e l y
and Ya.
it is true
the
partition
there exist_______ssa s t r i c t l y
different (see
general
way
[Liv and
setting.
for A n o s o v
[BOW]l).
It
PART IV THE E S T I M A T I O N OF ENTORPY FROM _ABOVE T H R O U G H L Y A P U N O V C H A R A C T E R I S T I C NUMBERS A. Katok and J.-M.
i.
Strelcyn
I N T R O D U C T I O N AND F O R M U L A T I O N OF THE RESULTS
i.i
In the p r e s e n t part we prove
for the broad class of smooth maps
w i t h s i n g u a l r i t i e s the i n e q u a l i t y w h i c h gives the e s t i m a t i o n from above of metric entropy of such m a p p i n g through the integral of the sum of its p o s i t i v e L y a p u n o v C h a r a c t e r i s t i c Exponents. T H E O R E M i.i.
Let
(C) and
from Sec.
(i.i)
M,N
serving the measure
and
~
i.i I.
satisfy the conditions Let
~:N + V
be a
~, s a t i s f y i n g the c o n d i t i o n s
C2
(A), (Ba),(Bb), m a p p i n g pre-
(1.3)i,
(1.4) I
and
such that f log+lld~xlld~(x) < + ~.
(1.1)
M
Then ki(x)xi(x))dz M where
-~ ~ Xl(X)
e x p o n e n t s of tively 1.2
(1.2)
< + ~,
×i (x)>0
~
<...< Xs(x) (x) a_~t x,
(see A p p e n d i x
denote the L y a p u n o v c h a r a c t e r i s t i c
{ki(X)}l~i~s(x)
2 and Sec.
their m u l t i p l i c a t i e s
respec-
2.21).
This i n e q u a l i t y was first found by G.A. M a r g u l i s around 1967-68.
He c o n s i d e r e d d i f f e o m o r p h i s m s of compact smooth m a n i f o l d s and absolutely c o n t i n u o u s proof,
invariant measures.
a l t h o u g h it is m e n t i o n e d in
M a r g u l i s never p u b l i s h e d his [Mil] and in Sec.
5 of
[Pes] 2.
In
1978 D. Ruelle p u b l i s h e d a proof of this inequality for a more general situation,
namely for an a r b i t r a r y Borel i n v a r i a n t measure of a
(not n e c e s s a r y invertible) self.
Our proof is an a d a p t a t i o n of Ruelle's proof to the f r a m e w o r k
of smooth maps with singularities. sence of singularities, other circumstances. case,
C1
map of a compact smooth m a n i f o l d into itBut in order to overcome the pre-
we apply a trick w h i c h can be useful also in
Let us note that unlike the compact m a n i f o l d
the finiteness of the integral in
(1.2)
is not obvious.
In A p p e n d i x i, using a v a r i a t i o n of Ruelle arguments,
we d e s c r i b e
the inequality giving the e s t i m a t i o n from above of the metric entropy
181
of skew product through its "vertical Lyapunov Characteristic Numbers". This answers a question of H. Furstenberg. The content of this part is in fact independent from the rest of the book.
We will only use the definitions from Sec. i.i I and the
Multiplicative Ergodic Theorem (see Appendix 2 and Sec. 2.2i).
Necessary
elementary facts concerning entropy can be found in [Roc].
2.
PRELIMINARIES
2.1.
The following inequality is basic for our proof.
PROPOSITION 2.1. spaces and let
Let
p ~ 2.
Ui c Xi
Let
be an open subset of
Hi: U i ÷ Xi+l, be a map of class H i o...o H 2 o HI, 1 S i S p mapping.
Let
u 6 U1
X , 1 ~ i ~ p+l i
X i, 1 ~ i ~ p.
C 2, 1 ~ i ~ p.
and bv
G0
be the Euclidean
Let us note
Let Gi =
let us denote the identity
be such a point that
Gp(U)
is well defined.
Then
IId2Gp(u) II ~ P " 0~isp-lmaxG lld2Hi+ll "'2(D-l) (Gi(u)) II1 • [max( Proof.
max HdHi+l( i(u))II, )] 0~isp-i
Let
U c ~m
and
be such mapping that at some point able in
a 6 U.
b = f(a).
differentiable
at
V c ~n
f(U) c V Let
be open subsets.
and that
g: V + ~ P
Then the mapping a
and for any
f
Let
f: U ÷ ~ n
is twice differentiable
be a mapping twice differentih = g o f: U + ~ P
is twice
Xl,X 2 E ~ m
d2h(a) (Xl,X2) = dg(b) (d2f(a) (Xl,X2)) + d2g(b) (df(a)Xl,df(a)x 2) (see [Car], formula
(7.5.1)).
Consequently one obtains the inequality Ild2h(a) II ~ Ildg(b)N'lld2f(a)II + lld2g(b)N'lldf(a)II one deduces now by induction that lld2Gp(U) II <~ r=l~ lldHD(Gp_ l(u)) II" .-. " HdHr+ I(G r(u)) II'lld2Hr(Gr_l(u)) II
•
(IldHr_l(Gr_ 2(u)) II" -.- "lldHl(u) If)2,
182
where
for
r = 1
and
as
follows
respectively
p lld2Hl(U) ll• ~ j=2 Our proposition 2.2
Let
T:
r = p
HdHj(Gj
follows
M ÷ M
Sec.
9.1 a n d
8.6 of [Roh].
of f i n i t e
followinq
l i m ~(A~) i+~
(2.2)
lim i+~
(
h~(T)
p-i Ild2H (G (u))[ I, p p-i j=l
formula
lldHj (Gj_ l(u)) H-
mapping
is an easy
preserving
consequence
i i i (Ao,Al,...,Ap(i)),
~i =
measurable
partitions
the m e a s u r e
of r e s u l t s
i = 1,2 ....
o_ff s p a c e
M
Proof.
be a se-
satisfying
the
= 0,
sup l~k~p(i)
2.2.
diam(A[))
Let
~
= 0.
be a finite
= lim H(T-k~I~ k÷~
First
partitions
=
v...v
let us r e c a l l
of
~ = T-I~
sk
measurable
partition
of s p a c e
M,
then
and
that
H(~v~)
B = ~ v...
H(T-I~I~
v...v
T-(k-l)~).
if
a
= H(~I~) T-(k-l~
T-(k-l)~)
and
B
are
+ H(~). one
finite
Applying
obtains
measurable this
equality
that
= (2.3)
= H(~
has
~.
from
Then h(T,~)
to
above
= lira h ( T , ~ i ) .
PROPOSITION M.
in the
conditions:
(2.1)
Then
Let
terms
immediately.
be a m e a s u r a b l e
Proposition
quence
the
(u))II and
now
following
2.1.
read
-i
The
PROPOSITION
we
v...v
T-k~)
On the o t h e r
hand,
s k >_ Sk+ I.
Thus
l i m su~ = l i m k+~ k~ =
lim
- H(~
v...v
it is e a s y from
(2.3)
s I +...+ k
sk
H(~vT-I~
v...v k
to see one
that
for e v e r y
has
T-k~)-H(~) =
h(T,~)
.
k >_ 1
one
183
As
s I { s k ~ Sk+ I, P r o p o s i t i o n
following
Corollary
COROLLARY
2.1.
which
For e v e r y
2.2 i m p l i e s
is at the b a s i s finite
immediately
the
of the p r o o f of T h e o r e m
measurable
partition
~
of
M
i.i. one
has h(T,~)
3.
CONSTRUCTION
3.1 in
~ H(T-I~I~)
For N
x 6 N
OF P A R T I T I O N S and
of r a d i u s
Let us n o t e h o o d of
r > 0
r
{~t}t>l
by
and c e n t e r
N(r)
B(x,r) at
= N~Ur(A),
one d e n o t e s
the c l o s e d b a l l
x.
where
Ur(A)
is the o p e n r - n e i g h b o r -
exists
a finite measurable
A.
PROPOSITION partition
3.1. Pr =
F o r any
r > 0
there
(C0'Cl'''''Cp(r))
of the s p a c e
M
such that
A c C O c U2r(A )
and for all
i,
C i n Ur(A)
(3.1.1)
1 ~ i ~ p(r),
: @,
(3.1.2)
d i a m C. ~ 2r
(3.1.3)
1
Ci
Proof.
contains
As
an
N(r)
open
ball
in
N
of
r
radius
~.
(3.1.4)
is c o m p a c t , one can find its f i n i t e c o v e r i n g by r ~, c e n t e r e d at p o i n t s of N(r). Let r = B ( Z ~ ( r ) , ~) be a c o v e r i n g of N(r) of
c l o s e d b a l l s of r a d i u s r B 1 = Bl(Zl,~) , .... BZ(r) minimal
cardinality
One can a s s u m e BI,...,B s
t h a t the b a l l s
are p a i r w i s e
s B~j n (k=l @ Bk)
~ 0.
a m o n g the b a l l s Now,
by such c l o s e d
disjoint
Let us d e n o t e
{Bi} s <.< +l_l_Z(r)
let us d e f i n e
ql C1 = B1 n u Blk. k=l
Let
balls. are o r d e r e d
and t h a t
by
J2'
j, s+l ~ j ~ Z(r)
Bjl,...,Bjq j
which
the d i s j o i n t
in such a w a y t h a t
for any
intersect
sets
2 ~ J2 ~ s
all the b a l l s
the b a l l
CI' .... Cq,
B.. ]
as f o l l o w s
be the s m a l l e s t
number
such
184
qj that J3'
C1 n B]2. = @.
C2 =
(Bj2 N (k=l U2 B j 2 k ) ) \ C I .
Now,
let
J2 +I <- J3 <- s
We define q(r),
be the s m a l l e s t m e m b e r such that (CIUC2) N Bj3 = @. qj (Bj n ( u 3 B k ) ) \ ( C 1 U C2 ) etc. A f t e r some steD 3 k=l J3
C3 =
1 < q(r)
any
ball
set
q(r) j=IU C.3
by
We d e f i n e
<- s, this p r o c e d u r e
{Bi}l
t. B~, .... Bj 3
intersects
can not be continued.
f r o m our c o v e r i n g some
is e i t h e r
of the set
all the ball a m o n g
At this
contained
{Cj}I< j<_q(r) "
the balls
Let us d e n o t e
{ B i } s + l < i < i(r)
C., 1 <_ j S q(r). ] N o w the d e f i n i t i o n of the sets C 1 ..... Cq(r)
step
in the
which
the set
tI
goes
as follows:
t.
C1 = C1
N k=l U B1k
Finally
we d e f i n e
It is e v i d e n t (3.1.1)-(3.1.3)
Cj = 4 9 N ( U3 B k )\Cj_ 1 k=l
and
q(r) C o = M\ U k=l that
for
2 < j <_ q(r).
C k-
for the p a r t i t i o n
Pr =
(Co,...,Cq(r))
conditions
are satisfied.
Let us remark
that
[]
in general
the e l e m e n t s
of the p a r t i t i o n
Pr
are not c o n n e c t e d . For follows:
t >- i, the p a r t i t i o n ~t = {Ao'AI .... 'Ap(t) } is d e f i n e d as P _ r ~t 2t w h e r e R comes f r o m c o n d i t i o n (Ba) of Sec. i.i I.
It is c l e a r
that the s e q u e n c e
the a s s u m p t i o n s Since not e x c e e d for e v e r y
of P r o p o s i t i o n
the set m
N(r)
is a c o m p a c t
4.1
F r o m n o w on
l.ii,
satisfies
all
subset
(C) of Sec.
of
l.ii).
N
its c a p a c i t y This
implies
does
that
~ > 0 - 0.
(3.1)
THE GOOD AND BAD E L E M E N T S
constants
{~t}t>l
2.2.
(cf. c o n d i t i o n
lim log p(t) t÷~ tB
4.
of p a r t i t i o n s
n
and
C, g, c3, a
conditions
Let us d e f i n e
(i.i) I
t and and
OF P A R T I T I O N S always d
denote
{~t}t~l the p o s i t i v e
come
from c o n d i t i o n
(1.4) I
respectively.
integers.
(Ba)
The
from Sec.
185
~(n)
1
=
,
2 (nd2+g)
(4.!)
and
bad
2 } <- t~(n ] .
Htn = {x ( M;
0(x,A)
For a fixed
n, let us d e f i n e
(noted
B(n,t))
elements
G(n,t)
= {S ( ~t;
B(n,t)
{S ( ~t;
the
good (nQ£ed
of p a r t i t i o n
for e v e r y
~t
G(n,t))
and the
as follows:
k, 0 ~ k ~ n, ~k(S)
{ H~},
(4.2)
Now,
the s t r a t e g y
stead of e s t i m a t i n g estimate
there
directly
=
+
2.1 a p p l i e d
[
~(A) [- [
A(G(n,t) using
the = o u n t e r p a r t
Secs.
5 and
(3.1)
and the fact
~(
6).
this
The
a
comes
i.i is as follows,
In ~
#n)
that
lo,g P(9-nBflA) ~ (A) ] "
Main
it into two parts~
be e s t i m a t e d
Lemma
in our
sum w i l l be e s t i m a t e d
for any
fixed
in a fine manner,
framework
very
crudely
(se e using
n,
1
~
from condition k,
we d i v i d e
sum will
of R u e l l e ' s
(4~3)
(1.1)
I" 0 <_ k ~ n, let us d e f i n e
Indeed,
for any
B0(n,t)
= {S ( {t; S c Ht}
and for
c H~}]
h (T) = lim h(T,~t) ~ we v}il!
U(%-nBnA) ]/(A)
sum,
first
The s e c o n d
U A) ~ 2a(n+l) A(B(n,t)
where
to
#k(s)
B({ t
to e v a l u a t e
~ . AEB (n,t)
of T h e o r e m
the e n t r o p y
AE~ t In o r d e r
k, 0 ~ k ~ ~,
of the p r o o f
(cf. C o r o l l a r y
H(~-htl~t~
exists
i <_ k <- n
Bk(n,t)
= {S ~ ~t;
~i(S)
n ~ Ht
for
0 <_ i <_ k-1
and
~k(s), c Ht} ~
n
Clearly BV t h e
B(n,t) definition
=
O Bk(n,t) k=O of
and
Bk(n,t)
n Bl(n,t)
{Bk(n,t) ]0
that
=
for
k ~ ].
B k ( n , t ! c ~-k(H n~t)
186
and thus
as
P(Bk(n,t))
U
is an
<_ z(Ht).
p(B(n,t))
invariant
measure
one o b t a i n s
that
Thus
n [
=
~
U(Bk(n,t))
<_ (n + l ) p ( H t) =
(n + I ) u ( U
k=0 N o w the c o n d i t i o n 4.2
The
2
(A)) .
t~r~ ]
(i.i) I i m m e d i a t e l y
implies
f i r s t s t e p to the e s t i m a t i o n
(4.3).
of the
is the p r o o f A(G(n,t)
of the f o l l o w i n g LEMMA
4.1.
for e v e r y k,
Lemma.
For e v e r y t ~ t(n),
n ~ 1 every
there exists S (G(n,t),
a number
every
such t h a t
t(n)
z (#k(s)
and e v e r y
0 <_ k < n, one has
(4.1.1)k
%k(s)
c {x ( M ;
~(x,A)
1 e(n) },
>_ t
(4.1.2) k
d i a m %k(s)
(4.1.3)k
UI(¢k(s))
<_ (2cd) k
c B(z, (k+2) (2dc3)k
c B(z, (2dc3)k
where,
recall,
(4.1.4)k
1 t I- (n+k) e (n) d'
R(z,N)
i ) c t I- (n+k) ~ ( n ) d
1 tl_(n+k)e(n)d2- ) c B ( z , R ( z , N ) ) ,
is d e f i n e d
(B(z,(2dc3)k
by
(3.12)
and
I
1 tl_(n+k)~(n)d2 ) c
c B(#(z) , (2dc3)k+l
Proof
For a fixed
induction.
Indeed,
(4.1.i)k-(4.1.4) k are true for L e t us fix (4.1.1) 0 diam
n ~ i, we w i l l we will prove are true
t ~tk+l(n),
for
now prove
(4.1.i)n-(4.1.4) n
(4.1.i)0-(4.1.4) 0 t > tk(n),
then
and t h a t
by if
(4.1.1)k+l-(4.1.4)k+ 1
0 ~ k S n.
S (G(n,t).
F r o m the d e f i n i t i o n
(S) < ~. 1
1 t I- (n+k+l) e (n) d 2") "
Let us fix
w
of the p a r t i t i o n 6 S\H t .
Thus
~t
for e v e r y
it f o l l o w s z ( S
that
and e v e r y
187
t ~ 1
one has
p(z,A)
because
~ p(w,A)
0 < a(n)
< I.
(4.1.2) 0 f o l l o w s
Let
is proved.
f r o m the fact that for e v e r y
z 6 S.
t ~ 1
and e v e r y
To finish, for
As
y ~ UI(S)
it r e m a i n s
t ~ t~(n)
t > 1
as
i.e.
and as
< 1
UI(S ) c B(z,
(cf.
for
t
then
for
one has
2 tl_n~(n)d).
t l(n) o
such
(3.12) I)
= min(p(z,A),R(z,V))
S E G(n,t),
na(n)d
o n l y to see that t h e r e e x i s t s
one has
2 < R(z,N) tl_ne(n)d 2 -
Indeed,
1 (S) % ~
diam
2 < 2 <_ ~ tl_n~(n)d;
p(y,z)
that
(4.1.1) 0
c 3 { i.
(4.1.3) 0 every
Thus
i > 1 t ta(n)
< _1 < 1 (S) _ ttl-na(n)d
diam
because
2 ta(n)
- p ( z , w 0)
= m i n ( C R , C [ p ( z , A ) ] g)
2
big enough
< CR
and
tl_ne(n)d22
C - tg~(n)
<
tl_n~(n)d2-
(4.1.4) 0 since
Let
diam
~ C[p(z,A)]g
z 6 S.
1 (S) ~ ~
Let us fix and
ne(n)d
1 B(z,
(4.1.3)0,
< 1
s u c h that
(w) ~ H nt.
mhen
we h a v e
2 n)d2 ) c B(w,
tl-n~(
From
w 6 S
B(w,
) tl-n~(n)d 2 "
2 tl_ne(n)d2 ) c B(w,R(w,N)) .
Moreover
(4.1)
and
(4.1.i) 0 i m p l y t h a t B(W, tl-n~
2 (n)d2)
c {X 6 M;
p(z,A)
- 7 >
1
}
"
(4.4)
188
Let
2
y E B(w,
d 2)
be an arbitrary point.
Let us denote by
tl-n~(n) F
the shortest
~(r)
geodesic
2
~
where
joininq ~(F)
w
and
denote
y; F c B(w,RN(w,N))
the length of
F.
and
Now, by Mean
tl-n~(n)d 2' Value Theorem, t ~ t0(n)
using condition
(1.4) I and
~ t~(n)
p(¢(w),¢(y))
<- 9,(1")
sup Z6F
11
Itd#zl I <_
2 c32dte (n) d tl_n~(n)d2 " < 2dc 3
i.e.
(4.4), one obtains that for
t I- (n+l) ~ (n) d
(4.1.4) 0 is true. Let us suppose that
(4.1.i)k-(4.1.4) k is true for
t >_ tk(n),
0 < k <_ n-l. (4.1.2)k+ 1 and
Let us fix
w 6 ~k(s)
such that
(4.1.2) k one easily sees that for
~k(s)
~(w)
~ H nt-
From
(4.1.i) k
t ~ t~+l(n)
1 c B(w, 2 (2 d c 3 ) k tl_(n+k)e(n)d)
c {x ~ M; p(x,A)
1(n--------~}.
> 2t ~
(4.5) Let
y 6 ~k(s)
~k(s)
be an arbitrary point.
c B(w,R(w,N)).
Let us denote by
ing w and y, F c B(w,R(w,N)). 1 Z(F) < 2 (2dc3) k t I- (n+k) ~ (n) d" Now, by Mean Value Theorem, one obtains
that for
p(¢(w),¢(y))
II
sup zEF
1 e(n)d (2dc3)k tl_(n+k)
F
(4.1.3) k it follows
the shortest
As follows
from
from condition
t >_ tk+l(n)
S £(F)
From
geodesic
that join-
(4.5),
(1.4) I and from
(4.5)
I
> tk+l(n),
lld#zlI <
" c32dt~ (n) d
(2 d c3 )k+l
1 6J tl- (n+k+l) ~ (n) ,
(4.1.2)k+ 1 thus follows. (4.1.1)k+ 1 t > t'" -
k + l
As
~(w)
(n) > tk+l " (n) -
~ H nt, then from
(4 1.2)k+l one obtains
that for
189
#k+l(s)
c {x E M;
p(x,A)
1 t l _ 2 n e (n) d } c
2 (2dc3 ) n t~(n)
c {x E M;
p(x,A)
> i } -
Indeed,
for
t
t~(n
)
•
1 te(n ) > (2dc3)n
big e n o u g h
1 tl_2n~(n) d •
Thus
(4.l.1)k+ 1 is proved. From
(4.1.3)k+ 1 z 6 ~k+l(s)
(4.1.2)k+ 1 and
it follows
the first
follows,
for
inclusion
of
inclusion
exists
(4.1.3)k+ 1 is proved. fact that
(4.1.3)k+ 1 follows
IV ~ tk+l(n)
tk+l(n)
(2dc3) k+l
of
!)
tl_(n+k+l)a(n) d + t
IV 1 >_ tk+ ,,,I, from the t ~ tk+
The third
1 tl-(n+k+l) e(n)d
such that
<
for
The p r o o f
(4.1.4)k+ 1
We
5.1
leave
is a l o n g
The second
one
d > 1
from
(4.1).
Indeed,
t ~ tk+l(n)
one has
C < min (CR, C tg~(n-------~tge(n-----~) -<
.% m i n ( C R , C [ p ( z , A )
5.
for e v e r y
1
c B(z,(k+2)(2dc3)k+l
and thus
(4.1.4) 0 •
that
one has
Ul(¢k+l(s))
there
(4.1)
exactly
]g) = R(z,N) .
the same
line as the p r o o f
of
it to the reader,
a
THE MAIN L E M M A When
E
is a linear
space of all Let
linear
II'II d e n o t e
T ~ [ ( a m,
~m)
Let us d e f i n e mapping
in
IITAII = 1 + induced
~m. m~ k=l
by
space,
mappings
by
of
E
the s t a n d a r d T~
Let us define lIT ~ii ' w h e r e
by the n o r m
ll'II.
n o r m of
the k - t h e x t e r i o r
), w h e r e
Id
the n o r m of
lITAkN
we d e n o t e
as u s u a l
the
itself.
Euclidean
we d e n o t e
m T ^ = Id • ( • T k=l
L(E,E) into
denotes
denotes T A
~m.
For
power
of
T.
the i d e n t i t y
by
the o p e r a t o r
n o r m of
T Ak
190
We know linear
f r o m Sec.
21 t h a t
there
exists
a measurable
family
of
isometries
{TX:
(TxV,!I'II x)
For
x =
~n
E N
5.2
The
part
of the key
Sec.
2 of
following
5.1.
there
exists S
n >_ 0, let us n o t e
"Main
remark
Lemma"
used
There
exists
a number
is in our
in the
framework
Ruelle's
E G(n,t)
it w i l l
variation
and
PROPOSITION
proof
the e x a c t (see r e m a r k
counter(a)
For such
that
such
that
for e v e r y
for e v e r y
t ~ t(n)
from
n > 2
and
for
x E S
~ ~} ~ KH (d¢~)hl I. from
following
5.1.
C 1 = Cl(m,a,r)
K > 0
such
every
be c l e a r
of the
a number
~(n)
E ~t; A n Cn(s)
As
(5.1)
[RUe]l).
LEMMA
#{A
and
T -1 o d~ n o T . %n(x ) X X
x
every
(m m, li.H))x( v.
÷
the proof,
this
geometrical
any
for e v e r y
lemma
is a n o n l i n e a r
fact.
m ~ l, a ~ 0
that
(5.2)
T
and
r > 0, t h e r e
E i(~m
, ~m)
exists
one has
V o I ( U a [ T ( B ( O , r ) ) ]) <_ C IIIT All
where
Vol
Proof
L e t us n o t e
operato~ axis over
5.3
denotes
the
It is w e l l
of the e l l i p s o i d AkL k !IT I >- ~ s. i=l 1
T(B(O,r)) for
V o I (_U a [ _ T(B(O,r))
]) <-
<_ C l ( m , r , a ) I 1
m ~ k=l
We p a s s
following (3.13)) I
now
+
for
all
known
m ~ i=l
k E i=l
w
on
are
~m.
eigenvalues
that equal
1 < k ~ m.
the
of the
lengths
to
linear
of p r i n c i p a l
r s l , . . . , r s m.
More-
Consequently
(rs. + a) 1 s i] <_ Cl(m,r,a)IITA}}.
to the p r o o f
notations
measure
s I > • ..~_ s m
by
(T'T)1/2•
Lebesgue
E S
of L e m m a and
5.1.
n ~ 0
Let
us
(do not
introduce confuse
the
with
191
kw
-i o ~ o exp k exp~ k+l (w)
T~ k+l (w)
~nw = ~'(n-l)w o...o
The m a p p i n g s hoods
to
d2%~w(y)
y.
As
When
~m
sides Of
Now,
for
~(#k(w)), of
is w e l l
t ~ t(n),
Let us d e n o t e
we w i l l
~k
for all estimate
~nw(0)
f r o m Sec.
derivative
of
~nw
l.ii,
from
#
plays
q
Formula
t h a t for
of
4.1,
x, w
( S
the
1 tl_n~(n)d
•
the q u a n t i t y
one o b t a i n s
2.1,
x, w
( S, t h e n f r o m c o n d i -
(4.1)
it f o l l o w s
that
<
_
t 5/4
the s e c o n d
role. 1 llzll < t l _ n ~ ( n ) d ~
z ( ~m that
Jn(W,X)
to the i n t e r v a l
conditions
(1.3) I,
then
2 2 ~ lld ~nw(Z)II'llvll joining
(1.4) I and
v
and
0.
(5.1), one
t ~ t(n)
~ Nd2~nw(Z)II'llvll 2
Jn(W,X)
• max
respect
the s u b s e t
z ( ~ m , llzH _<
in t h i s p a r t w h e n
for all
is some p o i n t b e l o n g i n g
n •
with
are g i v e n by
from Lemma
for
As
a substantial
defined
using Proposition
obtains
The s y m b o l s
we w i l l n o t e
(4.1.2) 0 and
to the o n l y p o i n t
is w e l l
u s i n g the T a y l o r
Now,
w
= 0.
v = ~o _ Qo = ~o.
N o w we a r r i v e
z
T -i
(1.3) I r e s p e c t i v e iv.
_
where
C 2.
then the e s t i m a t i o n s
As f o l l o w s
defined
o
in some n e i q h b o r -
the d i f f e r e n t i a t i o n
llvll ~ d i e m ( S ° ) 5 q diam(S) < t l _ n ~ ( n ) d
As
are d e f i n e d
llhll < R ( # k ( w ) , N )
by
~m.
Let us n o t e by the w a y t h a t
(Bb)
for
eXPw
o
= llCnw(X °) - d~nw(O)x°ll.
Jn(W,X)
tion
Cnw
assume
(1.4) I and
(T k ( w ) o exp -ik(w) ) (Z) ~nw(Z)
~ w
and are all of c l a s s
always
IId2~' (h) ll kw
Z c Q(~k(w),
mapping
and
are the i s o m e t r i e s ,
and of
the r i g h t h a n d
in etc.,
{T w}
lld¢~w(h) ll
(w)
~' = T o exp-~ o ~n ow #n(w ) # (w)
{ kw}0~ksn_l
of the o r i g i n
d~w(y),
-i o T k , 0 < k < n-l,
max OSiSn-i (
max O~iSn-i
lld2¢]w(~iw(z))ll.
IId~]w(~iw(Z)ll,l)
2(n-l)
llvll2
_<
(5.3)
192
_< n ~ c ~ ( n _ l )
. 2b+2d(n_l)
. t[b+2d(n_l)]~(n)
d(n) h[b+2d(n-1) ] ~ ( n ) . t l ~
where
~
Thus O~e obtains that for all
C(n)
t ~ ~(n) d~f max(t(n), [C(n)]4)
and
O~e has
!i~nW(~ °) - d~nw(0)x°ll
diam
=
2 2 (n-l) 2b+2d (n-i) = q nc2c 3
C(n)
W,M £ S
. ~q2
1 <_ ~ •
NOw We can finish the proof. As A ( ~t (A) .~ [, 1 then from (4.1.3) n one obtains
R(%n(w),N)). Consequently
Thus for such
#{A ~ ~t; A n #n(s) #
A, ~n
a~d A n cn(s) ~ ~, as that A c Q(#n(w),
-1 (A) = eXP~n(w)
is well defined.
@}
#{A £ ~t; ~n N Ul[d~nw(0) (sn) ] ~ ~} d~f inW(S) " ^
From the condition Moreover
(Bb) from Sec. 1.11 it follows
that
diam(A n) ~ ~.
from the Proposition 3.1.3 it follows that A n contains an R ~ ~n a 12qt" As diam(S n) ~ , then c B(0,~) and
open ball of radius thus /nw(S)
<_ #{B ~ ~t; An c U l+q[d~nw(0 ) (~n)] @ @} < t
Vol(UI+ q[d~n w(0) (B(0,q)_) ]) t = Vol (B ( 0 ,±~HL ~ )R) Vol (Ul+q[d~nw(0) (B (0,q)) ] R
Vol (B (0,iy4)) ~- C211 (d~nw(0)) All <- KII (d¢~)
All,
where the constants C 2 and K do not depend on t, but only on q and R. Indeed, the second inequality follows form Proposition 5.1 and the last one from condition (Bb) of Sec. i.i I. D
193
6.
THE ESTIMATION
6.1
OF ENTROPY
We can now finish the proof of Theorem
From Propositon > 0
2.2 and Corollary
there exists
nh
(~) = h (¢ n)
element
of
~t
z(x)
= -x log x
z(x)
>_ 0 q
for
a t(n)
H(#n,~t ) + e
.~
containing for
0 <_ x <- i.
If
n a i.
t ~ t(n,e)
H(%-n~tl~t ) + e.
.~
the set
0 < x < 1
Let us fix
that for every
such that for all
~t = {A0'AI'''" 'Ap(t) }
Let us denote
then
t(n,e)
1.1.
2.1 it follows
A.
and
where
A0
Let us denote z(0)
(6.1) is the also
= z(1) = 0.
a i > 0, 1 <_ i <_ q
"singular"
and
q Clearly [ ai = 1 i=l
[ z(a i) <_ log q. i=l Let us proceed
consider
H(~-n~tI~t)
=
now to the estimation
of
H(¢-n~tI~t).
p(t) [ i=0
=
~(Ai ) [_
p(t)
p(t) [ j=l
~(~-nA. N A.) ~ z ~(Ai)
~(~-nA log
N A i) J ] = ~(Ai)
~t)
~(A i) [P z(#-nA N Ai) ] = i=0 j=0 J
=
Let us
t >_ t(n,e)
(A0) [ j=0 p(t)
z(~-nA
J
(6.2)
N A 0) ] +
p(t)
[ ~(Ai) [j=0
z(~-nAj
N Ai)]
= HB(n,t)
+ HG(n,t).
i=l
We will estimate HB(n,t)
separately
<_ log p(t) A.6B(n,t) 1
Thus,
from
HB(n,t)
HB(n,t)
and
HG(n,t).
~(Ai).
(4.3) one obtains <_ 2a(n+l)
Let us estimate
log p(t) ta~(n)
HG(n,t).
(6.3)
194
HG(n,t ) <_
Now
[
~(Ai).log#{ j ; %-nA
A . -~,~,~Cn,*~ 1
3
N A. ~ ~}.
#[j;~-nAj N A i ~ ~} ~ #{j;Aj n ~nA i ~ @}. HG(n,t) S
1
Hence one obtains that
~ ~(Ai)log #{j;¢-nA. N A. ~ @} AEG(n,t) ] i .|
log #{j;Aj N #nA(x) # @}d~(x),
G(n,t) where by A(x) we denote the element of partition x. Thus using the Main Lemma 5.1 one obtains: HG(n,t)
log K +
~t
which contains
log ll(d~)Alldp(x) M\A 0
log
÷ I log 11
(6.4)
1
M
Now, (6.1)-(6.4), t ÷ ~, one has hu(# ) ~ lOgn K + !n
(4.3) and (3.1) imple that for any
f
n ~ i, when
log If(de )AIId~(x).
M
To finish the proof it is sufficient to prove that lim nl I l°gll(d~x)Alld~(x) = f ( ~ 0ki(x)xi(x))d~(x) n+~ M M Xi(X)>
< + ~
Now, as noted by Ruelle in [Rue] 1 (see also Appendix 2) everywhere on M, one has the following equality:
(6.5)
almost
lim ~i loq!i(d¢~)Aii = [ ki(x) xi(x) n÷~ Xi(X)>0 Indeed, this follows easily from Oseledec Multiplicative Ergodic Theorem in the form given in [Rue] 1 (see also [Rue]2). Thus the equality from (6.5) is equivalen to 1 logll(d~n) lim 1 / logil(d~x) n Alid~(x) = f lim [~ ~ - x Aii]du (x) n+~ n M M n÷~
(6.6)
195
But
(6.6)
is a direct consequence
of Kingman's
Subadditive
Ergodic
T h e o r e m ( s e e T h e o r e m A . 1 . o f [Rue] 2 and [ D e r ] , s e e a l s o T h e o r e m 1.8 of [ K i n ] ) a p p l i e d t o t h e m u l t i p l i c a t i v e cocycle (x,n)
÷ (d¢~) A, n ~ 0.
Taking into account (6.5),
it suffices
(6.6),
in order to prove the finiteness
of
to prove that
f lim [~i logil(d~)Ail]d~(x)
< + ~.
(6.7)
n÷~
M
This turns out to be true because log ll(d~)All
1](d¢~)AIl- ~ m(l + IId¢~I])m
and thus
~ log m + m log(l + lld~!])
< log m + m log(l + l]d~x11.11d¢~(x) l]....-lld~ n 1 -
If) <-
(x)
n-i <- log m + m k=0~ log(l + Hd~ k(x ) [I).
Consequently n-i 1 logl] (de n) All <- log m + m log(l n n k=O Now,
(6.7)
applied to
¢
follows and
from
(6.8) and from Birkhoff
~(x) = log(l + IId~xlI)
6.2
let
into itself. Indeed,
¢
be a
For such
belongs
(1.1) to
~, inequality
argument B(n,t)
Theorem
that the
LI(M,~).
(1.2) remains
concerning
(4.4)
a
= @.
(see
M [RUe]l).
from the simple
d~.
In particular,
m o r p h i s m of a smooth compact manifold
true
follows
in this case the whole d e m o n s t r a t i o n
simplified because
Ergodic
implies
mpapping of smooth compact manifold
in this case the validity of
uniform continuity In fact,
C1
(6.8)
!I). (x)
~, because condition
function Now,
+ lld¢ k
is considerably
when
into itself,
¢
is a diffeo-
then for every
n > 1 1 <_ II(d~n) AllA
<- m(l + maxlld~x]l) nm m6M
Thus in this case, Dominated
Convergence
(6.6)
follows directly
Theorem.
form the Lebesgue
APPENDIX
1.
VERTICAL
LYAPUNOV
Let
ESTIMATION
(X,~)
OF E N T R O P Y
CHARACTERISTIC
be a L e b e s g u e
OF S K E W P R O D U C T
space
(cf. Sec.
(at l e a s t of c l a s s
C 3) c o m p a c t
Riemannian
probability
defined
M.
measure
w i l l be s i m p l y n o t e d by {Tm:
(TraM, If'If) ~
be a m e a s u r a b l e
on
If'If
(M a i m
sup x(X,m(M
be a B o r e l
n o r m in
T m, m
=
6 M,
let
f a m i l y of f
x
C1
differentiable
preserves
mappings
the m e a s u r e
9
and (A.I)
IIdfx(m) fl = [ < +
the s k e w p r o d u c t (A.2)
(TX,fx(m))
It is e a s y to see t h a t on
2.21
~
be a s m o o t h
isometries.
s u c h that e a c h
Let us c o n s i d e r T(x,m)
manifold,
M
M, If" If) }m6M
be a m e a s u r a b l e
of
ll.2Ii),
The R i e m a n n i a n
As in Sec.
f a m i l y of l i n e a r
Let { f x } x ( X M i n t o itself,
FROM ABOVE THROUGH
EXPONENTS
~
preserves
the p r o b a b i l i t y
measure
px~
XxM. With
defined
the s k e w p r o d u c t on
XxM
with values
X × M x Z + ~(x,m,k)
where
Z+
~
denotes
we a s s o c i a t e in
From Theorem
(A.2)
it f o l l o w s
Lyapunov
Characteristic (x,m)
UI,--.,U q • x ( X for e v e r y exists
i
fx(ml)
( Uj
and and
(A.I) and
M
Id
(A.3).
The L y a p u n o v
the v e r t i c a l
of the s k e w p r o d u c t {Xi(x,m)}
Ergodic Characteris-
~.
The e x p o n e n t s
and t h e i r
corres-
by a f i n i t e n u m b e r of local c h a r t s
it f o l l o w s
that there exists
m l , m 2 6 M, such t h a t
j, 1 ~ i, j ~ q fx(m2)
fx0 =
{ki(x,m) }.
Let us c o v e r the m a n i f o l d From
and w h e r e
(A.3) w i l l be c a l l e d
Exponents
by
(A.3)
t h a t the O s e l e d e c N o n c o m m u t a t i v e
w i l l be n o t e d by
ponding multiplicities
( GL(dim M,~),
> 1
to the c o c y l e
of the c o c y c l e
at the p o i n t
integers
cocycle
g i v e n by the f o r m u l a
k odfxk(m) OTx I) f (m) x
the n o n - n e g a t i v e
can be a p p l i e d
tic E x p o n e n t s
GL(dim M,~)
+ (Tk(x)'T
fxk = fTk_ I (x) o . " °.f T ( x. ) ° f.x ' k
the m u l t i p l i c a t i v e
( Uj.
such that
~ > 0
such that
d ( m l , m 2) ~ B, t h e r e ml,m 2 ~ U i
The use of local
and
coordinates
in
Ui
197
and
U. gives the p o s s i b i l i t y to d e f i n e the n u m b e r s (the norms) 3 IIdfx(m I) - dfx(m2) ll if o n l y d ( m l , m 2) ~ ~; we l e a v e the d e t a i l s
to
the reader. T H E O R E M A.I defined every sup x6X i~£
Let us s u p p o s e on
[0,B]
ml,m 2 ( M
that there
such that
¢(0)
exists
a non-negative
= 0, l i m %(r) r÷0
= 0
function
and t h a t
o n e has
(A.4)
lldfx(m I) - dfx(m2) lI ~ # ~ d ( m l , m 2 ) )
d ( m l , m 2) <- B. h ×~(~)
<_ h
Then
(T) +
From condition sup x6X,m6M
for
I X ×JM
(A.I)
(A.5)
( [ k i (x,m) Xi (x,m)) d~d~ Xi (x,m) >0 it e a s i l y
]Xi(x,m) I ~ [ < + ~.
follows
Thus
that
for all
i,
f r o m T h e o r e m A.I one o b t a i n s
di-
of T h e o r e m A . I
then
rectly COROLLARY h xv(T)
If the a s s u m p t i o n s
A.I.
< + ~
The proof
iff
h
(T)
of T h e o r e m A.I
p r o o f of T h e o r e m
if o n l y one u s e s
entropy
of a s k e w p r o d u c t
of c o m p l e t e n e s s e
of
M
lim n+~
exists
Hv(~n) n
almost
the e n t r o p y
def
[Abr], formula.
let us d e n o t e
(i.e.
that
if
[Adl],
(A.2).
by
H
an
formula
[Pet]).
lines
as the
as the R u e l l e ' s for the
For the sake
For e v e r y m e a s u r a b l e the p a r t i t i o n
(~) < + ~
then the l i m i t
V
h (x, a)
everywhere
of p a r t i t i o n
The e n t r o p y
(see this
the s k e w p r o d u c t
n-i -i V (fxk) ~- One p r o v e s k=0
a l o n g the same
of s i n g u l a r i t i e s
the A b r a m o v - R o h l i n - A d l e r
we r e p o r t
Let us c o n s i d e r partition
goes e x a c t l y
2.1 in a b s e n c e
proof),
are s a t i s f i e d ,
< + ~.
and B
is finite.
H e r e by
computed with respect
of the s k e w p r o d u c t
(A.2)
H
(B)
we d e n o t e
to the m e a s u r e
is g i v e n by
~.
198
h ×~(~)
: hp(T)
+ [ h({fx})du(x), M
where h({fx})
= sup{h(x,a) : ~ is a m e a s u r a b l e p a r t i t i o n of
This formulas allows to reduce h({fx})
~
I X×M
to the f o l l o w i n g inequality (i.6)
( ~ k' (x'm) x i ( x ' m ) ) d ~ d ~ Xi(x'm)>0 1
Now for the e n t r o p y of results
(A.5)
M, H (a)<~}.
from Sec.
h({fx})
2.2.
one can prove the exact c o u n t e r p a r t s
Thus if instead of
%n
one considers
Fnx the proof of T h e o r e m i.i can be r e w r i t t e n with minor m o d i f i c a t i o n s in such a way that one obtains the proof of
(A.6).
We leave the
details to the reader. Let us note that it is also p o s s i b l e to formulate a c o u n t e r p a r t of T h e o r e m A.I and of C o r o l l a r y A.I for the case when the m a p p i n g s {fx}xEX
are the mappings with s i n g u l a r i t i e s
The following easy P r o p o s i t i o n f u l f i l l m e n t of
(A.4)
(A.2.1)
In every one of the two f o l l o w i n g cases,
All m a p p i n g s
X
for the
the condi-
is fulfilled. {fx}xEX
sup Iid2%x (m)I] < + ~, where xEX,mEM (A.2.2)
gives sufficient c o n d i t i o n s
2.
(A.4).
P R O P O S I T I O N A.2. tion
in the sense of Sec.
are of class
C2
and
~ x (m) = e X P f x ( m ) O f x o e X p m-
is a compact metric space and the m a p p i n g
is continuous with respect to both v a r i a b l e s When the condition
(A.2.1)
x E X
(x,m) ÷ dfx(m) and
m E M.
is satisfied and when the mea3ure
is a b s o l u t e l y c o n t i n u o u s with respect to the Lebesgue measure on then in the e s t i m a t i o n
(A.6) and c o n s e q u e n t l y
(A.5) we have an equality, skew p r o d u c t holds.
i.e.
in the e s t i m a t i o n
the Pesin Entropy Formula for the
The proof of the fact can be o b t a i n e d along the
same lines as the proof of e s t i m a t i o n given in Part III.
M,
from below in Pesin formula
PART PLANE
BILLIARDS
AS
WITH
SMOOTH
T h e a i m of t h i s
the results billiards in t h e
which
in the
sense
all
of r e a l
part
allow
all
[ B U n ] l _ 4.
Consequently,
entropy In t h i s
theless
our
billiards,
are not
The
now
indicate
space
M
The
singular
The
conditions
should
hypotheses
quently,
and
the validity of t h e
7.2
Never-
horizon
case.
i0.
represent to w h i c h The
a very
large
the m e t h o d
limitations
developed
of this method
at present. general
Nevertheless
" R l'e m a n n l a n
there
is no d o u b t
them.
the e x a c t
(A)
A
(1.1)
places
where
the conditions
from
~
in Sec. (C)
is d e f i n e d
is d e f i n e d -
3.2.
follows
(1.4)
From
in Sec.
in Sec.
follows
its d e f i n i t i o n
the
directly. 4.4.
6.1.
from Theorem
6.3,
Theorem
respectively.
note which
the billiard same b u t
for
billiards.
for o u r b i l l i a r d s .
measure set
here
and more
here.
is d e f i n e d
invariant
not the
[Sin] 2
the validity
bounded
in Sec.
applied.
billiards
are t r u e
of c o n d i t i o n s
5.1 a n d T h e o r e m
able
in
to t h e u n b o u n d e d
exist many
to o v e r c o m e
considered
The
One
number
of c l o s e d
such billiards
the plane also
considered
be directly
results
We will
only
there
difficult
i.i I a r e v e r i f i e d
validity
to w h i c h
in p a r t i c -
of a f i n i t e
in p a r t i c u l a r
b y an e x a m p l e
The multidimensional
1.2.
singularities
contains
images
of
of p l a n e
of b i l l i a r d s
part
considered
for
I-IV,
results
the b i l l i a r d s
I-IV cannot
similar
class
proofs
class
with
is t h e u n i o n
the billiards one obtains
is e x p l a i n e d
s e e m to b e v e r y
billiards"
The
of r e a l - a n a l y t i c
part we consider
of p l a n e
in P a r t s
detailed large
formula.
Although class
I-IV.
in P a r t s
one can apply
The method
and
of t h e
smooth mappings
boundary
(i.e.
as all
proved
full
of the p r e s e n t
whose
curves
as w e l l
Pesin
of the
in P a r t s
the billiards analytic
of all r e s u l t s
is to p r o v i d e
the r e s u l t s
intervals)
Sec.
Strelcyn
the c o n s i d e r a t i o n
framework
explained
can apply
ular
that
SYSTEMS
INTRODUCTION
i.i.
one
DYNAMICAL
SINGULARITIES
J.-M.
1.
V
that we usually assure classes
they become
give
the w e a k e s t
the v a l i d i t y considered more
but
still
of our assertions. in the v a r i o u s
and more
restrictive.
reason-
Conse-
sections
are
200
2.
T E R M I N O L O G Y AND N O T A T I O N
2.1.
An interval means always a straight line interval.
ferent points
A, B ( ~ 2
one denotes by
r e s p e c t i v e l y the closed,
open,
intervals connecting
and
A
[A,B]
(A,B),
intervals,
[A,B)
h a l f - o p e n on the left and on the right B.
By d i f f e r e n t i a b i l i t y of m a p p i n g s d e f i n e d on closed closed)
For two dif(A,B] and
(resp. half
we u n d e r s t a n d the d i f f e r e n t i a b i l i t y in the i n t @ r ~ o r
of the interval and the e x i s t a n c e of the right and
(resp. or)
left
d e r i v a t i v e s at the ends of interval. An arc intervel
(resp. closed or open) means a h o m e o m o r p h i c
image of ~n
(resp. closed or open).
A closed curve means a Jordan curve, i.e, an h o m e o m o r p h i c im~g@ of a unit circle
S I.
A closed arc of class closed interval, Let
F
Ck
F
Ck
diffeomorphic
F
is a point
is not an arc c o n t a i n i n g
z
z ~ F
~. in w h o s e neighbo~-
or b e g i n n i n g at
z.
By points of n o n d i f f e r e n t i a b i l i t y of the b o u n d a r y the points in w h i c h
F
we u n d @ r s t ~ D d
and the points b e l o n g i n g to
the b o u n d a r y c o n s i d e r e d as a one d i m e n s i o n a l F
F
has no u n i q u e l y defined two sided tangent8 ~S
well as the points of ramification,
of
image of
1 s k ~ ~.
be the b o u n d a r y of an open plane region
A ~ a m i f i c a t i o n point of hood
means a
space.
~F
All other poiDts
will be called the points of d i f f e r e n t i a b i l i t y . A smooth b o u n d a r y piece
any smQoth arc
(resp. b o u n d a r y piece of class
(resp. arc of class
c o n t a i n i n g any r a m i f i c a t i o n point of
C k) F
b e l o n g i n g to
F
C k)
means
and not
except perhaps its own ends,
The smooth b o u n d a r y pieces will be also simply called b o u n d a r y pi@~@8, The b o u n d a r y wise
Ck
F
of an open plane region
~
will be called pi@~@-
if it is a union of an at most countable number of b o u n d a r y
pieces of class
C k.
We will call a smooth b o u n d a r y piece strictly c u r v i l i n e a r if ~t does not contain any interval. A b o u n d a r y piece
y
is of the first kind if for any point
any pla~e n e i g h b o r h o o d of this point contains an open subset of of
~2\D,
Otherwise
y
~ ~ ~
is of the second kind.
A s t r i c t l y c u r v i l i n e a r b o u n d a r y piece of the ffrst kind will b@
~n~
201
called strictly c o n v e x
(resp~ strictly concave) if the interval con-
necting any two points of this b o u n d a r y piece belong to ~2\~),
except its ends.
the notion of c o n v e x i t y or c o n c a v i t y are meaningless. venience,
we will call
~
(resp. to
For the b o u n d a r y pieces of the second kind Thus,
for con-
a strictly c u r v i l i n e a r b o u n d a r y piece of the
second kind s t r ~ c t l y e o n v e x if it is visible as strictly c o n v e x from some points of ~. Let
L
be a smooth o r i e n t e d arc of class
{x(~),y(¢) } a ~
b
with the o r i @ n t a t i o n of Z =
(x($),y($))
~ L
IYl
L.
The c u r v a t u r e of
L
at the point
Y"Y'(~))(~)
(2.1.)
means %he length of arc If
and let
is given by the formula
k(¢) = d~t ( x' (~)'x''($) ,
tively).
C2
be the p a r a m e t r i s a t i o n by the arc length w h i c h agrees
y
(or of closed curve
such that
r # s, then, by definition,
lrl
=
~
3.
THE PLAN~ ~ L L I A R D S .
the length
IFI
of
is equal
GENERALITIES
3.1.
C o n s i d e r ~ bounded open c o n n e c t e d subset C1
bo~Dd~ry
Denote,
F.
aS Dsual,
The region by
~
~
~ 6 ~2
rally i d e n t i ~ @ ~
with
the b i l l i a r d flow
the closure of
z ~ ~
t
in
~.
Let
v
X
be the set of
~, w h i c h is natu-
We p r o c e e d now to the d e f i n i t i o n of
Q.
If
v 6 X
and d i r e c t e d inside
iS o b t a i n e d by ~ o v i n g during time
~ x S I.
{T t}
w i t h piece-
will be called billiard.
all tangent veQtors of length one at the points of
~ime
F
Iwil.
wise
aisely,
respec-
U Yi' where all {Yi}i(i are arcs or closed curves i(I Y~ ~ Ys have at most one common point for all r, s ( I
where
~t point
y
F =
is a unit tangent v e c t o r
~, and if
t ~ 0, then
along the straight line with unit speed
with elastic c o l l i s i o n at the b o u n d a r y
when th@ vector
Ttv ~ X
v
reaches the b o u n d a r y
s, an ~ D ~ t a n t a n e o u s r e f l e c t i o n from
F
F
F.
More pre-
of the region at
takes place a c c o r d i n g
to the rule "tb@ angle of incidence is equal to the angle of reflection",
i.e. ~he c o m p o n e n t of the vector tangential
~Dd the n o r ~ !
~ o m p o n e n t changes
to
F
is p r e s e r v e d
its sign; then the motion inside
e o n t i n u e s al~n~ a straight line until a new c o l l i s i o n with the boundary F.
By d e f i ~ t i @ n ,
at the time
s
one has
TSv = v, where
v
is the
vector o b t a i n @ ~ at the moment of c o l l i s i o n by the r e f l e c t i o n d e s c r i b e d above. where
Moreov@~, w = ~W
if
z ( ~, for all
t ~ 0
one defines
Ttv = T-tv,
is the tangent vector at the same p o i n t as vector
w
202
with
opposite
direction.
T t v = T-(t+r) (Try), While of
v
trying
to d e f i n e
for a n y t i m e
A.
For
some
where
When
the Ttv
for all
t <
is s u f f i c i e n t l y
correctly
t ( ~
t ~ ~,
z E F,
r ~ 0
the b i l l i a r d
following
reaches
three
a point
0
by d e f i n i t i o n
small. trajectory
{Ttv}
obstructions
may
occur:
of n o n d i f f e r e n t i a b i l i t y
of t h e b o u n d a r y ; B. to Fig.
r
The b i l l i a r d which
i.
This
trajectory
is a b o u n d a r y interval
piece
of
v
contains
of t h e
c a n be r e d u c e d
first
an i n t e r v a l
kind
to a p o i n t
of
belonging
as p r e s e n t e d inflection
on of
the b o r d e r .
Fig. Tsv
i. P o i n t s A, B a n d C b e l o n g to a s t r a i g h t line. For s > d (A,C), is n o t d e f i n e d in s p i t e of t h e f a c t t h a t rI is a s m o o t h curve.
203
z/
F2
FI
z3
<-iF3 / z4
Fig. 2. {z.}i~ 1 denotes v with thelboundary, lim
a sequence z.
•
half-trajectory smooth curve.
C.
Either
fact
first
by
Z
Tt
measure flow
in
zero
this
Instead X = ~
See
of t h e
fact
the
that
{Ttv}tz0 for
of
positive
[Hal]
F1
is a
or the n e g a t i v e
t ~ t(v) occur
where
> 0,
in s p i t e
a t the p o i n t
this
phenomenon
v 6 X
for w h i c h
o n e o f the
above
occurs.
for a l l
vectors
of
preserves
flow
in
Q
the
v 6 Y
that
occurs
a n d all
v the
that
measure.
the
the L e b e s q u e on
This
billiard measure
studied
6 X
see
t 6 ~ , the
Then we have
for a f i x e d
when
apply
to
T t.
Lebesque
boundary
directly
for a n y
the L e b e s q u e
at some point
studying
for
of b i l l i a r d s
to the c o n c a v e
that
or b r i e f l y
4.4.,
to n o t e
× S I, w e w i l l
Y
It is e a s y
is i n v a r i a n t
situation
of
that
is b i j e c t i v e .
for c l a s s e s
is t a n g e n t
space
2).
described
in C o r o l l a r y
be n o n d i f f e r e n t i a b l e Indeed,
of
defined.
which
It is i m p o r t a n t
v
of a t r a j e c t o r y
length
the b o u n d a r y
set of a l l v e c t o r s
× S1
the b i l l i a r d
to
with
(Fig.
It is c l e a r
: Y ÷ Y
be p r o v e d
equal
in s p i t e
f r o m the d e f i n i t i o n
X = ~
{Tt}t( ~
called
F
A - C
is w e l l
It f o l l o w s
will
of
the
Y = X\Z.
t 6 ~ , Ttv
mapping
total
is n o t d e f i n e d
collisions
possibilities
Let
the
described.
Denote three
of c o l l i s i o n s
and
half-trajectory
{T-tv}t>0_
of d i f f e r e n t i a b i l i t y was
,
is f i n i t e
the p o s i t i v e
t h a t all
z
1
{Ttv}
half-trajectory of t h e
=
flow
in of
Y
~. Z
a is
As
it
is a l w a y s
in t h i s p a p e r . t,
the m a p p i n g
at which
straight
line
Tt
may
it is c o n t i n u o u s . passing
through
piece.
the b i l l i a r d the m e t h o d
of
flow
{T t}
"section"
in its p h a s e which
is q u i t e
204
natural
for this
ergodic
properties
properties
problem.
F
section
used
the
is w i d e l y
F
could
section
what more ing t w o
3.2.
in Chap.
smooth
involved.
These
subsections
that
VI
of
by that
induced
descriptions
from the definition
of
smooth
closed
curves
of s m o o t h
closed
arcs
is n e v e r
contains
arcs.
Moreover,
Dividing
the closed
purely
of
particularities
arcs
of
rather
when
then we will
{Lj}j~ 1
class r (jUIF j)=
C k , then we will
Li
belong z ~ L.. l tangent of
and
a finite
Fj
follow-
under
arcs
number
these
curve,
or infinite
points curves
F =
then
number
of
( U L~) U ( U F ) i~l ~ j~l J eliminate
F i, w e c a n
of
F.
Further-
at all.
The
in o u r c o n s i d e r a t i o n s
given
to o n e of t h e s e
one depends
the billiards
only
with
that all curves
If the b o u n d a r y
of c l a s s
consider
of a t
a
repre-
o n the
consideration.
assume C k.
and
2).
closed
representation
plays
of a t
without
t w o of
is no u n i q u e n e s s
to a n o t h e r
we c o n s i d e r
of c l o s e d
F. a r e of c l a s s ] W e w i l l say t h a t
any
there
than
a r e of c l a s s
number
all
curves
...
...
1 and
representation
from this L
always
finite
with
the
of t h e p r o b l e m
In p a r t i c u l a r , boundary,
in the
r2,
Any
finite
role and the preference
F
of
is s o m e -
is a u n i o n
L2,
(see Figs.
the c l o s e d
curves
FI, LI,
ends.
either
1 (i~iLi) U (j~IFj)_
F =
secondary
sentations
point
F
a nondifferentiable
as a u n i o n
in the d e f i n i t i o n
representation
for t h e i r
one common
F
unique.
completely
perhaps
this curve
smooth closed
more,
except
h a v e at m o s t
we consider
in o u r
description
with
Q, t h a t
of
if
as
of
respectively.
number
that
section.
Nevertheless,
are dealt
same
the m e t h o d
transformation
number
Note
f l o w o n the
the c o r r e c t
countable
or a r c s
study of many
study of the
curve,
[Bir].
most countable of r a m i f i c a t i o n
the
to t h e
closed
complicated,
and of the corresponding
It f o l l o w s
most
induced
is a c o n v e x
be m u c h m o r e
known
be r e d u c e d
of the t r a n s f o r m a t i o n
In the c a s e w h e n
case
It is w e l l
of a f l o w c a n
only
of a r c s
~
piecewise {Fi}ia I
is a u n i o n
Ck
a n d of c l o s e d
the
representation
and curves,
curves F =
where
all
Ck
and all of a of
(i~lLi) Li
U
and
C k. F
has a fixed orientation
is fixed.
Let us
if t h e o r i e n t a t i o n
fix an o r i e n t a t i o n
of
F.
Let
of z
to some
L. or to s o m e F . Consider first the case when i 3 As L. is a n o r i e n t e d arc, o n e c a n c o n s i d e r t h e o r i e n t e d I ~z . Notice that when z is an e n d to L I at z, d e n o t e d h e r e
L i, o n e c o n s i d e r s
is o f t h e
first kind
kind.
In the
latter
to the
interval
the one-sided and
let
case we will
[0,2~).
tangent.
Let
e 6 S 1 = ~/2~
We denote
consider by
0 S 8 S ~
if
Li
if
9
(z,e)Li
L. is of the s e c o n d 1 as a n u m b e r b e l o n g i n g the unit
tangent
vector
205
at
z
which
has an oriented
angle
0
with
the o r i e n t e d
straight
line
£ . In a s i m i l a r m a n n e r , for z E F and 0 Z 0 < 2~, w e d e f i n e t h e z 3 v e c t o r (z,e)F . F r o m n o w o n w e w i l l a l w a y s s u p p o s e t h a t t h e o r i e n t a 3 t i o n of £ s a t i s f i e s the f o l l o w i n g c o n d i t i o n : the v e c t o r s (z,e)Li and
(z,@)Fj
uniquely kind
defined
defines
above
are directed
the orientation
and all closed
curves
of a l l
F
inside closed
(see Sac.
Q.
This
arcs
2).
If
L. 1 L
3
kind,
then
vector
for a l l
(z,0)Li
f r o m n o w on w e w i l l
omit
lead
to a n y a m b i g u i t y .
used
in
[Sin] 2
tangent
also
Define depending tively.
MFj
and
M =
two different the
= £j
MF~)
L. I
are metric
second
instead
and
F 3 with
angle of
the c o r r e s p o n d i n g the n o t a t i o n s ,
but
this will
oriented
the a n g l e
with
never
normal
is
an o r i e n t e d
[Bit]. x
[0,~] L• l
MFj
and
is of the can
= L1• x
MLi
first
be n a t u r a l l y
( U ML•) U (•~IMFj) ; q if ial z eiements o~-this summs
same vector
z 6 L 1 To s i m p l i f y
~.
that the
[BUn]l_ 4
in
for a l l
indices
Notice
on w h e t h e r MLi
Define
and
used
and
inside
the
first
is o f t h e 1
0 S 0 < 27
is d i r e c t e d
condition of t h e
TI~2,
then we
spaces
with
where
d(z,v)
[0,7]
or
or of the
second
considered
as
p =
(z,9)
q =
and
if
identify
them
the metric
and
p
p
and in
given
M L i = L.1 × S 1 respec-
o f T1 ~ 2
(v,~)
q
M.
kind
subset
belong
correspond All
by t h e
MLi
. to
to
(rasp.
formula
P(p,q)=
I
/[d(z,v)]2 on
Li
+ ~2
(rasp.
rectly
Fj)
and where
to t h e m e t r i c
considering
the p
p(p,q)
for a n y
component We
of M.
Notice
that
on the
the
is c o n n e c t e d
space
M
depends
representation
of an a t m o s t
countable
number
sional two
compact
(cf.
Sac.
Clearly, any
along
us n o t e
that
with
is a g l o b a l
smooth
to s o m e
boundary,
MLi
of
and
leads of
always
returns
3.3.
L e t us d e s c r i b e {Tt},
to
di-
M,
by
such a component. M
if o n e d e f i n e s
to t h e F
same connected
is c o n n e c t e d .
on t h e
region
a boundary manifolds and
finite
MFj,
number
then capacity
of
b u t a set
with i,
boundary j a i.
o f two d i m e n M
is e q u a l
for the b i l l i a r d
flow
in
in
and,
after
Q
meets
M
~,
i.e. finite
M°
now the i.e.
flow
the
v
( U~I F 4 J ). wi~h
compact
is a u n i o n
section
of the b i l l i a r d
time,
flow
M
U
z
l.ii). M
trajectory
billiard
of
(iUILi)~
This
of
iff
not only
a manifold
belonging
when
manifolds
F =
is n o t
arcs
between
component
space
belonging
M
M
together
and not
that
speaking,
Let
two p o i n t s
to the w h o l e
6 M
Generally
glued
on any connected
between
c a n be e x t e n d e d
stress
but also
path
p, q
the d i s t a n c e
~ = min(Io-~I,2z-10-yl).
defined
shortest
The metric = 1
p
denotes
transformation
induced
on
M
transformation
defined
by the
by t h e time
of
206
the
first
return
w =
(z,~)
6 M
line By
beginning zI
to
straight [z,z I] not
Let of
F.
= (Zl,@l).
same
6 M
trajectory
all
flow
denote
{Tt}.
the h a l f
to the v e c t o r
the n e a r e s t such that
contained
and
point
z ~ zI
in
S
TSw =
¢
with
with
by
z I = Zl(W)
and
that
trajectory
contains
parallel
L(z,e )
s = d ( z , z I)
L e t us d e n o t e S
and denote
of t h e
we will
~.
w =
If straight
(z,e).
of
F
and
such that
Note
that
belonging
to
the
such a point
a t all.
This means
a billiard the
line
(z,0)
Let
z
is e n t i r e l y
exist
w =
trajectories
L w = L (z,@)
we w i l l
interval may
along
at point
= Zl(Z,8)
the h a l f
zI
M
t h e n by
F
of d i f f e r e n t i a b i l i t y
We d e f i n e
~(z,~) =
to the c o l l i s i o n the
(see Fig.
the
the p o i n t s
associates
boundary P
be a p o i n t
(Zl,@l).
subsequent
(z,%)
of
collision
of
3).
s e t of e n d s of a l l
arcs
of n o n d i f f e r e n t i a b i l i t y
{ L i } i a I.
of
Clearly,
F.
Fl
Z
Fig. L1
3.
~(z,0)
and
We will
F1
belonging F=
( U L i) ial
(Zl,81).
at p o i n t s
consider
the point
=
z ~ S to U
S.
z
~(z,0)
Zz and
and zI
£z
are the 1 respectively.
not defined
or t h e p o i n t So,
the domain
( U F.). j~l 3
Generally
for a l l
z I = Zl(W) of
~
(z,@)
is e i t h e r
depends
speaking,
w =
oriented
o n the
the d o m a i n
not
tangents
E M
to
for w h i c h
defined
or
representation D~
of
¢
and
is
207
the d o m a i n liard
D~0
flow
{T t}
call
~
4).
De0
~
are not
.
Thus,
the mapping
It is e a s y
is a c o n v e x
transformation
closed
~
¢0
exactly
to see t h a t
same,
on
M
on
M
by t h e b i l -
but
it is f o r m a l l y
induced
: D ÷ M
induced the
although
a transformation
In g e n e r a l , Fig.
in
= ~0ID~ N
¢ID¢ N De0 we will
of t h e
by t h e
not correct, flow
is n o t a c o n t i n u o u s
the m a p p i n g
~
{Tt}. one
is c o n t i n u o u s
(see iff
F
curve.
L1 v
Fig. 4. The points t i n u i t y of m a p p i n g
Denote and
such
that
: M1 + M
4. 4.1.
by
M1
the
subset
e I @ 0,7.
is a
THE M A P P I N G
(z,e), (v,~) ¢ : D% + M.
THE
F r o m n o w o n we w i l l
next
two
with
the computation
subsections
and
of all
Clearly,
homeomorphic
%.
L~
(w,~)
(z,e)
M1
are points
E D~
is an o p e n
such
of d i s c o n -
that
subset
e ~ 0,z
of
M
and
imbedding.
COMPUTATION study
w
OF
d%.
the mapping
we deal with
9.
In t h i s a n d
the d i f f e r e n t i a b i l i t y
of t h e d e r i v a t i v e
d%.
This will
of
in the ~
l e a d us
and to the
208
well known
~
invariant
G.D.
Birkhoff.
Sec.
8 of
THEOREM
absolutely
[Bit] or be e a s i l y d e d u c e d
4.1.
Let
is of c l a s s
(P0,e0)
6 M1
Let
4, a < ~ < b
~i'
¢(P0,e0)
r
of c l a s s
be a p a r a m e t e r defined
Denote
by
(F(~),G(~)),
(FI(41),GI(41)). metric
(F(40),G(40)) Let
P of
(resp.
We will x-axis
an d
p0
angle to
of
pl = P1 )
(resp.
denote
by
between P1
r
T
the x - a x i s 5).
by
in Chap.
(Pi,01).
of points C k-I
VI,
p0
r
and
Ck
pl,
corresponding
of p o i n t
in a n e i g h b o r h o o d
F, G 6 ck(a,b)
P.
Let
of p o i n t
pl.
the c o r r e s p o n d i n g
in the n e i g h b o r h o o d
F 1 G 1 6 ck(c,d)
If
in some n e i g h b o r h o o d
of class
p0
and by
the c o r r e s p o n d i n g
in the n e i g h b o r h o o d
of
pl.
Let
para-
p0 =
(FI(~I),GI(41)). denote
% = e - T
(see Fig.
r
pl).
and the o r i e n t e d
to see that
of
41 6 (c,d),
representation
borhood
~ 6 (a,b),
representation
=
in a n e i g h b o r h o o d
c < ~i < d, be a s i m i l a r p a r a m e t e r
parametric
P
of
discovered
f r o m it.
c k, k ~ 2, in some n e i g h b o r h o o d s
to the o r i e n t a t i o n
measure
f ound e x p l i c i t e l y
and let
then ¢ is a local d i f f e o m o r p h i s m of (p0,~0).
Proof.
continuous
All this can be e i t h e r
a point
(resp. tangent and
f r om a s u f f i c i e n t l y
We s u p p o s e
t hese n e i g h b o r h o o d s
~i ) the o r i e n t e d to
F
in
eI = T1 - e
P
disjoint.
angle between
(resp.
where
and the t r a j e c t o r y
small n e i g h -
~
Pl ) .
the
It is easy
is an o r i e n t e d
of the b i l l i a r d
going
from
209
T1
£PI
Fig. 5. @ = ~ - T, 01 = T 1 of F at the n e i g h b o r h o o d s is of no importance for the d i f f e r e n t i a b i l i t y of F in is required.
Rotating, suppose that
if necessary, F' (~0) ~ 0
-~. On this figure the b o u n d a r y pieces of points P and P1 are convex. This proof of T h e o r e m 4.1 where only the C2 some n e i g h b o r h o o d s of points P and P]
the region
and that
g e n e r a l i t y it can be supposed that s u f f i c i e n t l y small n e i g h b o r h o o d s of It is also easy to see that G i (~i) Fi (~i) .
Then :
~
on the plane, one can always
Fi(~')
~ 0.
F' (~) ~ 0 ~0
and
G' (~) tan T = F - ~
W i t h o u t any loss of and
#i
F~(#I)
~ 0
in
respectively. and that
tan T 1 =
Thus, t a k i n g i n t o a c c o u n t t h a t
-
8 = a
T and
el
G(m)
def
-
-
T~
-
a , we
o b t a i n from ( 4 . 1 . ) :
G; =
Arctan
From ( 4 . 2 . ) equations f o r
(ml)
-F; (Q1)
and (4.3.)
$1 = $ l ( Q , 8 )
G1(Q1)
Arc t a n
F1(Q1)
-
F($)
M(m,ml)
(4.3.)
we o b t a i n t h e f o l l o w i n g i m p l i c i t e f u n c t i o n and
el
= e1(@,8):
A c c o r d i n g t o t h e I m p l i c i t F u n c t i o n Theorem, f o r t h e . e x i s t e n c e o f Ck-l = $ 1 ( $ , and e l = 81($,8) of c l a s s satisfying -
functions
t h e equations (4.4.)
-
(4.6.)
i n some n e i g h b o r h o o d o f
sufficient that:
where but
"$1 -
aA ,
Ael=- aA, etc.,
A&1 = Lelr Ael = 0 , Bbl = M+l Thus, we o b t a i n t h a t :
and
Bel = -1.
($Or8O)
it i s
211
Let us compute respect
to
¢i
L~I(¢0,80).
By differentiation
F{ (~l)
G~ ( ~ I ) )
F(~0)-FI(¢I) L¢I(~0,80
(4.8.)
det
L~I(~0,80)
{ kF(~0)-FI(~I)
This is equivalent
pl
to
¢
what has been proved
pl.
of
This
above, C k-I
arcs
{L i}
of
and because
F
of class
and all closed curves
: M1 ÷ M 4.2.
4.1.
obtains
If the boundary
is a C k-I
F
diffeomorphic
It is quite natural
obtains ¢-i
from
is also
(pl,@l).
This means
in some neighborhood •
C k, 2 ~ k S ~, then all
{rj}
are also
the following
C k.
From
is piecewise
in the neighborhoods
C k, 2 s k S ~, then
imbedding.
suppose
of points
p0
and
pl
on
length of arcs
from some fixed points
F
measured
formula
that the parameters
fixed arc length parameters Let us proceed
corollary.
then to find an explicit
From now on we will always defined
that
in some
j~l
4.1. one immediately
COROLLARY
and
one considers
that
of C k-I
be a piecewise
i~l Theorem
(p0,80),
G 0 # 0,z,
P
we assume C k-I
then one immediately
in some neighborhood
that ¢ is a local diffeomorphism of (P°,80). Let now the boundary
of class
and
one obtains
through
is so because
is a mapping
(Pl,81),
Finally,
line passing
in a neighborhood
in a neighborhood of class
This
(F{(¢I),G{(¢I))
are not parallel.
~
of
zero.
0
iff the straight at
a mapping
is never
to the fact that the vectors
F
If, instead
(4.8.)
GiIl) 1
(P0,00) 6 M I. Therefore, neighborhood of (P0,80). ¢-i
of
, G(~0)-GI(~I) /
L~I(~0,80 ) ~ 0 is not tangent
[G(#0)-GI(¢I)] 2
~ 0 iff
(F(~0)-FI(~I),G($0)-GI(¢I)) that
+
p ~ p1, the denominator
that
, G(~0)-GI(# I)
= [F(¢0)-Fl(#l)]2
Since
(4.2.) with
one easily obtains: det (
implies
of
formula
for
~
d#. and
respectively
F, i.e. the parameters
now to the Benettin
for
of
~i are
given by the F.
de(P,@)
which
the basis of all future considerations. From now on we will speak very often without
any distinction
of
is
212 Points
P
dering
d~(P,8)
and
PI'
and
Let us i n t r o d u c e k = k(4),
k I = k(4 I)
at p o i n t
z.
points
P
THEOREM 91
By
and
as b e f o r e
9~ and
91
we w i l l c o n s i d e r the f o l l o w i n g where,
i.e.
k(z)
Instead
of c o n s i -
etc.
notations:
recall,
i = Z(P,8) Pl'
respectively. d ~ ( 4 , 8),
d = sin 8, d I = sin @i'
denote
we w i l l d e n o t e
the c u r v a t u r e
the d i s t a n c e
~ = /[F(4)-F(91)] z +
of
between
[G(4)-G(41)] z
F the
where,
¢(p,o)=(Pl,el).
4.2.
Suppose
respectively
¢(90,80 ) =
F
(41,8 l) .
small n e i q h b o r h o o d
t h a t in som____~en e i q h b o r h o o d s is a
C2
curve.
Then,
for
Let
(9,8)
o_~f (40,80 )
of p o i n t s
(90,80 ) ~ M 1
belonqinq
40
and
and let
to a s u f f i c i e n t l y
one has:
~(9,e) , ~(9,e)
d~(9,@)
~e I
(4.9.)
~e 1
-~--$--(¢,e)
, -f~-(¢,e) %
kS - ed el dl ed
-
elklk~ where
Z ' el dl kl£ --k , e I dl
dl
parametrised
e I = el(90,@ 0) are the c o n s t a n t s In a p a r t i c u l a r
arc or s m o o t h l y
We p r e c e d e
4.1.
o_~f r.
(4.10.)
1
e = e(90,@ 0) = ±i and
on the o r i e n t a t i o n
LEMMA
)
parametrised
the p r o o f of T h e o r e m
Under
the c o n d i t i o n s
4.2.
case w h e n
F
c l o s e d curve,
4.2.
e = e I = i.
lemma:
one has:
L¢
=
d sT - k
(4.11.)
L¢I
=
d1 el-~-
(4.12.)
M9
=
d -s~
(4.13.)
Me i
=
d1 kI - eIT
(4.14)
where
e = ±i
and
the o r i e n t a t i o n metrised Proof.
of
E 1 = ±i F.
arc or s m o o t h l y To p r o v e
are a b s o l u t e
In p a r t i c u l a r parametrised
the f o r m u l a s
(4.11.)
constants
case when
F
c l o s e d curve, -
(4.14.)
only
i__{ss m o o t h l y then
by the f o l l o w i n g
of T h e o r e m
dependinq
dependinq
o n l y o__nn
i__sss m o o t h l y then
para-
e = e I = i.
one uses the c o n s e c u t i v e
213
differentiation
of formulas
Let us prove for example
(4.2.)
- (4.3.)
the formula
and the formula
(4.12.).
The other
(2.1.).
formulas
are
proved analogously. We will use the formula
(4.3.) which gives the value of
in the form of a fraction. to Z2.
The numerator
The denominator
is equal
of this formula
to the oriented
gram spanned by vectors (F~(¢I),G~(¢I)) and The length of the first vector is equal to 1 length parameter Thus,
on
one obtains
r.
, G' (¢i)
F(¢)-FI(¢ I)
, G(¢)-GI(¢ I)
is a constant
that
Proof of Theorem a) 0
Using
(4.15.)
depending d = c i ~.
L¢I(¢,@) 4.2.
Calculation
Let us calculate
of
-~.
I.
L¢I-~- = 1
one finally
(4.15.
ClZ sin 01 = el£d 1
only on the orientation
of •
3¢1 3e 3e ' 3¢'
Differentiating
3¢1
one obtains:
to
) =
gl = ±i
(F(¢)-FI(#I),G(¢)-GI(¢I)). because ¢ is an arc
that:
F' (¢i)
This implies
is equal
area of the parallelo-
The length of the other one is equal
det
where
L¢I(¢,0)
and by consequence
3¢1 3¢
(4.2.)
3¢1 38
361 3e "
and
with respect
to
1
- L¢I(¢,¢ I)
obtains:
3¢ 1 3e (¢,e) = £ i ~ b) respect
Calculation to
361 _ 3¢
(4.16.
¢
of
381 3¢ "
and using Lemma
1
det/ L¢I
L¢I
I L¢
Differentiating 4.1.,
(4.4.)
/ L¢I
' Me1) =
dl)
Me
=
L¢
=
glk I (ki - ed
91
c) ¢
Cl~ll det ( ~ id-~-
=
k~[ - k , - gd
, Me
)- k.
Calculation
one obtains
' kl - £i-~-
' Me1
t
(4.5.) with
one easily obtains:
d1 Z = gl~ll det
and
that
of
3¢ 1 3¢"
Differentiating
(4.3.) with respect
3¢ 1 L¢ + L¢l ~ = 0, and by consequence
3¢1 3¢
to L¢ L¢ 1
214
Thus
from
(4.11.)
and
(4.12
) one o b t a i n s :
,
d) O
~($i~)
= elk£ - Ed dl
•
Calculation
one o b t a i n s
~91 ~.
of
that
~6
Differentiating
- M$1
~O "
Thus,
(4.4.)
from
with respect
(4.14.)
and
to
(4.16.)
one
obtains: ~91 5-6 (¢1'6)
4.3.
=
i.
Let us fix o n c e and
for e x a m p l e Theorem
pieces
of
•
for all a n o r m
(a
,
b)
c
,
d
the n o r m
4.2.
COROLLARY
4.2.
Let
F.
P ( I; e < 6( (P,8),
=
we i m m e d i a t e l y
Let
o f the ends of
near
klZ di
s
I
and
P
J.
obtain
J
Let
~
following
and w h e r e
•
= ~i.
Then :
the b o u n d a r y
at
closed
Let (P,@)
boundary
~i
(P,9)
From
consequences.
C2
a n d let
be one
( M 1 where
is s u f f i c i e n t l y
Let lim IId}(¢,e) JI = + (P,e) ÷ (P,O) P6I, (P,6) (M 1
~i.
9 > 6.
through
A completely (See Figs.
--
P
--1
an_~d P
analogous
6a - 6e).
J
Fig. 6a
Idl •
matrice,
e<~
li___nep a s s i n g
J
( J"
Icl +
important
I
be a f i x e d angle.
hold_____ssif__~ft h e s t r a i g h t
vali____~di n t h e c a s e o f
Ibl +
be two d i s j o i n t
P1 = P1 (P'e)
lim PI (P '@ (p,e) ÷ (P,e) P(I, (p,8) (M I e
piece
Ia[ +
be one of the ends of
6 > ~respectively) so t h a t
the
in the space of 2 × 2
--I
Fi~. 6b
.
l_SS t a n q e n t assertion
is
t_~o
215
F
J
~E ~~-
~I
r
F
Fib. 6c
Fig. 6d
P1 P
Fig. 6e Figs. 6a - 6 e s h o w five t y p i c a l s i t u a t i o n s d e s c r i b e d in C o r o l l a r y 4.2. Figs. 6a, 6b and 6e c o r r e s p o n d to the c a s e 8 > ~ a n d Figs. 6c - 6d to the c a s e 9 < 8 respectively. On Figs. 6a a n d 6c F c o n t a i n s an i n t e r v a l b e g i n n i n g at ~I and b e l o n g i n g to the s t r a i g h t line g o i n g through P and E I. On Fig. 6d the b i l l i a r d t r a j e c t o r y is o n e - s i d e d t a n g e n t to r in E I. On Fig. 6e the i n t e r v a l g o i n g f r o m P to ~i b e l o n g s to T.
THEOREM
4.3.
Let
be one
a way
E
that
for
Let
Y
of the
be a s t r i c t l y ends
sufficiently
of
y.
small
convex Suppose e > O
C2
boundary
that one
has
Y
piece
is o r i e n t e d P1
= Pl (p'@)
of
r.
in such ~ Y.
216
Then
lim (P,8) ~ (P,0) PEy, 6>0
zero
at
To
simplify
Consider and
the
are
tangent
going
orientation
hood
of p o i n t
U
function
and
such
at
P
g
that
iff
of
y.
suppose
have
just
respect system
P =
P that
P
convex that
and
the
(x,y)
x-axis
the y - a x i s
defined
small of
0 S x S x~
The c u r v e
where
y
the
is the o r i e n t e d
is an o r i e n t e d
y is a g r a p h for
lim (P,0) + (P,0) PEy, 0>0
to the x-axis.
in a s u f f i c i e n t l y
the c u r v e
C2
P ~ P
and
divides
7
Exactly
6 y+. with
y+ N U.
fp
to
let
P
straight For an
neighbor-
strictly where
con-
x~ > 0
is of c u r v a t u r e
be d i f f e r e n t y+
construction
and
y
y+ N U
at
can
7).
P1 = P! (P'0)"
from y_,
zero
be a l s o
for
the ends and we
as that made
we with
orthogonal
is a g r a p h
C 2, d e f i n e d
(See Fig.
/
y
and
same
in the c o r r e s p o n d i n g
the c u r v e
= 0.
P
two parts, the
of c l a s s
!
= fp(0)
is t a n g e n t
to
Then,
(x,y),
function fp(0)
into
respect
P=(0,0)
£PI
of c o - o r d i n a t e s
P
and
strictly
7.
is of the c u r v a t u r e
l i m i n s t e a d of
perpendicular
= g' (0) = 0.
of c o - o r d i n a t e s
such
(0,0),
p ~ y N U,
and
Fig.
y
= 0.
described
to
P and
use
follows:
of the y - a x i s
g(0)
Point
will
as
at P
of c l a s s
g"(0)
Let n o w
y
point
we w i l l
system
defined
to
through
adequate
vex
notation
the o r t h o g o n a l
the y - a x e s
(one-sided) line
iff
P.
Proof.
x
holds
IId~(P,e) II = + ~
of a
0 S x S Xp
217
It is e a s y small
to see
x > 0
EoXTI ,
,
h(0)
and
that
0 < i _< 2.
g,, (0)
Here
=
that
fo,r all
convergence
on %he
h (i)
is u n i f o r m
denotes
the
tity
Z(P,8) sin 81
and
the c u r v a t u r e
from below
where,
Pl = P1 (p'8)"
6 - tan
~(P,8)
interval
i-th derivative
of h,~
of
y
at
i.e.
P
~S zero,
=
81
Clearly,
81
sin
81
tan
Z(P,8)
small
This
£ ( P , e ) > ! . tan tan 81 - 2 sin
implies 81
let
81
X(X+fp(X))
1
x2 xf~(x) - fp(X)
Applying
to
that
for
some
This
implies
that
i(P,8) ~
l im
fp(X)
= g"(0)
= 0,
in v i r t u e
immediately
that
if
0 < ~ < x
tan > sin
81
sin
81
< -
fp(X) x x2
Mean
xf~(x)
- fp(X)
Value
Theorem
2 x xf~(x) - fp(X)
eI 81
P
x + fp(X)
tan
has
between
that
81
the C a u c h y
one
the quan--
tan ~ - tan @ 1 + tan ~ t a ~ - 8
!
tan
8 > 0
is the d i s t a n c e
e I = tan(~-8)=
fp (x) x
fp(X)
tan
sufficiently
as we recall,
!
8 =
for
fp(X) 1
sufficiently
0.
Let us e s t i m a t e
sin
this
l i r a .(i) (x) = g(i)(x) P~P rp
de f h. L e t us s u p p o s e
tan
that
1 fp(X)"
As
Qne
obtains
2 = f!'p(~)" tan l i m sin 8+0
81 @ 1
1
and
8>0 of
formula
(4.10.)
for
de,
one
P(Y+ deduces lim
IId#(P,8)ll
Let n o w Then,
as
y
tan
0 -< lim
> £(P,8) - sin 81
y
is of c u r v a t u r e
1 l i m . . =
~p~x~
Z(F,O)
N o w we w i l l
< ~
O 1 - g" (0)
prove
that
< + ~.
at
i.e.
let
P,
then
+ ~.
be of non zero c u r v a t u r e 1 01 > - ~ ( t a n {3 - tan 8) one
sin
zero
at
P,
g" (0) ~ 0.
obtains
(4.17.)
218
lim
sin
sin 8 01(8,P)
=
tan l i m tan 8 8÷0 s-~n ~ - lim sin
As
(4.18.)
i.
81 (PI,9) 81 (p,9)
= l, to d e m o n s t r a t e
(4.18.)
it is
9>0 sufficient
lim
to p r o v e
tan
that
tan 8 81(P,8)
=
(4.19.)
i.
fp(X) But
tg 8 -
and
X
tan
fp(X) x
f~x
From
fp(X) i + f~(x)---~
this,
fp(X) =
as before,
xf~(x)
=
- fp(X)
fp(x)
lim
tan e tan 81(P,9)
quently
= 0
and
Now,
lim
-
X
-
f'(x) = lim P _ i _ _ x
(4.18.),
(4.17.),
has
tan
tan 8 81(P,8)
fp(X) f~(~) •- x ) "~ "
(l+fp(X)
are
proved.
(4.18.)
let us
and
= i.
N o w our
assertion
formula
another
4.3.
If in the a s s u m p t i o n
small
neighborhood
to
U
preserves
measurable
Proof.
The
subset
U
(4.10.)
of
~0(S)
du O =
one
But
has
(4.19.),
and c o n s e -
immediately for
follows
d~ .
of T h e o r e m
Theorem
of the p o i n t
the m e a s u r e S c U,
Then
consequence
ficiently stricted
this,
1 lim f ~
from
formulate
From
g"(O).
COROLLARY
every
one
f~(x)
!
lim
4.4.
just
fp (x)
( l + f S ( x ) ' ~ )
from
tan 8 - tan 9 = 1 + tan ~ tan 8 =
81 = tan(6-@)
4.2.
we c o n s i d e r
(P0,80), Isin
4.2.
then
81dSd%,
%
i.e.
for
f o r m u l a (4.10.) for d% immediately implies that sin 8 . ~ Now, the C o r o l l a r y 4.3. f o l l o w s f r o m
of c h a n g e
From always end of P0(M)
of v a r i a b l e
the d e f i n i t i o n ~0(M\MI)
Sec.
3.3.
< + ~ iff
COROLLARY
4.4.
in d o u b l e
of m e a s u r e
= 0, w h e r e We
recall
IFI
< + ~.
If
IFI
M1 that
< + %
~
then
the
integral. one
is the IFI
re-
= ~0(¢(S)).
Idet d%(¢,@) I = formula
a suf-
obtains
subset
denotes
of the
the p r o b a b i l i t y
immediately M
defined
length
of
measure
that at the
F.
Clearly
219
d~
-
1
Isin 8 I d e d ¢
~0(M)
S c M
subset
one
Corollaries Sec.
8 of
det
formulas
imply
(4.1.)
Proof.
invariant,
go b a c k
Birkhoff
to G.D.
proved
of the m a t r i x using
every measurable
the
Birkhoff
Corollary d~(¢,8),
(see Chap.
4.3.
without
VI,
explicit
but he computes
formulas
obtained
by d i f f e r e n t i a t i o n
4.4.
the B i r k h o f f
of
(4.3.). that
that
Corollary
m(Z)
If
= 0, w h e r e
IFI
for w h i c h
< + ~,
the c a s e s
respectively. that
For
for
= U(~(S)).
4.4.
and
~
then
Z = Z A U Z B U Z C, w h e r e
occure
i.e.
denotes
the
Ergodic
Lebesgue
Theorem
measure
× S I.
4.5.
v ( TI~
prove
-
note
easily X : ~
COROLLARY
G.D.
directly,
Finally,
~
u(S)
and
of e l e m e n t s
d¢(¢,~)
in
has
4.3.
[Bir]).
computation
is
Y(Zc)
this
~(Z)
Z A,
A,
Clearly,
B
=
ZB
and
and
~(Z A)
0.
C
ZC
are
sets
described
= ~(Z B) = 0.
of
such
in Sec.
3.1.
It r e m a i n s
only
to
= 0.
purpose
it is e n o u g h
to p r o v e
that
U ( Z c N M I) = 0, as
U (M I) = i. For
w =
of e a s e
C
~(~kv)
(z,0)
6 M
we will
it f o l l o w s < + ~.
that
note
for
v
In p a r t i c u l a r ,
~(z,8)
E M1
for
= ~(w).
one
has
By definition
v ( ZC
v E Z C N MI,
one
iff
has
k=l 1 n lim ~ ~ ~(¢kv) n÷~ k:l that
= 0.
~(ZcNMI)
to the
space
to the
%
to the
= c > 0. ZC N MI,
invariant
function
everywhere,
Clearly, Then, to t h e
/
M1,
I J
~du c > 0.
invariant
= I
Ergodie
~(~kv)
Z*du c.
set.
Suppose
Theorem
applied
% : Z c N M 1 ÷ ZC A MI,
1 ~c = c U
measure
lim n [ n÷~ k=l
ZcNM 1 on
~
transformation
gets
ZdUc
is a
from the Birkhoff
probability
i, o n e
and
ZC N M I
=
As
on
Z C N MI,
and
(v) , U c - a l m o s t
£
is s t r i c t l y
positive
ZcNM 1 This
is in c o n t r a d i c t i o n
with
the
fact
that
ZcNM 1 ~*(v)
= 0
Thus, where
on
measure
on if M ~.
Z C N M I. IF1 and
< + ~, ~
In t h e
So,
U ( Z c A M I) = 0.
the mapping
preserves future,
¢
•
is d e f i n e d
the a b s o l u t e l y
when
speaking
u-almost
continuous
about
ergodic
every-
probability properties
of
220
~, we w i l l
4.5.
consider
Let us m a k e
{F i}
and
n o w a few r e m a r k s
{Lj}
ial (4.10.) If H3,
them exclusively
with
about
, a n d of the n u m b e r s
respect
to the m e a s u r e
the o r i e n t a t i o n e
and
eI
u.
of the c u r v e s
(see
formula
jal associated F
... and
orienting
them. of m u t u a l l y {Hi ~ 22
if all c u r v e s
all
boundary
with
is the u n i o n
the c u r v e s
pieces)
one o b t a i n s
disjoined are
inside
closed
curves
the c u r v e
{H i }
(and t h e r e f o r e a l s o i~2 in the d i r e c t i o n o p p o s i t e to t h a t of
HI,
H 2,
H I, then their H1
smooth
(see Fig.
8),
e = e I = i.
H3
Fig. 8
Nevertheless,
generally
a
such
that
eI
are
(z,8)
ively,
6 M e
precisely
and such
speaking, in some
in o t h e r
neighborhoods
of o p p o s i t e
a situation).
cases
signs.
of
one z
(See Fig.
could and
always zI
9 which
find
respectshows
221
z ¸
Fig. Notice ments
also that
if one changes
of the m a t r i x
approp r i a t e l y . complete
on
elk I r
4.6.
5.1.
the m a t r i x
of
r, then
(4.9.)
change
with
their
of signs
is in
the rule of t r a n s f o r m a t i o n
the o r i e n t a t i o n
do not depend
of
r.
the o r i e n t a t i o n
Indeed, of
r
of m a t r i x
the elesigns
(4.10.)
the q u a n t i t i e s
but i n t r i n s i c l y
ek depend
only.
Finally,
defined which
the o r i e n t a t i o n
i.e.
It is easy to see that this c h a n g e
agreement
when one changes and
d¢(~,0),
9
let us note that very
in this
section
the results remain
true
large classes
do not yet e x h a u s t
described (see Fig.
in this i0).
section
of plane
the class as well
regions
of regions
for
as those of Sec.
222
-
Fig.
i0.
F o r all
k,
H
is t h e o n e p o i n t
5.
APPLICABILITY
5.1.
From
sidered
Appendix then
n o w on w e w i l l have
we w o u l d
first
to t h e m a p ¢.
C2
in Secs.
5.2.
billiard
whose
of c l a s s
Ca
boundary. - 5.4.
of
Since
Pesin
However,
not
theory
always
satisfy
an e x a m p l e C1
it,
THEOREM that
all con-
to the b i l l i a r d s ,
the O s e l e d e c
such a class
describe
rectangles.
length.
in the O s e l e d e c
is of c l a s s
does
of
almost
ERGODIC
repeating
finite
the
disjoint
U Fk k~l
without
consider
we w i l l
boundary which
assume,
to a p p l y
of
the a p p l i c a b i l i t y
f r o m n o w o n we w i l l
piecewise
are mutually
MULTIPLICATIVE
boundaries
like
to d i s c u s s
2)
Hk
compactification
OF T H E O S E L E D E C
billiards
Since have
1 ~ k < ~,
Tl
and
theorem
theorem only
one
uses
billiards
is too
large
the a s s u m p t i o n
for o n e of
d~,
with
because
of a s i m p l y
except
we
(see
connected point
also
the Oseledec
theorem.
THEOREM
5.1.
a r y of f i n i t e boundary
us c o n s i d e r
length.
is u n i f o r m l y
to t h e m a p p i n g
Proof.
Let
We are
If the a b s o l u t e bounded,
and
that
then
with
value
a piecewise
C2
of t h e c u r v a t u r e
the O s e l e d e c
theorem
boundof
its
is a p p l i c a b l e
¢.
to p r o v e
I log+]jd~(¢,0)ljjsin M
a billiard
that
01d~de
< +
(5.1.)
223
fMlOg+H[d%(¢,8)]-iNlsin
where
log+a = max
Denote
by
h(%,8)
=
h
81d%d0
(5.2.)
< +
(0, log a).
the m a p p i n g
of
M
onto
itself
given
by the formula
/
Clearly,
~(%,~-0)
for
0 ~ 8 ~
< (~,3~-8)
for
z ~ 8 S 2z
h = h -I
perty").
and
Moreover,
ho~o h = ~-i
as
dh(~,8)
=
(o)
(the so c a l l e d
0 all
(%,8)
6 M.
From
the time
lldh(~,8)II : 2, it follows (5.2.).
Therefore,
F r o m the has
formula
IId¢(%,8)I; <
i~ediately
(4.10.)
[sin I
bounded, (¢,8)
one d e d u c e s
( M1
one has
f f M l o g + H d ¢ (%,8)N]sin
log ~ l [ I s i n
property
that
one o b t a i n s
that
value
is e q u i v a l e n t
for all
Ikl(kZ
(~,8)
- sd) [ + of
C, C < + ~
isi~ 81 I •
to
(5.1.).
that
for some n u m b e r
F
6 M1
Idlk[
one
+
is u n i f o r m l y
and for all
So, we o b t a i n
that
Old~dO E ]3 [[MlOg + (Isin C 0 l]-),sin 81d¢d8 OldCd8
for
and from the e q u a l i t y
of the c u r v a t u r e
IId~(},8)ll ~
pro-
lldh(~,8)II = 2
(5.1.)
to p r o v e
{]ki - edl + 1 +
+IklIZ+Z} . As the a b s o l u t e
reversal
, -i
reversal
it is s u f f i c i e n t
then
"time
<
Isin OId*d8 +
S flog c I M
IJM
(loglsin
011) Isin @Id%dS.
ff Isin
As
@Id~d0
=
~0(M)<
+ % hence
to prove
(5.1.)it
is
M enough
to p r o v e
ffM (loglsin However,
as
that
el(~,@)I)I sin @Id~ d8
d~0 = Isin 81d~d@
> -~
is a
#
invariant
ff (l°glsin81(~'8)I)Isin81d~dO= ff (loglsin M
As
measure,
one has
81) Isin 01d~de.
M
sin @d8 > - ~ one has also
ff M
(loglsin
81) Isi n 81d0d#
> - m.
224
COROLLARY
5.1.
For
any
billiard
number
of c l o s e d
arcs
curves
of c l a s s
C 2 t the e .l.e d e c . . . . .O . s . .
If i n s t e a d 1 S k ~ ~,
of a r c s
a class
will
be c a l l e d
5.2.
In the
C~
of c l a s s
class
boundar~
a n d of theorem
C2
of b i l l i a r d s
one
as t h a t
is the u n i o n
finite
numbers
is a p p l i c a b l e
considers described
of a f i n i t e
of c l o s e d
to the m a p p i n g
arcs
of c l a s s
in C o r o l l a r y
C k,
5.1.
Pk"
following
of a s i m p l y
connected
and,
for one
except
of c l a s s
whose
three
subsections
billiard
point,
~
whose
is a l s o
I log+lld~ (~, @) IIsin 0d~d@
=
we w i l l
construct
boundary
of c l a s s
C~
r
an e x a m p l e
is of c l a s s
such
c1
that
+ ~
(5.3.)
M
In this one:
example
the a b s o l u t e
liard
all a s s u m p t i o n s
value
is not u n ~ o r m l y
To p r o v e
bounded,
the e q u a l i t y
We w i l l of p o s i t i v e
val
this
I.
If
(~,@)
of
r.
so,
(¢,e)
6 H.
all
II.
If
IIl.
~
curve subset
~
~i of
then
except
of this
bil-
suplk(z) I = + ~. x6r
will
occur
such
that
conditions
belongs
formula
k(~)
(logk(~))sined~d~
shape
bounded
of a r e c t a n g l e boundary
valid
to p r o v e
the e q u a l i t y
in o u r for
to the
(5.1.)
some
will
example. subset
interior
one
H c M
be s a t i s f i e d :
has
of
some
201 ~ - -k(#)
interfor
~ 1
= + ~
(5.5.)
H
A possible region
then
E H,
are
(5.4.)
following
by v i r t u e
(¢,e)
~ JJ
6 H,
5.1.
the b o u n d a r y
it is s u f f i c i e n t
that
a region
the
of
= + ~
equality
construct measure
i.e.
(5.3.),
M(lOg+l-~--l)sinSd*d@
It is p r e c i s e l y
of T h e o r e m
of the c u r v a t u r e
and
by some
exactly of set
of
such
a billiard
is g i v e n
by the c u r v e
r
obtained
by r e p l a c i n g
the
interval
curve
L
in the n e x t {(~,8) ; ~ L } .
as
shown
subsection.
on Fig. H
by
smoothing
AB ii.
will
for e x a m p l e
the a n g l e s
belonging We w i l l
be c h o s e n
by the
to the
define
this
as a s u i t a b l e
•
225
%
1
#
y-
L Fig.
The w h o l e
5.3.
easy
to p r o v e A.
The
construction
series
B.
The
sequence
C.
If
rn
=
f (x)
is b a s e d
on the
following
well
known
and
facts.
n=l
Consider
ii
1 ~ n ( l o q n) ~ n=l
is c o n v e r g e n t .
[
1
n(log
the
1 is d i v e r g e n t . n log n
n) z' n ~ 2, t h e n
following
function
rn log rn
f 6 C~[-I,I]
/i - X 2
for
Ixl Z 3
0
for
Ixi
a
n=2
....
(see Fig.
12).
4
= on the
[-~5' -~] 4
(respectively
[3 4. ~,~1)
increasing
(respectively
decreasing).
interval
f is s t r i c t l y
, y
-I
4 5
3 4
3 4
>
4 1 5
x
Fig. 12 Let us d e f i n e fa the
6 C~([-a,a]) same
for and
structure
0 < a < 1 in the
as the
fa (x) = a 3/2
interval
function
[a-,a] f
on the
f(~). the
Clearly,
function
interval
fa
[-i,i].
has The
226
only
difference
tion
is an arc From
is t h a t on the of an e l l i p s e
the u s u a l
h"(x) (l+[h, (x)]2)3/2, absolute less
that The
Fig.
values 1 7~"
will
interval
I =
For
interval U 13 U J3
of
that
of
the g r a p h
of
func-
t h a t of a c i r c l e . of c u r v e
for all
the g r a p h
3 3 (-~a,~a),
x E
of
y = h(x);
function
k(x)
piece
L
laying
between
be c o n s t r u c t e d
in the
[0,4d]
d =
I
where
by
can
In
and
~ n=2 Jn
be r e p r e s e n t e d
J~
13
the points
following
U I n U Jn U ....
I
the
f
is n o t
r
and where
n
!
n
-
intervals
as a u n i o n
of
as
shown
..........
B
(see
1 n ( l o g n) Z" of
intervals
on Fig.
In
!
r
and
L e t us t a k e a n
the closed
J3
!
A
manner.
length of
2r n.
12 U
13.
Jn
I
"
¢ ......
Fig. 13
Define
o n the
interval
I
the
function
g
as
follows:
n-i
~ rk-rn) Ii 3n/2frn k=l (x-4
g(x)=
where,
by d e f i n i t i o n ,
function
Fig.
2
x E In,
n { 2
for
x E Jn'
n { 2
the c u r v e
The
shape
It is e a s y not have
the this
the c u r v e
L
. . . . . . . . . . . .
I3
J2
14.
Indeed,
Define
r I = 0.
for
L
as a g r a p h
of
g.
.3~k 1
=
a
U ...
12 L
easily
of t h e c u r v a t u r e
n { 2, d e n o t e
The J2
instead
for the c u r v a t u r e
it f o l l o w s
boundary
ii)
formula
3 3 [-~a,~a],
interval
J3
of
to c h e c k
second follows
the g r a p h
that
of
immediately
in the c e n t e r s
of
: .......
In Jn
function
g E cl(I)
left derivative
:/h:
g
but
g
~ C2(I),
at the r i g h t
from
the
intervals
fact In
e n d of
that tends
as
g
does
interval
I.
the c u r v a t u r e
of
to i n f i n i t y
when
227
n
tends
to t h e
Note
5.4.
To
of o u r and
that,
infinity. since
finish
II
the c u r v e
the c o n s t r u c t i o n
rectangle,
to v e r i f y
g (CI(I)
the
size
of
of
F
L
is of f i n i t e
it r e m a i n s
smoothing
of
length.
to d e f i n e
the c o r n e r s ,
the
the
size
set
H,
that
(log k(~)
sined~d~
= + ~
(5.5.)
H
L e t us m a k e
following
class
C1
on
one can consider
L
of
the
F
which
to
see t h a t
finite
associates
length,
remark. instead
Since of a n a r c
an e q u i v a l e n t
a number
x
L
is a c u r v e length
parameter
parametrization
to t h e
point
of the {4}
of c l a s s
(x,g(x)).
C1
on
It is e a s y
fr
where
and
by
k(x)
is e q u i v a l e n t
we denote
L e t us d e n o t e
by
that
e n d of
the
val
In .
that
the
point
left
Consider ray
there
x 6 Kn,
Kn
an K
between exists
Denote
one has
A =
7 > 0 6(x)
(log k(x))
such
by
of
L
with [
3 ~r n
= +
of
15). L.
inter-
vector L
We denote
n ~ 2
K n ¢ In
the
unit
to t h e c u r v e
and vector for a l l
(x,g(x)).
such that
the c e n t e r
the u n i q u e
(see Fig.
that
0dx dO
at point
length
is t a n g e n t
(0,i)
sin
H
of
coincides
y 6 In+ 1
vector
~ JJ
interval
n
x E K n.
where
to
the c u r v a t u r e
{ (x,g(x))+tk,t~0}
(y,g(y))
the a n g l e that
(5.5.)
by
6(x)
It is e a s y and
such
at a
to see
for a l l
~ y.
A
~
(x) >
!
x
,
k,
J<
V
K %
n
J
Y
In+l
n
J
I
Let
J Fig.
15
l'l
us n o w d e f i n e
exactly
the c u r v e
F.
Consider
any
rectangle
228
with
sides
longer
than
10d w h e r e
d =
!2rn
=
[ n=2
n be a n
interval
tangle side L
and
of l e n g t h
such that
(see Fig.
(see Fig.
that
16).
Ii).
from the
4d
belonging
its r i g h t L e t us
substitute
Moreover,
interval
A
smoothed
dividing
the middle,
the
l o s s of g e n e r a l i t y
arcs
are
lower
with
s i d e of o u r
the center
interval
AB
of t h e
seen at an angle
rec-
of t h i s
by the curve
such a smoothing
the h e i g h t
AB
Let
i
of
the c o r n e r s
rectangle
smaller
than
through Y 1-00"
o n e c a n a s s u m e t h a t t h e l e n g t h of a n y 1 1 of t h e f o u r s m o o t h e d a r c s is s m a l l e r t h a n -- • From the 4 2 ( l o g 2) z" last condition we assert that this smoothing takes place outside curve L
any
the
l e t us m a k e
Without
to t h e
end coincides
1 n ( l o g nj 2
(see Fig.
16).
It is c l e a r
just o b t a i n e d ,
satisfies
which
the c u r v a t u r e .
concerns
all
that the
region
the a s s u m p t i o n s
bounded
by the curve
of T h e o r e m
5.1.
except
F one
C m
~
(Z)
Z
•
to
H
,.J A
B
Figure 16
Denote which
by
[C,D]
has not been
Define
H
as the
6 {(x,y(x))(L;x(~ is t h e
interval
any
fixed
~.
Thus,
t h i s p a r t of t h e u p p e r
smoothed subset for
defined
x ( K n,
(see Fig. of all
some after
n~2}
one obtains
that
(~,e) and
(5.6.).
the Lebesque
ff
s i d e of o u r
measure
6 F x
that
#i
Define
K
of
(log k(x))
sin
Bd0)
~ ~---- • 16
(log k ( x ) d x ,
(0,~)
such that
( [C,D] =
where
~ K • n=2 n
{e; (x,@)(H} sin
edxd~
H
•( ~0
rectangle
16).
~
Kn
Clearly, is b i g g e r
ff
for
than
log k ( x ) d x ) .
k
because
for
0 < y < ~
one
has
229
i
s i n y a ~y.
because i.e.
However,
IK log
k(x)dx
~ r log r = - ~. n=2 n n
the O s e l e d e c
theorem
l
=
n 2 n(log
ff
Therefore,
n) z ' l ° g [ n
(log k)
sin
log
o11j2
ed~d6
= + ~
= + ~,
H
is n o t a p p l i c a b l e
to t h e
just constructed
billiard. L e t us n o t e is l a r g e l y
that
in the
arbitrary,
example
except
for
described
the c u r v e
above
L
and
the b o u n d a r y for the
F
interval
[C,D].
5.5.
It w o u l d
be v e r y
h~({Tt})
of the
finite
infinite
or
interesting
billiard and
flow
to k n o w w h e t h e r
in the r e g i o n
if it is f i n i t e ,
~
the m e t r i c
described
is it s t r i c t l y
entropy is
above
positive
or is
it z e r o ? The c u r v e strictly example will
L
convex
was
chosen
boundary
just described
have a sinusoidal
in s u c h a m a n n e r
piece.
Clearly,
in s u c h a w a y curve
of
the
that
as n o t
to c o n t a i n
it is p o s s i b l e instead
"damped
a
to m o d i f y
of the c u r v e
oscillation"
type
L
the we
(see Fig.
17).
A
B
Figure 17
The
same
problem
liard.
Are
6.
SINGULAR
6.1.
THE
After
we now pass
as above
the a n s w e r s
SET.
the general
the
THE
also makes
BILLIARDS
OF C L A S S
considerations
to t h e c e n t r a l
sense
for t h i s m o d i f i e d
bil-
same?
point
of
of t h e
t h i s part.
P preceeding
two
sections
230
To a p p l y class will P.
P2 thus
For
the
define
class
An
Let of
of
be a b i l l i a r d
number
13.1.
or in some
class
prove of
the
first
results
F 2.
namely
specific of P a r t s
We w i l l
the We
a class
result
which
I-IV.
now define
a subset
ends
3.1.
in at l e a s t
straight
contains
of
The
F,
A~, 1
y
...,
and c l o s e d
is not set
...,
by
some
S
L r.
Denote tangent
curves,
as d e s c r i b e d
we a l w a y s
in some
by
8~
suppose
Li,
that
1 S i ~ p,
other
F, and
of
point
S.
which
set of all
one-sided)
or t a n g e n t
of
points
of all the
(perhaps
simultaneously
to
to
F
The c a s e s
F either
when
in p a r t i c u l a r
a
an
excluded. A~
A~5
is d e f i n e d of
M,
as
of w h i c h
{(z,9)EM:
zES}
2 A~
=
{ (z,0) ~M:
0=0,~}
3 An
=
{ (z,O)EM:
ZlES}
4 A~
=
{(z,O)EM:
L(z,0 ) is
A~
=
{(z,0)EM:
L(z,0 ) is p a r a l l e l
of Sec.
that
subset
or at the p o i n t s
=
the b e g i n n i n g
the
either
An1
stress
section
is c o n t a i n e d
points
inflection
singular
subsets,
L I,
in the p l a n e
of
interval
piece
two d i f f e r e n t
line
arcs
the p r e s e n t
let us d e n o t e
of a r c s
lines
closed
from
1 S j ~ r.
at the p o i n t s
Let us
¢, e v e n
is too wide.
of b i l l i a r d s ,
the
the
of c l a s s
smooth
boundary
Fj,
in Sec.
straight
of
Beginning
any considered
(cf.
already
5.1.
w h i c h we w i l l call the s i n g u l a r set of the b i l l i a r d ~. p r (i=iUL i) U (3UIFj)= be a r e p r e s e n t a t i o n of F as the sum
finite
are
restrictive
we can
to the m a p p i n g
in C o r o l l a r y
F =
in Sec.
As
I-IV
for the a p p l i c a t i o n
~
M
Let
of P a r t s described
a more
P
is n e c e s s a r y
6.2.
results
of b i l l i a r d s
the
the u n i o n only
(one sided)
of f o l l o w i n g
A2 n
is a l w a y s
tangent
to F at
to a s t r a i g h t
line
five
nonempty:
z I} from
0~}
3.3).
singular
set
A
depends,
just
like
the
space
M,
not o n l y on the r e g i o n ~ but a l s o on the r e p r e s e n t a t i o n p r F = (i=llJL i) U (3UIFj),= fixed o n c e and for all. The
set
~
is not of c l a s s than
A~
=
4
i
U A i=l Q C k-I
is f o r m e d if
F
by all
is p i e c e w i s e
points C k.
of
M
in w h i c h
It is m u c h
more
A~
w h i c h d e s e r v e s the n a m e of s i n g u l a r set of }. The r e a s o n s to c o n s i d e r AQ5 as a p a r t of the s i n g u l a r set of # are of p u r e l y
231
technical
nature
To a p p l y
one dimensional of c l a s s
P2
are always
and will
the
results set
of
finite
be of i n f i n i t e
P2'
as w e w i l l
~
strictly
Clearly,
of
THEOREM
M.
6.1.
singular
set
~
is a u n i o n
numbers
and of
finite
that
of c l o s e d
1 ~ k S ~, Ck
and
a n d of a f i n i t e
num-
of c l a s s
the
set
8~
is
An
is a l w a y s
a compact
eharacterises
Pk'
number
the
2 S k S ~, of a r c s
length,
finite
set
then
Ag.
the
a n d of c l o s e d
a n d of f i n i t e
num-
the
1 A~,
sets
arcs
2 A~
and closed
and curves,
5 An
are
all
unions
of c l a s s
of
C k-I
length.
us e x a m i n e
presented
set
already
finite
Pk'
points.
It is c l e a r
finite
C k-L-i- a n d of
A~
curves.
Pk'
of a f i n i t e
and
of class
is of c l a s s
intervals
concave
singular
is a b i l l i a r d
of c l a s s
of
A~
finite.
to t h e c l a s s
1 _ < j _ < r,
that the
for t h e b i l l i a r d 1 A ~2 a n d A~, A ~3
to see t h a t
is a l w a y s
belongs F3,
theorem
However,
in the b i l l i a r d s
of c l a s s
the
__If
isolated
isolated
Pk
to k n o w
Although
easy
A~
number
following
A~
all
Let
of
or s t r i c t l y
to see t h a t
length.
the case.
it is v e r y
length
9.
it is n e c e s s a r y
Nevertheless,
of a f i n i t e
The
bers
Proof.
finite
of c l a s s
convex
curves, of
the
in Sec.
I-IV
not
for t h e b i l l i a r d s
it is e a s y
subset
only
L i , 1 ~ i ~ p, a n d
if it is a u n i o n
and
length,
length.
prove,
billiard
if a n y c u r v e
b e r of
is of
it is g e n e r a l l y
can
The
A~
appear of P a r t s
points. o n Fig.
the
set
An example 18.
3 A n-
First
of a n
note
isolated
that point
3 An
can also
(z,8)
of
contain 3 is An
232
Y
L
(z,0)
Fig. 18. The b o u n d a r y p i e c e s Yl and 72 L(z,8 ) are all m u t u a l l y t a n g e n t in w~
Let us a l s o this
is so if
of w h i c h
one
one
that
has
w
T
a closed to
case.
z, w h e r e
Clearly,
S, b e c a u s e
in such
convex
straight
=
for
piece
¥
{(z,0) ;z~:T,~I~W}
of g e n e r a l i t y
that
~ine
Indeed,
boundary
A3(w,¥)
a loss a way
0(z)
A3(w,T)
the h a l f
is not c o m p a c t .
strictly
Without
is o r i e n t e d near
= w.
A~3
in g e n e r a l
belongs
in t h i s
0(z)
Zl(Z,@(z))
that
contains
end
is not c o m p a c t suppose
note F
and
we m a y
z E y
near
W
is the u n i q u e a n g l e s u c h that def (w,~) = B3(w, Y) is c o m p a c t and
U
B 3 ( w , y ) E An. Fix exists
w ~ S. at
w
of a f i n i t e sufficient
~xcept some
number
for the c a s e
"sector of
to p r o v e
of v i s i b i l
smooth
that
of
closed
if
y
to
some
Li
it is v i s i b l e
w,
then
B3(w,T)
Here
"y
interval
is v i s i b l e going
subdividing parts,
one
of m u t u a l
from
from
w
reduce
position
respectively.
of
system
of c o - o r d i n a t e s function x-axis
w
f when
arcs
and
Fj,
F.
has
no c o m m o n
T
into
to c o n s i d e r i n g
and
T
of
(x,y);
~ ¥
w = Ck
while
one
(0,0), one y
uses 7
of the
is t a n g e n t
T
B~
cases
19 a-c
following
of
the
smaller
the t h r e e
on Figs.
~
C ~=~..
F\(wUz) ~ of
is the g r a p h ends
bound=
z 6 T
with
only
the
it i s
closed
of G l a s s
number
as p r e s e n t e d
figures
curves,
points
points
a finite
aiwa~s
! ~ J ~ r, and
arc
for all
there
is the u n i o n
small
1 E i ~ p,
that
F
of c l o s e d
problem
three
As
is a c l o s e d
means
z
of c l a s s w
of
or
piece
our
In t h e s e
convex
positive
to
the b o u n d a r y can
w"
ty
3 AD,
points
is a s u f f i c i e n t l y
ary piece belonging fro m
isolated
orthogonal
of a P o ~ i g i v 8 is on the
to the x - a x i s
in
0
233
if
w
6 y,
i,@~
f' (0) = 0.
d
a)
y
b) y W W
\
w=(o,o)
> x
w=(O,O
x
f
w=(0,0)
x
Fig. 19. Th@ e ~ s e (a) c o r r e s p o n d s to w 6 y. The cases c o r r e s p o n d t@ W ~ Y. O n e d o e s n o t e x c l u d e the c a s e of interval.
We will (b) a n d
(c)
suppose
that
[0,A]
for
Let to
w.
~Qn§ider
f
som@ ~
in d e t a i l
i§ ~ i m i l a r .
is a s t r i c t l y A > 0.
~@ ~ h e arc
length
that we
length
~arameter
oriented
angSe
such that
hand
z 6 ¥
if
Dition
@(t)
then
z =
~ ~ - e(~).
= {(~,@(¢)),0~Iyl}
to p r o v e
U(0,0).
it,
To p ~ v @
20).
~.
parameter
of on
the point For any
(t,f(t)) Clearly,
= w
y
calculated
we will
same
for
express
that
with
(Fig. 19). respect
the u n i q u e
On the other
0 ~ t ~ A. B3(w,y)
C k-I
using
interval
to W
the c o r r e s p o n d -
@(~). t,
C3(w,y ) def @(t)
w
¢ ~ 0, d e n o t e
by
is an arc of c l a s s the
and
we will
on t h e
is f r o m
some
to p r o v e
defined
z ( y
of c a s e s
situation
y
~ { y,
for
the p r o o f
a trivial
function
z I = Zl(~,8)
U(0,~)
it is e q u i v a ~
Fig.
convex
identify
(a) only;
with
The orientation
Reca~l
ing a r c
the case
To d i s p a t c h
(b) a n d (c) y b e i n g an
of
By defi-
=
finite
length,
{ (t,@(t)),0
the
function
f
U (see
Fig. 20.
R,
denotes the oriented tangent to
y
at point
-
Clearly, B(t) = a(t) - B(t); tan a(t) = f l(t) and Consequently, for 0 < t r A one has -
0 (t)
=
Arctan f' (t)
-
z.
tana(t) =
f (t)
t '
f(t) . Arctan t
), Moreover let us define, O(0) = 0. As f 6 c ~ - ~ ( I o , A Ithen ck-l. 3 i c ~ - ~ ( ~ o) ,, Ai.e. ] C3 (w,y) is a closed arc of class the . As in the case of set A Let us consider now the set '4; 4 set An is in general not compact and may also contain a finite num3 ber of isolated points. Like in the case of the set An to prove our assertion it is enough to consider only two boundary pieces, y and 'fir fulfilling the following conditions: to every
z
€
y
corresponds the
unique angle 0 = B(z) such that the half straight line tangent to yl at the point z (perhaps one 1 = zl(z,B) Let (F(@),G(@)) such that a < @ < b and (F1($l) < d are the parimetrisations by arc length that c < 0 0 pieces y and y1 respectively. If z = (F(@ ),G(@ ) )
L(Z,B) is sided). such of the boundary and
zl = (F1(@'),G1(@')), then
By the virtue of the Implicit Function Theorem, for the existence of the function = Ql ( @ ) of class cr in some neighborhood of
m0
ml
235
such that
$i($0)
neighborhood,
~R
Indeed, tor
: $i
R($,@I($))
(@0,@i)
=
det / FI(@I)-F(@0)'
the vector
and, as
(F~($I),G[(@I))
$I
sided)
This is so because
GI(@I)-G(%0)
is parallel to the vec-
is the arc length parameter,
is o r t h o g o n a l to the vector
then the set
tangent to 71}
from this
for all
(F($1)-F($0),G($1)-G($0))
This i m m e d i a t e l y implies that if boundary pieces,
= 0
~R ($0,$i ~ ) ~ 0.
it will suffice that
(F{($1),G~($1))
vector
and
7
and
7i
(F{($1)),G{(@I)).
are two closed d i s j o i n e d
A4(7,7 I) : { (z,8),zEy,L(z,8)
is a closed arc of class
To finish the proof,
having a common end
vial situation,
is
(one
C k-l.
it remains only to c o n s i d e r the case of two
s u f f i c i e n t l y small closed convex or concave b o u n d a r y pieces yI
the
w
as a unique common point.
we will c o n s i d e r only the case when
7
and
To avoid a tri71
is strictly
curvilinear. If
7
and
situation.
71
are s u f f i c i e n t l y small we are in the following
Let(x,y)
plane such that
w =
be the o r t h o g o n a l (0,0)
Then
7
that
f(0) = f' (0) = 0
and
7
system of c o - o r d i n a t e s
is tangent to the x-axis at
is the graph of a convex function
or concave function
and
71
f 6 ck([0,A]),
w.
A > 0, such
is the graph of some strictly convex
g (ck([B,0]),
B < 0.
We will c o n s i d e r the mutual d i s p o s i t i o n of sented on Fig.
in the
7
and
71
as pre-
21 but we note that in all the other cases the proof
follows along the same line.
236
~y
z1
I
I
I
I
I
I
B
v
--
Fig. 21. One does not exclude ing to t h e x - a x i s . 0 ~ Y < ~.
From half
the preceding
o p e n arc
finish
is of ~(u)
This
finite =
6(v)
Moreover, C k-l.
is e q u i v a l e n t
the
- ~(u)
where
from above
L e t us n o t e
same
is t r u e
Notice
for
Let
X by
= {z
Ug(E)
6 X;
It is w e l l
exists
it f o l l o w s
that
the g r a p h
def
e(z)
e(u)
= Arctan that
= ~(u).
functions ~ of
the convex
(see Fig. f' (u)
and
of
u
are
of
C > 0
p(z,e)
< g
> 0
defined
is a arc.
To
finite
{ (u,~(u)) ; 0 < u Z A Clearly,
B(v)
are
finite
functions
convex
= Arctan
of c l a s s
bounded,
continuous,
on interval
length
g' (v).
[0,A].
and consequently
described
with metric
g
in Sec.
of c l a s s
This means
p. of
For E,
that
E c X
7.8.
it
C ~,
such
in g e n e r a l
and
E > 0
i.e.
e E E}.
to p r o v e ,
for e v e r y
a
function
g-neighborhood
such that
is o f
is a f u n c t i o n
~
belong-
~ = 6 - ~.
for s o m e
and easy
such that
g 0 = g0(E)
space
the o p e n
interval
21).
and
v = v(u) 6
;
is a c l o s e d
to s h o w i n g ~(u)
~
A4(Y,yl)\(v,~)
A 4 ( Y , y I)
A 4 ( Y , y l)
the g r a p h
known,
an
that
a n d of
be a m e t r i c
being
to p r o v e
to f i n d a s t r i c t l y
we denote
constant
for
y
f E 0, l i m ~' (u) does not exist. u+0 is n o t a c l o s e d arc of c l a s s C I.
A4(Y,yl)
Ug(E)
~
that using
is p o s s i b l e
6.3.
of
of
Moreover,
it f o l l o w s B(v(u))
decreasing
the g r a p h s
that
where
A
u-
the c a s e
C k-l.
has o n l y
length,
monotonically Thus
one
-
considerations
of c l a s s
the proof
length.
-
w=(O,O~/"/
that
there
rectifiable
for e v e r y
~,
arc
exists
an absolute
E ¢ ~2 , there
0 < g < g0'
one
has
237
VoI(Us(E)) Theorem
THEOREM
6.1.
6.2.
a closed
and
If
subset
for e v e r y
Let
E CSlE 1
P(UE(A~))
E CIE.
us r e c a l l
that
and
~
there
It is e s s e n t i a l
first
and
apply
this
of c l a s s
exist
is the
}
second
[.
for the
derivatives
theory
THE R A T E theory
An
is the
always
and
singular
be a s s u m e d . set of
of the c l a s s
imbedding
of c l a s s
The o n l y
is the c a s e everywhere
case way of
when to the
strictly
to a d i f f e o m o r p h i s m itself.
This
(~,z).
the
strictly
is s y s t e m a t i c a l l y
Except theory can
{F
of 4.1.
N~, and
to the concave convex
Theorem
boundary boundary
set
A~
i__ss
such
that
measure
OF
defined
lid%If a n d
in P a r t Thus,
I-IV
us
recall
Corollary
} : N~ ÷ M
IId2¢II
to use
if one w a n t s
to c o n s i d e r
} of c l a s s J l_]_r<'< that 4.1.
the
to
the bil-
C 3.
In Secs.
N n = M\A~, we o b t a i n
where
for
the
is a d i f f e o m o r p h i c
curve
C k-I
4.3.,
piece
of the
with
study
described
of the
C k,
of
in
M
in the
IId~I[ set
growth
situation: first first
kind
to a p p l y
of such
of
the P e s i n
an o b s t a c l e of c o m p a c t near
it f o l l o w s
either
lld%II F
(see Sec.
kind with
by
curvature
properties
meets
is n o t b o u n d e d
unbounded
(%,0)
bounded
non-zero
any attempt
As
onto
[Dvo].
inevitably
A n.
torus
the p o i n t s
region
ergodic
and
with
an e x t e n s i o n
two-dimensional
some
above
k ~ 2
admits
as a d i f f e o m o r p h i s m
singular
of the
reduces manifold,
everywhere
[Laz], [Dou]
~ : N~ + M
this
~
by i d e n t i f y i n g
curves
in the
¢ : N~ + M of c o m p a c t
of c l a s s case
billiards
geometrical piece
F
of the
speaking,
the
of the m a p p i n g
In this
be c o n s i d e r e d
near
OF G R O W T H
of d i f f e o m o r p h i s m s
convex
smooth
used
generally
following
study
study
is o b t a i n e d
for the c a s e
i.e.
the
for e x a m p l e
in no way
From
curvature.
to the m a p p i n g
Moreover,
~. that
property
convex
See
the
s0 > 0
probability
described
Let
P3
of c l a s s
torus This
billiards.
theorem.
C 2.
non-zero
and
then and
it is n e c e s s a r y
{L i}
billiards
in a n a t u r a l
P2'
of the m a p p i n g .
to b i l l i a r d s ,
curves
will
following
C1 > 0
invariant
iEiEp 7-8 this
the
has
OF C L A S S
7.1.
with
directly
4.4.
THE B I L L I A R D S
liards
imply
is a b i l l i a r d M
0 < s < s 0, one
in the C o r o l l a r y
7.
(6.1.)
~ of
(6.1.
the b o u n d a r y from Corollary is e q u i v a l e n t
contains 2),
a point
of
that
manifold.
a strictly
or a s t r i c t l y zero c u r v a t u r e ,
238
or a s t r i c t l y
convex
Nevertheless, class and
H
which
which
are
near
the
fast
results This
from
enables
are
The
class tion
of
a particular
set.
I-IV
us
deduce
true
convex
of
curves
on w h a t
denote
by
s ~ t,
non-zero
f
the
for
f' (s)
and
Let on
> 0
of
E Via).
interval
every Rf(s
s, t)
Any
such
val
t E
f(s)
positive
symmetrical implies
that
for
[0,a] and
instead and
f
To Sec.
F
7.2
theless, can with
on a
in
replace only
results
class
of
of
E
~
all ~.
from
a particular
now.
This
F.
set
strictly
all
that
class
Pesin
uses
= f' (0)
too
as w e l l .
to d e f i n e
of
grow
immediately
billiards
condition
one
defini-
convex
=
0.
For
such
and
if
s,t
E
E
satisfies
the
a number
C > 0
+ f(t)
condition such
F that
> C
f' (t)
(7 i.)
-
be c a l l e d
s,
t
[-a,a],
has
(s-t)
will
f
exists
every
s
an
and
exponent t
conditon
[-a,a],
play F,
s ~ t,
one
the
of
f
(on i n t e r -
a completely inequality
(7.1.)
has
1 Z ~ < + ~
i7.2.)
0 < C ~ i. the
interval
and
the
such
intervals
only
strictly
[0,a] made
the
[-a,a]
that
f(0) or
the
Proposition interval
7.1., [-a,a]
modifications.
F
considers
convex
functions
=
f' (0)
= 0t o n e
on
the
leave
the
intervals
f E cl(i[0,a]) obtains
the
con-
respectively.
repetitions,
Theorem by
We
one
I-a,0]
to a v o i d
condition
the
minor
follows the
f E V(a)
that
of
framework
- 7.4
thus
definition
of
P
class
f(s) - f(t) _ f'(t) are both s - t (s-t)f' (s) - f(s) + f(t) > 0 f(s) - f(t) - (s-t)f' it)
in t h e
[-a,0]
fix
if
as
E cl([-a,0]),
dition
-
C
of
and
- f(s)
remark
of
IId2¢II c a n n o t
it
f(0)
that,
necessarily,
If
f' (s)
us
0 < C S Rf(s,t) Thus,
that
say
-f(t)
billiards
0.
s ~ t
number
Let
role
the
that
the
billiards
to
going
a > 0,
if t h e r e
[-~a],
de f (s-t)
[-a,a]).
are
kind.
for
of class
the
sign;
We will
the
of
call
clear
same
7-9
principal
billiards
x ~
[-a,a]
'
this
- f(s) - f(t) s - t
the
of
billiards
we
second
IId¢ll a n d
the
such
for
geometrically
then
the
V(a),
f(x)
It is
that
which
functions
case
7.8.
we will
f E cl([-a,a])
the
applicable
the
functions
of
in Secs.
From
are
for
definition
us
in Sec.
singular
to
piece
is p r o v e d
Parts
is b a s e d Let
it
defined
[PeS]l_ 3 remain
7.2.
boundry
as
the
7.1.
and
interval this
we will
interval
to
consider
[-a,a].
Corollary [0,a]
the
or
reader.
in
Never-
7.1.
one
[-a,0]
239
Let us n o t e
that
2 ~ i s k - i, a n d f(k) (0)
condition
F
of a f u n c t i o n is g i v e n
I f(x)
For
from
by a w e l l
V (a) known
e I/x2
for
0 <
0
for
x
if
f~i)(0) = I
necessarily
0
k
for
is e v e n
and
which
does
not
satisfy
the
function:
]x I ~ a (7.3.)
function
interesting
(7.2.]
=
is not
examples
of
0
satisfied
such
when
functions,
t = 0.
will
Other,
much
be d e s c r i h e d
in
7.8. Let
us n o t e
also
that,
x g(x)
does
not
4
satisfy
limR (-t2,t) t++0 g
x ~ 0
for
x > 0
condition
In the p r e s e n t
sibly
largest prove
class
the
7.1.
Then,
there
exists
tion
F
on the
7.1.
on any
we do not
interval
f (V(a)
a0,
Let
the
[-a,a].
such
function
Indeed
f 6 ck+2([-a,a])
Corollary.
COROLLARY
7.1.
interval
[-a,a],
If
f
satisfies
the pos-
F.
We w i l l
a > 0, and f
let
satisfies
be r e p l a c e d
f
a > 0, k { 2, Then,
there
the c o n d i t i o n
in T h e o r e m
f (V(a), then
for
that
f(k) (0) ~ 0.
compactness
following
such
the
Theorem.
N ck+2([-a,a]),
let
whether can
simple
and
search
the c o n d i t i o n
f"(0)
~ 0.
the c o n d i -
[-a0,a0].
f ~ V(a)
that
with
N C2([-a,a]),
0 < a0 ~ a
and
deal
satisfying
Proposition
interval
[-a0,a0]We do not k n o w
By the
paper
Let
2 E i Z k - 1
0 < a0 E a
F
of f u n c t i o n s
following
PROPOSITION
THEOREM
for
by M. M i s i u r e w i c z ,
= + ~.
7.3.
only
as r e m a r k e d
= X
for
N ck~-a,a],
=
this
Sec.
f 6 V(a)
f(k) (O) ~ 0, t h e n
> 0.
An e x a m p l e
more
if
if
7.1.
F
on the
a > 0
satisfies
one
and
if
f(i) (0) = 0 a 0, interval
the a s s u m p t i o n s
by the a s s u m p t i o n
argument
let
exists
obtains
f
f ~ ck([-a,a]). from
Theorem
is r e a l - a n a l y t i c
the c o n d i t i o n
F
7.1.
the
on the
on the w h o l e
240
interval
Proof
of
exists one
[-a,a].
Proposition a0,
f" (s) f"(t)
has Let
us
-f(s) + f(t) Value
G(t) H(t)
- G(S) - H(s)
As
f E C2([-a,a])
such
that
for
all
s E [ - a 0 , a 0]
and
let
us
and
s,
f"(0)
t E
and
H(t)
Theorem
= f(s)
one
has
G' (tl) H' (t I)
- f(t)
-
note
(s-t)
G(t)
=
to t h e
there
s ~
f' (t).
(s-t)
t
Then,
open
Rf(s,t)
f' (t I) - f' (s) tI - s
interval
linking
f' (s) + from
Cauchy
(s-t)f' (s) - f ( s ) + f(t) _ = f(s) - f(t) - (s-t)f' (t)
t
G(t) H(t)
_
1 • ~
for
some
f' (t I) ing
> 0,
[-ao,a0],
> ! - 2"
fix
Mean
7.1.
0 < a 0 S a,
and
s.
tI
belong-
- f' (s)
But
= tI - s
f " ( t 2) s.
for
Thus,
Rf(s,t) The
the
belonging
to t h e
open
of
Theorem
if o n l y 7.1.
s,
7.1.
If
k > 2
condition
F
Let
has
s,
Rf(s,t)
k
is e v e n ,
r ~ i. other
on
t E =
The hand
use
is b a s e d
is e v e n , the
then
interval
[-i,i],
on
the
the
continuous
~(r)
> 0.
and
we omit
The
the
and
7.1.
and
7.2.
for-
t ~
0
that
function
Rf(s,t)
> 0.
de ~Hospital
rule
function
= x
k
satisfies
[-i,i].
s ~ t,
= k - i.
f(x)
and
Thus,
¢
o n ~-- = ~
s r = ~.
For
f(x)
= x
(k-l)rk - krk-I + 1 def k r - kr + (k-l) Consequently, gives can
~(r)
> 0
l i m ~(r) = i. r÷l
On
be c o n s i d e r e d
U {-~}
U {+~}
k
~(r)
for the
as a s t r i c t l y
and
consequently I
following
7.2.
such
s ~ t, the
l i m #(r) r÷±~
inf rE
be
tI
•
Lemmas
function
(k-l)sk - ktsk-i + tk k - kst k-I + (k-l)t k
for of
positive
LEMMA
linking
t E [-a0,a0]
s As
interval
below.
Proof. one
t2
f"(t 2 ) ! > - f . ( t l ) - 2'
proof
mulated
LEMMA
some
finally,
lemma
is a s i m p l e
consequence
be e v e n
let
of
the
Taylor
Formula
its proof.
Let
k ~ 2
f(i) (0) = 0 g
given
by
for the
and
f E V(a)
2 S i ~ k - 1 formula
and
N ck+2([-a,a]), f(k) (0)
~ 0.
a > 0, Then,
241
is o f c l a s s The class
function
the
proof
As
v
of
(
the
b = g(t),
a,
In v i r t u e
h
some
q0'
~
inverse
=
As we
know
condition C1 > 0
F such
RF (a'b)
as
the
function basis
Lemma
7.2.
we
write
k[~(s)]k-lg' [g(t)] k -
bk
-
-
of
from
for
that
= ,
> O.
exists
> 0.
and
Moreover,
for every
us n o t e
a = g(s)
and
write:
(s)
-
[g(s)]k
+
[g(t)] k
k[g(t)]k-lg
' (t)
ak + bk
k b k - i [h (a) _ h (b) ] h , l (b) -
( a k - b k ) h ' (a)
def
- kb k-l[h(a)-h(b)]
. Tf(a,b)
Lemma
the
h = g
-1
=
h' 1(a)
( a k - b k ) h ' (b)
h' (b)
g,
can
g'(O)
[h(a)-h(b)~
kak-l[h(a)-h(b)]
h' (a)
where
Let
now
h' (a)
on
=
We
h' (b)
that
the
is t h e
so s m a l l < 2.
[-q0,q0 ] .
kak-l[h(a)-h(b)]
=
f
[ - q 0 ' q 0 ]' q 0
-< 1
h' (u) < 2 - h' (v)
[g(s)] k -
k
7.2.
idea
to
interval
0 < q0
[h(a)-h(b)]
a
in L e m m a
C k+l.
This
and
!
has b
of
(s-t)f' (s) - f(s) + f(t) f(s) - f(t) - ( s - t ) f ~ (t)
=
that
function
z = g(x).
E C2([-a,a])
on
one
=
Rf (s,t)
7.1.
g
We chose
where
the
> 0.
7.1.
C2
[-q0,q0 ]
by the
to c o n s i d e r
function
class
> 0.
class
where
> 0,
of
shows
replaced
Theorem
k
k/f (k) (0) k'
g' (0) =
variable
Theorem
[g(x)]
g ' (0)
h' (0) u,
of
Proof
is a l s o
-a _< x < 0
= x k + x k + l l Ix "
be
L e m m a 7.2. a l l o w s k = z in t h e n e w
=
for and
f(x)
cannot
F(z)
f(x)
0 _< x _< a
C2([-a,a])
C k+2
7.4.
for
7.1.
the
interval all
a,
function
[-i,i]. b
(k-l)a k - kba k-I bk (k-l) - kab k-I
6
This
[-i,i], + bk k + a
F(x)
means
a ~ b,
-~ CI-
= xk
satisfies
that
one
has
there
the
exists
h
242
[1 ~ h' h' (a) (b)
As borhood ql'
of
-< 2,
zero
0 < ql
is
~ q0'
to
prove
the
equivalent
such
that
to
for
condition
prove
all
F
that
a,
b
for
there
(
f
in
exist
[ - q l q l ],
some
neigh-
C2 > 0
a ~ b
one
and has
Tf (a,b) (7.4.)
RF(a,b ) { C 2 .
Tf(a,b) _ M(a,b) RF~,b) N(a,b) N(a,b)
=
where
(ak-bk)h'
(b)
= ka k - I [h (a) - h (b) ] - ( a k - b k ) h ' (a) (k_l)a k _ kbak-i + b k
M(a,b)
- kbk-l[h(a)-h(b)]
(k-l)b k - kab k-I
Instead C 3, a,
of
0 < C3 < ~ b
6
Clearly,
(7.5.)
kak-1 M(a,b)
and
[ - q 2 , q 2 ],
C 3 S M(a,b)
for
proving q2'
a ~ b
implies
h(a)
belonging
to
= ,izlil =
h' (z) ~Ii + 'l(z) h
=
h' (z)
h
(C2([-q0,q0]),
the
open
h' (a)
for
kak-lh =
interval
- bk
- b h'(z) k bk a a - b
there
' (z) kak-1
linking
a
and
b.
Thus,
=
(k_l)a k _ kbak-i
+ bk
z)
(ak-bk) (z-a)
h'
ak - bk h' (a) a - b k bk a a - b
h'(a)
h' (z)
bk
- h' (a)
z - a
7
(k_l)a k _ kbak-I
(ak-bk) (z-a) ( k - l ) a k :- k b a k - I
to
exists
all
ak
h" (z I )
As
that
that
h'(a)
1 + h' (z) belongs
ak - bk a - b k bk a a - b
a
a
= h' (z)
zI
such
prove
has
k
kak-1
where
~ q0
will
(7.4.).
- h(b) a - b
kak-i M(a,b)
we
(7.5.)
kak-1
z
directly
0 < q2 one
= M(b,a).
+ ak
< Cl~ _
=
some
(7.4.)
and
the
open
h"
is
interval
uniformly
+ bk
linking bounded
a on
and the
b.
interval
[-q0,q0 ] .
243
Moreover,
as w e know,
h"(z 1 ) 6 > 0, h' (z)
small
to f i n i s h
the proof
l i m L(a,b) a~0 b÷0 a~b
where
L(a,b)
L e t us
l i m h' (z) = h' (0) a+0 b÷0 is u n i f o r m l y
of
(7.5.),
The use the
=
that
last
Let
z E 7
z.
the
hand,
Y
together
and
real
< 6
to p r o v e
gives
CA,
r o o t of
l i m T(r) r+l
the d e n o m i n a t o r of
2 = k-l"
is b o u n d e d
where
L(a,b)
the
if
straight
exists
with
Zl(Z)
L(a,b).
on e v e r y
interval
Consequently
way
z
the curve
f .
f
proves
tangent of
passing line
Zl(Z)
only
for
7
one
~2(z),
or
point can always corres-
in a n e i g h -
of a s m o o t h x a 0
at p o i n t £2(z)
through
(x,y)
•
curve.
Let
Z2(z)
and
graph
to
7).
This
(7.6.).
plane
of c o - o r d i n a t e s
is t h e
]r I ~ A 0,
strictly
x S 0
if
z
say t h a t
T-
Y
z E y of
Zl(Z)
lines Y
if
smooth
ends
straight
is d e f i n e d
is o n e of the e n d s of
some n e i g h b o r h o o d
the
system
straight
(7.7.)
oriented
the
that
IL(a,b) I S 2[b].
the o r i e n t e d
to
z
if a t a n y p o i n t
convex
line orthogonal
in a s u i t a b l e
such
inequality
of
in t h e o r t h o g o n a l
of p o i n t
of
Thus,
0 < A < +~.
then
~ A 0,
denote z
A0 > 0
is o n e
to the o r i e n t e d
=
(7.7.)
~
let
function
and
that
Then,
of t h e n u m e r a t o r
function,
if
tangent
ponding
We will
rule
a strictly
that
z
Ibl
b r = -a •
l e t us n o t e
a root
constant
Thus
denote
Orienting
convex
is a l s o
there
assume
borhood
< @,
IL(a,b) I ~ a C A
(one-sided
denote
and
~, a s a c o n t i n u o u s
inequality
Let
z
r = 1
l~(r) I S 21r I .
7.5.
sufficiently
.
The u n i q u e
by s o m e p o s i t i v e
On the o t h e r then
+ bk
a ~ 0
of t h e d e l ' H o s p i t a l
sup a~0,a~b
lal
it is t h e n s u f f i c i e n t
(ak-b k) (a-b) (k-1)a k _ kbak_l
r = i.
function
[-A,A]
when
for
(7.6.)
suppose
is
bounded
Thus,
= 0
(l-r k) (l-r) = a~(r) k (k-l) - kr + r L(a,b)
> 0.
In p a r t i c u l a r ,
satisfies the
zero
fz(X)
the condition
function
(at t h e e n d s
fz of
F
satisfies Y
one
> 0 with this
for all
x ~ 0.
exponent contition
considers
the
C in
244
respective
one-sided
neighborhoods
of zero).
A billiard of class
Pk' 1 _< k _< co belongs to the class Hk if o r there exists a representation F = (i=iUL i) U (3UIFj).__ and a constant C > 0
such that for all strictly convex boundary pieces
(see Sac.
2) belonging
condition
F
with exponent
The class billiards
to some
H3
Li(l_
C
y
of
F
Fj (l_<j_
is satisfied.
will be denoted
belonging
or to some
simply by
to the class
~
H.
Natural examples of
are the billiards
for which all
the curves
{F. } and {L.} are real analytic; this is a 1 l_
given by the billiards boundaries
of class
P3
such that the curvature of their
is always non-vanishing.
This follows directly
from Propo-
sition 7.1. The class sidered in
7.6.
Z
contains
[Sin]2 and
in particular
p(w,~)
fix the norm
w =
(~,e)
6 M
and
is defined at the end of Sac.
Recall also that
con-
[Bun] 1-4"
Let us recall that for
distance
all the plane billiards
~[~' =
Nd2¢(¢,e)lJ
=
~ =
(¢,~)
6 M, the
3.2.
lal + Ibl + Icl + Idl.
Analogously,
~2~ 1 d~
+
~2~ii + ~
32,1 + 2 3~-~}
22 81 ~
+
we ~ J i ~e2,
+
_2~ + 21~
,
where
~(~,@)
= (~i(~,0) , el(~,e)).
The aim of the present and the next two sections following
is to prove the
theorem which is the main results of the present paper.
THEOREM
7.2.
w 6 N~
the following estimates
IId# (w) II
If
~
is a billiard of class
then,
for all the points
hold:
C1 p(w,A~)
(7.8.)
C2
IId2~ (w) II
(7.9.) ~(w,A~ 3
where
C1
p(v,A~) suffices
=
and
C2
are constants
inf p(v,a). ~A~ that
~
In fact,
depending only own
for the validity
belongs to the class
Remark that the choice of norms
~
and where
of estimate
(7.8.)
it.
H 2.
;I II has an influence only on the
245
size
of
cI
and
C2
but not
on the e x i s t e n c e
or n o n - e x i s t e n c e
of
such
constants. Let
us note
estimate among
also
(7.9.)
strictly
The p r o o f ing two
LEMMA
convex
points
w
billiards
where
Let
~
6 N~
in Sec.
7.9.
is d e s c r i b e d .
of T h e o r e m
lemmas
7.3.
that
fails,
7.2.
with
is an
r I = rl(w)
the
followin9
C~
of b i l l i a r d s
billiards
for w h i c h
c a n be
found
consequence
Idl(W) I =
of c l a s s
estimation
Isin
follow-
~i (w) I-
Then,
P3"
of the
for all
the
holds:
< el lld~ (w) II -
even
boundary.
immediate
=
be a b i l l i a r d
a class Such
(7.10.)
r1
a2 Ild2~ (w)II < --~
(7.11.)
rI where
aI
and
In fact, to the c l a s s
LEMMA
7.4.
points
w
a2
are c o n s t a n t s
for the v a l i d i t y
depending
of
(7.10.)
only
on the b i l l i a r d
it s u f f i c e s
that
~
Q.
belongs
P2"
Let 6 N~
~
be a b i l l i a r d
the
following
of c l a s s
inequality
HI"
Then,
for all
the
holds:
r I ~_ bQ(w,A~) where
b > 0
The p r o o f d~
and
9.
dependinq easily
has a p u r e l y
a really and
is a c o n s t a n t of L e m m a
formal
geometrical It s h o u l d
tiability
of all
7.3.
one.
only
character The p r o o f
be e m p h a s i z e d the c u r v e s
on the
follows
that
from
unlike of L e m m a this
{L i}
and lsisp
differentiability
7.7.
Proof
of t h e s e
of L e m m a
7.3. k£ ~i
curves
Let
which
7.4.
{F
7.4.
is g i v e n
uses
only
} J 1zjsr
Since
Q
belongs
(4.10.)
whose
for
proof
8
differenof
Lemma
7.3.
for
is
in Secs. C1
instead
C3
that c
'
i~ 1
(4. i0. )
d¢(¢,8) k£ - ed elk I dl
~.
formula
is n e c e s s a r y
us r e c a l l
- ed dI
Lemma
proof
billiard
the
kli E 1 dl
k,
to the c l a s s
P3'
then
from
formula
(4.10.)
the
246
estimation
(7.10.)
obviously
it follows
immediately
follows.
From the rule
that to prove the estimation
(~)' (7.11.)
to show that the absolute values of all the quantities ing table are less that
--, a where rI
on
a > 0
f'g -2 fg' g it suffices
from the follow-
is a constant
depending
only
] ~k I
Sk~ .
~d
--(z)
~d I
(z)
Dk]
This is evident zero. _
~k --
For
Sdl ~61 ~81 • ~ - -
~d --
and for
the formula {L i}
Sd(z )
~@ (z)
~(z)
(4.10.) and
<' < l_±_p
to examine
~
and
~~( @ , @ )
[F(@I)-F(})]
~kl ~d I ~8
=
for
de
~6( z )
~d I ~@ (z)
~(z)
~d 75-
and
=
{F } J l_<j<_r
In the notation Therefore,
~k
for
--(z)
~kl
~@i
~d I
~91
~@i
because they are always equal to ~kl
this follows
As
~
trivially
and from the condition are curves
of class
P3'
from
indeed,
all
It remains
C 3.
of Sec. 4.1. one has £= //[F(¢I)-F(@)]2+
= ~~ ( ~ , @ i ( $ , @ ) )
F' (~i)+
[G(@I)-G(%)]
P3
Analogously,
[F(,I)-F($)]
~dl
~--~.
from condition
follows.
~$i
~8 '
~@
G' (¢i)
~@i
[F' (¢i) - ~ -
belongs to class
lows that there exists an
because
the required estimation
one has
~(@,8)
F' (@)]
P3'
+
~%1 $8 _ el ~f
for
~
immediately
=
[G(¢I)-G(,)]
from the formula
~ > 0
[G($1)-G(@)]
~$i _
s1 ~ii { [F(@I)-F(@)]F' (%I)+[G(%I)-G(~)]G' (%1)} Thus,
Skl
[G' (,i) -$@
(4.10.)
such that for all
for (@,8)
de ( No
- G'(¢)].
it folone
247
has:
IF
(~i)
-F'
this we obtain _~(%,$Z 8)
7.8.
now describe
F is n o t
below
these
(~i)
- F(%)I+~
IG(%I)
the Misiurewicz
for
f. of
construct Ca
From
(%) I < rl
In fact, such
of
strictly and
7.9.
Using
exmaples the
remark
convex
that
the
the c o n s t r u c t i o n
functions.
for w h i c h
construction
< r12~ _ rla
= f' (0) = 0
in Sec.
boundary
- G(~) I
example
f(0)
class
to t h i s
-
estimation
a whole
with
we pass
IG
such that
we will
billiards
Before
and
satisfied
provides
functions
convex
required
f ~ C~([-I,I])
condition cribed
< rl
_< .~__rl . IF(%l)
We will
function
the
(#)I
of s t r i c t l y
estimate
that
des-
s o m e of
(7.8.)
if for
fails.
f 6 cl([0,a]),
a > 0, lira inf xf' (x) - 1 x÷0 f (x) then
f
from
can not
(7.1.)
(7.12.) which any
satisfy
when
also
fails
= 0
Precisely,
Notice
to s a t i s f y
increasing
and
the c o n d i t i o n
t = 0.
is s a t i s f i e d .
strictly
f(0)
(7.12.)
convex
for e v e r y
x,
Let
fix two arbitrary
positive
every
such
and
b lim n b - a n÷~ n n
that
such that
-
L e t us c h o o s e
X
= 0
{0}
for
U
x
except
F.
described
for
On the o t h e r
f 6 cl([0,a])
one
here,
function
(7.3.)
hand,
for
such that
has
sequences,
{a n } n~l
lima n = limb n÷ ~ n÷~ a n l i m ~-- = O, o r
n
and
{b n }
, of n>-i
= O,
< bn
an
equivalently
< an-1
that
(7.14.)
an a r b i t r a r y 6 {0}
U
function
U [an,b n] n=l
h
and
If s u c h a f u n c t i o n
g 6 C~([0,1])
perhaps
for
n
1
U [an,bn]. n=l
find a function x ~ 0
true
immediately
(7.13.)
n --'-ce
h(x)
examples
is n o t
function
follows
1
numbers,
n ~ 2
for t h e
this
0 < x ~ a,
>
us
This
the condition
xf' (x) f(x)
-
that
F.
an at m o s t
such
that
countable
E C
([0,i])
h(x)
h
> 0
= 0
that
then
one can
for
is fixed,
g(0)
such
and
set of n u m b e r s ,
g(x) and
> 0 that
if
248
I~
ng(t)dt
lim n÷~ I b n h ( t ) d t J0 L e t us d e f i n e Let us d e f i n e
-
Let us fix s u c h a f u n c t i o n
0.
;x
n o w the s e c o n d d e r i v a t i v e
f' (x) =
f"(t)dt
and
f(x)
f"
f (C~([0,1]),
of
=
f
hy
f' [tldt.
0 that
g.
f" = h + g. It is c l e a r
0
that
f(0)
= 0, a n d that
f
is s t r i c t l y
convex.
We w i l l p r o v e n o w that b n f ' ( b n) l im n÷~ This
-
implies
dition
F
(7.15.)
that the f u n c t i o n
f
so d e f i n e d
does not
satisfy
the con-
H' (tldt,
G' (x) =
(tldt
(see 7.12.).
Let us d e f i n e and
i.
f(bn)
G(x)
=
H' (x) =
;x
h(t)dt,
H(x)
=
0
G' (t)dt.
0
Remark
that
H
and
G
are c o n v e x
on the inter -
0 val
[0,i]
and
bnH' (b n) lim n÷~
-
1
7.16.)
H(bn)
Indeed,
by v i r t u e
diately
that for e v e r y
of the d e f i n i t i o n n ~- 1
H(b n) > I bn H' ( t ) d t
=
of f u n c t i o n
h, one d e d u c e s
imme-
one has
(bn-a n) g' (b n)
7.17.)
a n
because
the f u n c t i o n
sequently, (7.14.).
H'
bn H ' ( b n) H(bn )
bn bn _ an
E
Unfortunately,
not s t r i c t l y
convex
is c o n s t a n t
although
on the i n t e r v a l
and thus the
(7.16.)
function
in any n e i g h b o r h o o d
follows
H
(7.15.).
1 E
it is s u f f i c e
lim
to p r o v e
b n G ' ( b n) H(bn )
= 0.
By v i r t u e
of
it is
bnH' (bn)+bnG' (bn) <
f (b n) b nH' (bn) bnG' (bn) H (b n ) + H(bn )
7.13. ) and
from
is convex,
Con-
of zero. b nf' (b n)
Pass n o w to the p r o o f of
[an,bn].
(7.16.),
in o r d e r
H (bn) to p r o v e
(7.15.)
that
(7.18.)
249
Indeed,
by v i r t u e
bn bn
(7.17.)
one
obtains
that
0 S
bnG'(b n ) H(bn )
_<
G'(b n ) for all
an
H'(bn)
from
the v e r y
-
and
of
n ~ I.
definition
Thus,
of the
(7.18.)
function
follows
from
(7.14).
g.
7.9.
S u p p o s e that in a d d i t i o n to the c o n d i t i o n s d e s c r i b e d in Sec. a n 7.4. one has lim ~-~ = 0 a n d g(b n) = 0 for all n. Let us n o t e by n÷~ i ] n
y
the g r a p h
Rn =
of
(bn,f(bn))
Consider boundary
that lim n+~
the
~(P,8 n)
=
such
for
such
a billiard
8.
PROOF
OF L E M M A
8.1.
be c a l l e d Recall
that
number
bQ
if
that
> 0
for any
in
M
one
has
and
PART
and
a number
Isin
P =
(0,0)
us
in the
~
(
by
81n
easy
to v e r i f y
formula
such
every
inequality
neighborhood
that
C
that
(4.10.)
this
Consequently,
fails.
ELEMENTARY
following
that
for all
of
M,
element bp
CONFIGURATIONS
sections
the
elements
of
A~
A9
> 0
is a c o m p a c t imbedding.
subset w
of
E Q
P
exists
such
that
then
one has Lemma
there
This
N~,
to p r o v e
7.4.
for all
subset
of
M
and
immediately there
exists
a
r I ~ bQp(W,A~). it s u f f i c e s
its n e i g h b o r h o o d points
w
~ U(P)
As
to p r o v e U(P) D N~
(8.1.) (8.1.)
is not
evident
only
in the c a s e
when
for
U(P)
inf w6U(P)nN~
el(W )
The p r o o f
of L e m m a
consists
and
angles
r I a bpp(W,A~) The
Y
with
note
of the
(7.8.)
ONE:
billiard
8nl'IId~(P,Sn) l; = + ~.
homeomorphic
subset
singular
Let
It is v e r y
is a c o m p a c t
such
Let
elements.
is a
is a c o m p a c t
6 F.
estimate
7.4.
Q
f.
convex
In v i r t u e
lim
NQ = M\Ag,
~ : N~ ÷ M that
A~
the
singular
that
implies
y
n ~ i.
that
In the p r e s e n t
will
strictly
that
(Rn,8~),
implies
function
~ l.
~(P,8 n ) 81 I - + ~. Isin n
8nl
immediately
n
n o w an a r b i t r a r y
F = F1
Isin
corresponding
( y,
=
0
(8.2.)
7.4.
of a d e s c r i p t i o n
consists
of all
of t h r e e
"elementary
steps.
The
first
configurations"
step
of b o u n d a r y
250
pieces The
for w h i c h
"elementary
pose
(8.2.)
holds;
configurations"
any configuration
the r e d u c t i o n asserts
the v a l i d i t y
that
billiards
8.2.
of c l a s s
~ M
i
or
then j
plane
by
F
8.1.
is u s e d
canonical
either
w =
(v,8)
a n d by d e f i n i t i o n
section
we will
6 M L
and
or
=1 v.
vI
always
geometrically
distinct
and of
singular
elements
(z,G)
consider
in d e t a i l
in w h i c h
(8.2.)
(a) P =
z
belongs
(z,0)
or
(b)
z
P =
following
such that
z
z
to a s t r i c t l y
is t h e i r
is the b e g i n n i n g
X-
the
in
(a),
tangent
(b) a n d
(b)
If
then
in t h e
(Fig.
22). in the
B 1
B9
cases
of
y
at
X\{z}
noted
A I,
z
23)
Z(wfl
Indeed, for
if
some
denotes
(see Sec.
the
3.3.).
of b o u n d a r y
(8.2.)
holds,
exhaust
boundary
boundary such
T\{z}
pieces
we will
all
cases
piece
L
passing
~
and
pieces
Yl
T
and
piece
Which
T1
do exist.
T
and
z
is d i s j o i n t z
the b i l l i a r d
and
billiard
Tl\{Z}
through
yl\{~}
y
that the to
boundary
piece
to
and
z
is from
is
trajectories
do exist.
description
is p r e s e n t e d
neighborhood
convex
is the p o i n t
if (Fig.
various 71,
E MF.
of all
small
T
and
24)
cases
o n Fig.
of d i f f e r e n t i a b i l i t y
small
sufficiently
C1 ~ C5
and
Y1
of
is a s t r i c t l y
(Fig.
These
line
is a p o i n t
If
then -
to
sufficiently
y U T!
F.
onto
9.
the
cases
occurring
(c) r e s p e c t i v e l y ,
T h i s case,
when
M
which
and
of
of a c l o s e d
n o w to t h e d e t a i l e d
(a)
z
of
8.1.
that
closed
point
points
boundary
straight
from all points
L e t us p a s s
of Sec.
= z.
cases
convex
only common
of the c l o s e d
going
last
of L e m m a
(v,~)
for w h i c h
of two c l o s e d
from all
the b e g i n n i n g
unilaterally
The
(z,z).
starting
Moreover,
8.4.
configurations
three
of
which
place.
is the b e g i n n i n g
trajectories (c)
the
takes
com-
for the e l e m e n t a r y
is d e f i n e d
~(P)
f i n d all
P =
w =
vI
denote
To
of
Recall
if
8.1.
is the c o n t e n t
in the p r o o f
8.3.
can
in the d e f i n i t i o n
projection
z(w)
v
this
one
of L e m m a
(8.1.)
and
step consists
in Sec.
appears
only
the
between
and
which
8.2.
of w h i c h
The n e x t
inequality
~
distance
In t h i s
H
bricks
pieces.
is d e s c r i b e d
Lemma
the c o n d i t i o n
Denote
w
reduction
of p r o v i n g
like
in Secs.
to the d e m o n s t r a t i o n
of the m a i n
This
step consists
7.4.
is g i v e n
are
of b o u n d a r y
of L e m m a
configuration.
Notice
this
if
and by their
of
arc
z, e x c e p t
we h a v e
T
are and
~i
are
at
tangent
respective
position.
A2 - A5 of arc
z, w e h a v e
transversal
by t h e i r
relative
of
y U T I, for t h e c a s e
the cases
of n o n - d i f f e r e n t i a b i l i t y neighborhood
T1
differ
arc,
22.
of t h e
z, a n d at
T U TI,
cases the
z.
type of convexity
251
(c) and
We can classify
71
for w h i c h
taking well we
all
(8.2.)
into account
the mutual
a s the t r a n s v e r s a l i t y
restrict
cases
ourselves
D 1 - D4,
the configurations
holds
in a s i m i l a r disposition
or tangency
to s u f f i c i e n t l y
E1 - E4
and
of
of
in
L, of
~
and
small
F1 - F3
of b o u n d a r y
w a y as
~
y
L
and
at
and
represented
pieces
7
(b) b u t a l s o y1,
z.
YI'
as
Then,
if
we obtain
o n Figs.
the
25 - 27
respectively. To t h i s noted
by
for e v e r y L(v,9 )
l i s t of c o n f i g u r a t i o n s
G:
y
v 6 ~
tions".
Before
r
A1,
C1 P =
out
has a continuous
ing to
A~,
(v,0) to
singular will
8.3.
to
y
qv
ignore
71.
kind.
"elementary
line
configura-
of r a m i f i c a t i o n
if
exists
where
in a d d i t i o n
another
cases
Pv =
singular
in a n y
elements
Pv =
close
In C a s e
families
z, all
C 5 even
of
U(P)
(V'gv)'
to
(V,~v) , exist.
role
pieces.
neighborhood
singular
and
v
where belong-
two of
Finally,
the c o n f i g u r a t i o n
not play any
to t h e
element
of b o u n d a r y
v E ¥, v ~ z
except
to Figs.
notice
elements:
is s u f f i c i e n t l y
A1
and
C1
in o u r c o n s i d e r a t i o n s
z
22 - 27 w h i c h
To e x h a u s t that
or
all y
in t h e s e
and
is a p o i n t
show only
of
boundary
The c o r r e s p o n d i n g
pieces
always
of
are
these
and we
pieces
y
and
restrict
are
on t h e
A1
ramification
z,
relative
one
posi-
going
the boundary
from
pieces
of
defines
of the b o u n d a r y , 71
which
ourselves
are
to s u f f i -
all
boundary
pieces
y
either
strictly
convex
or
t y p e of c o n v e x i t y
appear
the
trajectories
in c a s e
a n d of
neighborhoods
intervals.
clearly
71
As we z
give
possible
an e x c e p t i o n ,
z
neighborhoods
laying
out that
singular
for o u r c o n s i d e r a t i o n s .
small
be a p o i n t
is s a t i s f i e d .
(v,z)
suppose
22 - 27 w e
will
half
begin.
two c o n t i n u o u s
To a v o i d
o n Figs.
Y1
the
de-
that
them.
Even
and
=
will
Y1 = 7-
ciently
one,
such
can
is s u f f i c i e n t l y
and
Nevertheless,
YI' w e w i l l
essential
the
configurations
many
of
(8.2.)
and
z
there
v
(V'gv)
contain
and
second
pieces
configurations,
pieces
these
and
L e t us t u r n n o w
t i o n of
the
Pv =
elements
simply
following
the e l e m e n t a r y
it t u r n s
in g r e a t
also
z.
C2
(z,0)
such that
families,
out that
family
v ~ z = ~(P)
close
above
l i s t of
boundary
with
that
v 6 Y,
qv =
the
boundary
v I = V l ( V , G ) 6 YI'
listed
to t h e
and
connected
We point
U(P)
add
YI"
36 c a s e s
various
element
(z,z)
that
we will
two disjoint
such that
let us p o i n t
In c a s e s
such
9 to
we pass
in w h i c h
singular
the
are
a few remarks.
Firstly,
one
and
call
let us m a k e
Pl =
Yl
is t r a n s v e r s a l
We will
of
and
figures
of t h e s e and will
boundary in p r i n c i p l e
252
not be d e s c r i b e d The a r r o w s pieces
y
is not
and
shown y
in such
a case
(8.1.)
Y1
22 - 27 show
fixed
by us.
on a s e p a r a t e
changing
as on the
separately.
on Figs.
into
Yl
figure.
and
Y1
It is c l e a r
is c o m p l e t e l y
If we c o n s i d e r
figure
the o r i e n t a t i o n
the o r i e n t a t i o n
but
into
from
y, we w i l l
always
independent
the
from
pieces
truthfulness
the c h o s e n
boundary
a configuration
is o b t a i n e d
of the b o u n d a r y that
of the
a given assume
remain of the
which one
by
that
the
same
inequality
orientation
of
7
and
YI" We w i l l and
z
tions
by
£
and
£i
the o r i e n t e d
to the c u r v e s
denote
y
and
Y1
respectively.
we w i l l
denote
elements
without
singular
elements
With
any
systems
figurations following system,
is the
interval;
note
if
(z, @)y =
=(z'8)F'3 '
(see Sec. pieces
y
z
and
except
is a p o i n t z, y
(z'~)L i , L (z,9)y
L(z,8) 7 = L(z,@)r. 3
the
captions
associate
piece
y
and
In all
71
verified.
are
if
function
is a l s o
to some
y c F..]
the
if
co-ordinate
that
when
L. or i y c L.l
con-
the
co-ordinate
of c l a s s
true
straight
elementary
so small
for the c a s e
= L(z,@)L i
only
the
In the d e s c r i b e d
same
z
capsingular
to the
the
of n o n d i f f e r e n t i a b i l i t y belongs
figure
described.
one c a n
some m o n o t o n i c
moreover,
of
In t h e s e
7.5.).
at p o i n t s
In the
the c o r r e s p o n d i n g
explicitly
to the b o u n d a r y
of
Pv
comment. are
to the y-axis,
neighborhood
and
Pv
configuration
is a l w a y s
graph
belonging
Moreover,
P1
i2(z)
the b o u n d a r y
on a c l o s e d
in some
and
related and
assumption
y
interval
P
elementary
£1(z)
P' PI'
any a d d i t i o n a l
(x,y)
lines
by
tangents
C1
Yl for of r.. 3 and
defined is an
~i" r
then we w i l l (z,e)y
=
253
Configurations
Ai)
A1 - A 5
~)
A2)
/ 7]
Fig.
22.
when
z A2
is
an
: y
of
A3
is
the a
arc
y
of
at
P =
but
is
the
convex
not
convex
begins
U YI"
A 2,
strictly y
strictly
which
: Like
is a
end
boundary z
boundary
excluded.
and
piece z
is
piece.
P : which
the
The
( z , 0 ) y ' P1 begins
point
of
case
=
(z,~)
at
z,
Y.
Yl
differentia-
(z,0)y. roles
of
7
and
71
are
reversed.
(z,~) A4
is
: y = 71 at
interval
bility
P =
A1 is
z
the
Y : 7
and
point
inflection. A5
: Like
Yl
of
are
two
boundary
differentiability
P = A4,
(z,~) but
y
and the
the
arc
of
family y U Y1
Configurations
Bi)
z×<
pieces arc
beginning
y U YI" P is
z
at is
z the
and
z
point
of
is
exists.
v
strictly
BI - B9
B4)
convex,
P =
(z,~)y.
254 BT)
.%1~i
BS)
->
z
z
Fig. Z
23.
and
71
At gI
point
z
denote
the
the
respectively.
~
B1
: P =
(z,~)y,
B2
: P =
(z,2~-~)y
B3
: Like
B 2,
pieces
half
straight
denotes
the
P1
= =
role
boundary
(z,~)
y
lines
angle
and
tangent
between
Y1
are
at i
z
transversal.
to
and
£i
y
and
: 0 < ~
< ~
.
( z , ~ ) y I.
of
y
and
y1
are
reversed.
P =
(z,0)y
. 1
the
B4
: P =
B5
: Like
B6
: P =
(z,~)y.
B7
: P :
(Z,e)y
B8
: P =
(z,~)
.
B9
: Like
B 8,
role
family
(z,e) 7
P
B4,
v
and
role
the
of
and
y
the
of
y
family and
Pv Yl
family
and
Pv
Y1
exists. are
reversed.
P =
(z,0)yl-
P =
(z,G)yl
exists.
are
reversed.
exists.
Configurations
C1 - C5
c3)
ci)
1
c4) ~i~
z
¥ Fig. and
24. the
At angle
and
point between
z
the them
boundary is
zero.
pieces
y
and
Y1
are
tangent
.
255
CI : P =
(z,0)y,
C 2 : Like CI, Pl =
PI =
(z'z) 7
role of
y
and the family
and
Y1
Pv
are reversed.
exists. p =
(z,0)
Yl
(z,z)
Yl C3 : P =
(z,0)y
C4
(z,0)y.
: P =
C 5 : Like C4, P1 =
(z,~)y
and the family
role of
y
and two families,
and
Pv Y1
Pv
exists. are reversed•
and
Configurations
Pv' D1
P =
(z,0)y,
exist.
--Dj
D2) ~z
L__ z
B-
BF
_ g
z
DB)
n4)
z_____
~
z
Z z
z
Fig. 25. line Z--
y is an a r b i t r a r y b o u n d a r y p a s s i n g t h r o u g h the p o i n t s
p i e c e t r a n s v e r s a l to the s t r a i g h t z and ~. ~ is the o r i e n t e d
Z
angle
between
the s t r a i g h t
lines
Zz
and
~.
0 < ~ < ~.
256
D1
: P =
(z,~-~)
D2
: P =
(z,~-~)
D3
: P =
(z,~) 7
D4
: P =
(z,~)7.
7 and
Y
and
the
the
family
family
P
P
>
26.
convex
E1 - E4
E3)
El)
Fig.
exists.
V
Configurations
exists.
v
E1
curve
z>
z
: y
is
tangent
an
interval
to
Z--
and,
in
seen
from
y,
Y1
is
a
strictly
z.
Z
case
E2
: Like
El,
role
of
7
and
E3
: Like
El,
but
71
is
a
E4
: Like
E3,
role
of
7
and
It
is c l e a r
of
consider
A 2, them
A 3,
that
all
these
B4
and
B5
71
are
strictly 71
cases
reversed.
concave are
might
curve.
reversed.
be
respectively.
considered Therefore,
as we
a particular will
not
separately.
Configurations
F1 - F8
F2)
F~)
z
~
z
-~
257
F3)
F~) Y]
z
Z
Fs)
Z
F6) 1 z
Z
-
>
z
Y
Fs)
F7)
z
z
> -
z
Fig.
27.
cases
Except
are
tinction
for the
exactly with
In c a s e s P =
(z,0) 7.
the
family
the
fact
same
to the
F 2, F3,
F4
always
that
as
respect
In c a s e s P
z
is t a n g e n t
D 1 - D 4.
F5,
to
Z~
Nevertheless,
t y p e of c o n v e x i t y
and
FI,
y
F8
there
F6
and
of
is o n l y F7
one
z,
we m a k e
Y
one has
at
and
a dis-
71 .
singular P =
these
element
(z,0) 7
and
exists.
V
8.4.
Fix now a billiard
and
LEMMA and w
Yl
8.1. _ f_ o
Y1
~ i
of
F
we
~
of c l a s s
associate
Fo___[ra n y e l e m e n t a r y F
there
exists
the
ZI"
To a n y
subset
iy,yl
confiquration
a number
by,71
two b o u n d a r y =
{ (v,8)ENQ;vET,vI(Y 1
of boundary > 0
pieces
such
pieces
that
y
for all
one has Y'YI
rI
=
r l ( w ) >_ b T , Y l P ( W , A Q )
(8.3.)
258
The
Lemma
implies
9.
8.1.
(8.1.).
PROOF
9.1.
From Lemma
8.1.
9.1.
For
and
Y1
exists
and
arguments
U(s)
it f o l l o w s
to p r o v e
singular of
the
w
6 U(s)
N i
it
s
immediately
that
Lemma.
of b o u n d a r y
pieces
element
and
INEQUALITY
following
s
related
a number
b
- -
for all
to see t h a t
OF THE ~IN
configuration
for e v e r y
a neighborhood
It is e a s y
PROOF
it is s u f f i c i e n t
any elementary
F
9.
for the r e a d e r .
P A R T TWO:
the c o m p a c t n e s s
of
in Sec.
leave details
O F LEM~LA 7.4.
prove
LEMMA
is p r o v e d
We
to
7
to it t h e r e
> 0
such
that
s
one has Y'YI
rI Thus,
=
rl(w)
to p r o v e The
proof
inequality singular ration
t bsP(W,A~) .
the T h e o r e m of L e m m a
(9.1.)
element G
and
related
since
to s t u d y
configurations Sec.
consists
for a n y to
it.
of
to p r o v e
We
have
nothing
denoted
are
cases
it r e m a i n s
In t h i s symbol
for a n y
for c o n f i g u -
particular
respectively,
by t h e i r
the
and
to p r o v e
E1 - E4
9.1.
that
configuration
31 c o n f i g u r a t i o n s .
be s i m p l y
only Lemma
the v e r i f i c a t i o n
elementary
A 2, A 3, B 4 a n d B 5
the remaining will
it r e m a i n s
the c o n f i g u r a t i o n s
of the c o n f i g u r a t i o n s only
7.1.
9.1.
is t r u e
(9.1.)
section
these
introduced
in
8.3. In fact,
few elementary proofs lated
we will
prove
separately
configurations
are analogous
or t h e y
because follow
the
inequality
for t h e o t h e r
from
(9.1.)
ones
only
either
the Proposition
9.1.
for a
the
formu-
below. All
the p r o o f s
condition
F,
them.
Notice
in
of
M
proved. denoted
only
a r e of v e r y the
that we will
singular
by the
same
define
for w h i c h
element
symbol
nature
geometrical
never
elements
For a singular
elementary
simplest
s
U(s).
explicitly
the e s t i m a t e
all
these
and,
except
considerations
in
the neighborhoods (9.1.)
will
neighborhoods
Nonetheless,
for the
are used
this will
be
will never
be lead
to
ambiguity.
9.2.
First
PROPOSITION mentary for a n y
of all
9.1.
let us p r o v e
Let
the b o u n d a r y
configuration. singular
the
element
If
Y1 Pv =
following
pieces
seen (V'ev)
from
y 7
6 fy,yl
proposition.
and
Y1
form an ele-
is s t r i c t l y
concave,
NAq 4
is n o t
which
an
then
259
isolated bp
point
of
4
[
N A~,
the
inequality
(9.1.)
holds
and
rigorously
with
Y'YI
= 1/2. v
Proof.
It is g e o m e t r i c a l l y
in the p r o o f v
on
that
7 P~v =
function
of T h e o r e m
such
of
that
(~,@v) ~
evident
6.1
that
for e v e r y
( t Y'71 (see Fig.
N AQ4
there
Q 6 Uv and
28 w h e r e
it was exists there
such y
that
demonstrated
a neighborhood exists @v~
is s t r i c t l y
an a n g l e
Uv
of
@~
such
is a c o n t i n u o u s convex).
260
Fig.
28.
Let has
not
!@v -@I,
us d e n o t e
that
only
y
6 :
sin
(v,0).
from
w =
p(w,A~)
6
0 < 61 < 7/2,
S p(w,P~)
is s u f f i c i e n t l y
depend
on
Notice
that
and
Y1
This
v E U
V
=
9.3.
A 1 - In this
7
one
has
proof
the
ends
of
y
of
y
only
it is s u f f i c i e n t neighborhoods that one
case
singular
it e a s i l y
one
81(w) I, if
where
60
does
v
is a c o m m o n
point
of
the v a l i d i t y
D 2, F 1 and F 2.
The
of L e m m a
remaining
7.4.
twenty
two
separately.
for any
point
elements, such
element
that
to p r o v e
to p r o v e
that
and (u,Q)
for
U(qv)
v
qv =
one
w =
when
implies
follows
U(qv)
for e v e r y
the c a s e
immediately
be c o n s i d e r e d
two
61 = 2 1 s i n
0 < 6 ~ 60
Y1
•
for B 4, B 7, B 9, C I, C 3, C5, will
if
of
excluded.
proposition
cases
i.e.
@l(W) I.
the c o n c a v i t y
6 S 61 ~ 2 sin
small,
Isin
.
in this
is not
Thus,
61 =
exists. the
every in
E U(qv ) n
belonging (v,0)y
M
From
point
and
of At
the c o m p a c t n e s s (9.1.)
v E 7
a number w =
interior
q v .= (v,~)y.
inequality
and
iy,y
to the
and
there bv > 0
(u,Q)
for
A1
exist such
E U(qv) A [ Y'YI
has rI
:
r l(w)
-> b v P ( W , A g )
rl
:
rl (~)
>- bvP (w,Ag)
As
qv
(9.2.)
and
and
q5
play
a completely
symmetrical
role
in the a b o v e
261
inequalities,
it is s u f f i c i e n t
To p r o v e Cv > 0 w =
and
(u,0) tan
(9.2.)
it is e n o u g h
a sufficiently
6 U(qv)
01(u,9)
to p r o v e
N
i
one
~ C v tan
to s h o w
small
Y,Y
only
(9.2.)
that
there
neighborhood
exists
U(qv)
a number
such
that
for all
has
9
(9.3.) C
From in
(9.3.) U(qv)
one
N
piece
any y
restriction
is so small
In the c o - o r d i n a t e and
Z2(v), Now,
v 6 y
duced
Y
~
related
in Sec.
7.5.
in some
of a s t r i c t l y F.
When
v
one-sided when
Let
w =
f(s)
< f(t),
(d)
s,
to the
(9.2.)
is true
with
some
an end
(u,8) We w i l l (b)
of
small.
bv -
v 2
y,
function
us
line
one
v =
Here
lines
il(V)
C I.
(v,9)
where
Z2(v),
In this
(0,0),
bound-
the c o - o r d i n a t e
and
F.
y
intro-
co-ordinate
is the
satisfying
considers
(0,0).
w =
introduce
g = fv
the
of c l a s s
Zl(V)
the c o n d i t i o n
then
v =
that
is v e r i f i e d .
straight
the p o i n t s
Let
straight
function
of of
convex
Consider
assume
condition
to the
of the p o i n t
smooth
is an end
we can
following related
defining
neighborhood convex
is not
(t,fv(t)).
of
(9.3.).
while
neighborhoods
v
(x,y)
graph
prove
the
is s u f f i c i e n t l y
(x,y)
system
that
of g e n e r a l i t y that
system
is the
we w i l l
and
system
immediately
Ly,y.
Without ary
deduces
graph
the c o n d i t i o n
the c o r r e s p o n d i n g
we w i l l
consider
the c a s e
y.
( /y,y.
Let
distinguish s < 0 ~ t
u = four
and
(S,fv(S)) cases:
f(s)
and (a)
{ f(t),
let
u I = Ul(U,G)
s S 0 < t (c)
and
s, t { O,
t ~ 0.
Consider
first
the c a s e
(a) w h i c h
is p r e s e n t e d
Y
29.
y=g ( x / U
v=(0,0) u
Fig.
on Fig.
29
/
t
1
=
262
Clearly, Thus
tan
+g(t)
@ =
~
+
and
one
considers
To
prove
(9.3.)
to
and
prove
=
tan
only
cient
~
tan(a+6)
- g(s) - s
t if
8 =
s
that
and
the
for
:
1
B-
6.
where
tan
f (t) t
6 =
-
f (s) s
t a n ~ + t a n 6 ~ t a n a + t a n .4 = -g' (s) i-tan ~ tan 6 tan(B-a) : tan 5 - tan ~ 1 _g(t)-g(s)) = 1 T-tan B ~n &-2(g' (t) t - s
@i
in
8
t
sufficiently
case
some
(a)
C
>
under 0
small.
consideration,
and
for
all
s,
it t,
s
is
suffi-
_< 0
< t,
I sJ
V
and
t
sufficiently
g' (t)
- g(t) t g(t) - g(s) ts
However the
(9.4.)
small,
-
g(s) s
g =
has
(9.4.)
v
(s)
states
function
£ C
, g
one
f
nothing .
but
Thus,
the
(9.4.)
validity
is
true
of
and
the
one
condition
can
take
F
as
for
C
the
V
exponent In In
all
y. cases
these
It A1
of the
follows y
A2 -
Let
related
to of
(b),
cases
iff
graph
V
from
the
=
proof
the
(d),
convex
the
to
line
that
the
are
We
completely
omit
the
inequality
analogous.
details.
(8.3.)
holds
for
F.
We
will
£1(z)
smooth
proofs
(9.4.).
condition
(v,8) ( /y,71.
straight
strictly
and
arrives
this
satisfies
w
(c)
one
use
and
function
the
co-ordinate
£2(z). g
For
defined
x on
system £
the
0,
y
(x,y)
is
a
interval
[0,A] .
%
£2 (x)
£
,
z=(0,0)
vI
F i g . 30. parallel
v to
=
(t,g(t)) £1(z).
for
some
t
6
A
[0,A] .
The
straight
line
£1 (z)
i
is
'
263
By virtue defined
on Fig.
On the other Thus,
of t h e d e f i n i t i o n
singular
reasoning
for
A2
elements
will
implies
appear
One
has h e r e
Let
w =
(v,e)
z/2
> el(W) that
> 81
5 A~.
that
in m o s t
{ Cz
b > 0
the
one
where
Y1
(8.3.)
(z,e).
is
a p((v,e),A~).
immediately
follows.
of t h e
modifications
the
use
same
to f o l l o w .
=
(v,~)
is a s t r i c t l y
w =
on
qv
~
0 < 81 < ~/2.
as an e x a m p l e
very minor
elements
tan ~
depends
where
inequality
typical
As
E A 95 c AQ w h e r e
p ( ( v , @ ) , (v,~))
of the p r o o f s
singular
) > 0
(v,~)
= ~ - eI
With
( iy,yl
tan el(W)
where
and
is a v e r y of
A3
has be
This
p ( ( v , e ) , (v,~))
~ - eI { p((v,e),A~)
The p r o o f of
30.
hand
5 A2,
of
where convex
However,
and
in c o n s e q u e n c e
Yl
only.
Thus,
the
then
(9.3.)
from
dl(W)
v 6 y. curve,
= sin
inequality
one
el(W) (8.3.)
is p r o v e d .
A 4 - Let the
w =
first when
(v,e)
E iy,71
0 < e < z
L e t us c o n s i d e r exists
a unique
(9.1.)
then
the
e
and
first
We will the
such that
P
V
follows
~ Fig.
31.
~
=
second
case
when
(see Fig. =
(v,e v)
separately:
~ < e < 2~.
31). ~ A4 n
two cases
In t h i s
case
and
inequality
the
there
V
from Proposition
9.1.
£v
e
consider
(see Fig.
31).
~Y1 ~v~ZI
-
ev ,
0
<
~
<
~
1
<
~/2.
264
In
the
second
case
(see
Fig.
32)
to
every
w =
(v,8)
6 i
we Y'YI
associate
w
[V,Vl],
and
=
(v,n-6)£z,
moreover
w
(i.e. and
~ w
( £z ), v are
belongs
to
the
interval
parallel.
1 £z
< Z
Fig.
32.
Then,
from
depending (9.1.)
~ =
A 3 we on
~ = w -
know
that
Y1
only.
the
minor
As
eI .
sin
Y
~ bB
8 > e
where
and
sin
b
> 0
~ =
sin
is a c o n s t a n t 8,
the
inequality
follows.
A 5 - With lines
8 - w,
as
modifications
the
proof
goes
along
the
in A 2.
B 1 - Here
there
is
only
one
singular
element,
P =
(z,~)y.
same
265
B
v
Fig.
33.
Let
0 < ~ < ~,
w =
(v,8)
Then,
if
8 - a
fig.
33).
The
B 2 - In Let
w =
one
when In
other
for
all
where close
w = to
has
is
where
Y'Yl cases:
the
]z,B],
v
is
p(w,A~)
(9.1.)
there
E i
two
first
hand, 8
p(w,A~).
case
=
where
Y'YI < ~/2 , one
this
convexity
the
( [
71
is
v first
_< p ( w , w )
one
is
(v,~)
=
close
81
to
_< 2 s i n
z. (see
81
proved.
singular
element
sufficiently
one
( A5Q.
sufficiently
then
only
w =
when
P =
close
0 < 8 S e,
to
z.
and
the
(z,~)y. We
will
second
~ < 8 < w.
the
strict
[z,A],
inequality
(v,8)
distinguish
7 :
Y
case of from
the 71 A1
sufficiently (z,8)y. P =
(z,~)y
Thus, one
proof one we
easily
0 < %1
know
that
close
to
finally, has
follows
has
Isin
~
<
there one
for 811
A I. (see
exists
has
w = :
from
6 < ~/2
(v,8)
sin
b
sin
%1 E
From Fig.
> 0
such
~ bp(w,p)
LY,71
61 ~ b ( ~ - 8 )
the 34).
On
that =
b(~-8)
sufficiently = b p ( w , w v)
266
z
>
v
Fig.
34.
[v,v I]
0 < e < ~, are
In t h e one Fig.
has
sin
parallels,
secon~
~/2
Y
61 = wv =
case,
811,
(v,a)
for
> 61 > 0 - ~
Isin
w =
the
intervals
iz,v I]
and
E A~.
(v,G)
in v i r t u e
of
sufficiently the
convexity
close of
to Y1
P =
(z,e)
(see
35).
L(z,~)
Zv I
£1
V
Fig.
35.
Moreover,
0 < e < 7,
if 1
B 3 - Here
61 >- 0 - e >0.
61 < ~/2, 21sin
there
then
GII.
is o n l y
sin
So,
one
the
61 =
Isin
inequality
singular
811.
Thus,
(9.1.)
element
P =
p(w,A)
S P ( w , w v)
is p r o v e d .
(~z,O)y.
Let
7
267
w :
(v,8)
( Ly,yl
and
let
Wv
Y
w
0 < ~ < 7,
p ( w , w v)
( An5
be as
defined
on F i g
36.
z
Fig.
36.
Then,
if
sin
61 =
and
61 ~ 2 sin
w
v
011 , a n d
proof
B 8 - The a
As
in B 2.
ciently 21sin (9.1.)
there far
As
close 811 ,
proofs
vector
for to
where
is p r o v e d .
Thus,
the
are
parallel
are
as
P PI'
PI' w =
close
two
the
P,
same
exactly to
has
then (v,~)
that
O(w,An) E An
as
in
the
same
~ p(w,w) (see
as
Fig.
is p r o v e d .
in B 2 if o n e toward
p =
proof
w =
_< p ( W , W v ') = (9.1.)
directed
the
0 < 61 < 7/2
B 2.
elements,
if
has
p ( w , A n)
Zl(Z)
singular
one
inequality
is c o n c e r n e d , one
to
consequently
811.
is e x a c t l y
to be
C 2 - Here
as
Isin
B 6 and
(z,~)y I.
sufficiently
61 = 2 1 s i n
B 5 - The
w
is
= S I.
(z,0)Yl
is e x a c t l y
(v,8)
( Ly,yl
= 6 ~ 2 sin 37).
Thus
defines
YI"
and
same
is
suffi-
6 S 2 sin
the
P1 =
the
61 =
inequality
268 y ~v
8
v
z
Fig.
Y1
37.
0 < 61 < 7/2,
C 4 - Here is the
same
as
we h a v e in
sin
one
C2
6 =
lsin
singular
(see Fig.
8 I , sin
element
61 =
P =
Isin
811 • and
(z,0)
the p r o o f
38).
7
I
V
vI
Fig.
38. Before
0 < 61 < z/2, proceeding
61 > 8, sin
to the p r o o f
61 = of
Isin
DI,
D3
@if. and
D 4,
let us n o t i c e
269
that
in
D1
and
always
either
of t h e
boundary.
D3
a point
is a p o i n t
In
In the
sition
Thus,
of g e n e r a l i t y ,
case
represented
line
Z~
D4
case
E 8n
or a p o i n t
two possibilities
latter in
is s u b s t a n t i a l l y
two cases
inflection
D4
that
D 1 - In t h i s proof
of
straight
of d i f f e r e n t i a b i l i t y
inflection. 9.1.
the
of
F
~
is h e r e
of n o n d i f f e r e n t i a b i l i t y
occur:
which
the proof
we can also
because
follows
suppose,
either
Z~
is n o t a p o i n t directly
without
E 8n
or
of
from Propo-
any
restriction
£[ E 8~.
there the
is o n e same
o n Fig.
as
singular
element
P =
(z,~-e)y.
in
Like
B2
we distinguish
B 2.
39 a n d Fig.
in
40 r e s p e c t i v e l y .
We omit
The
the
details.
Yl
vI
Y
vi
vI
8
z
Fig.
39.
w
v
~
0 < ~ < ~, the
0 < ~i < 61 < ~ 2 ,
sin
;~-
intervals
~i =
[v,v I]
Isin 81[ , w v
5 E An
and
[Z,Vl ]
is p a r a l l e l
are to
parallels, ~--. z
270
~v
~v
/z
z
Fig.
40.
sin
61 =
5 0 < e < z, w v ( A ~
Notice ~ 0,7. F3
and
that Thus,
in this
proof
the above
to
gz'
0 < 6 < 61 < 7/2,
we have never
proof
gives
the
used
the a s s u m p t i o n
inequality
(9.1.)
that
also
for
F4.
D 3 - in t h i s c a s e the
is p a r a l l e l
Isin @ii.
family
element
P
Consider
Pv =
v now
there
(V'@v)
is c o n c e r n e d , w =
(v,@)
are possible:
either
line
g--
not.
or it d o e s
singular
(see Fig.
the proof
E iY,yl
cases
z
is the
exists
the The
element
41).
follows
sufficiently interval first case
As
P =
(z,~)y
far as the
from Proposition close
[v,v I]
to
P.
cuts
the
is p r e s e n t e d
and
singular 9.1.
Now
two
straight
on Fig.
41.
271
7
1
v)
Z Z
/ Fig.
41.
In this v
0 < ~ < 7,
case,
4
.
does
81(w).
Notice Thus,
i~,
that
case
In this
I [Vl'A] I [v'A] I
=
deJ -from
in this
6 @i
6 8 l(w)
4
the
= 8 - 8 v = 6, sin
z, and
s sin
then
Thus,
the a b o v e
D4 This
to
I [V 1 , A ] I _< ~
not c u t
2 sin
close
P(W'Pv) ~ I
S
sin sin
because
sufficiently
@(w,A~) T ~
p(W,Pv)
for all
Proposition
proof
the
case
there
is o n l y
on Fig.
9.1.
42.
one
811.
v
E y, v # z,
v
one
l[Vl'A]l - 2 ~ - -
interval has
S
[v,v I]
O(w,AQ)
is p r o v e d .
used
inequality
Isin
for all
If the
(9.1.)
we n e v e r
gives
is p r e s e n t e d
8 > 8
C < + ~.
proof
then
sln @ sin 8 l(w)
S 2
inequality
61 =
the a s s u m p t i o n (9.1.)
singular
also
for
element
that
~ ~ 0.
F 7.
P =
(z,e)
Y
272
L~
vvv
Fig.
42.
sin
S1 =
0 < e < 7, Isin
Notice
that
0 < S 1 < ~/2.
Notice the
if As
< @i S 2 s i n
Thus,
that above
family
w =
(v,@)
usually,
in t h i s
P
6 M
for
61 = 2 1 s i n
911.
proof
is
such
Wv
6 An5
is p a r e l l e l
we
the
case
there
is t h e
exists.
one
the
never
gives
always
sufficiently w
Thus
proof
F 5 - In t h i s the
0 < 6 < @l < 7/2,
has
close p(w,Ag)
inequality
used
inequality
the
Consider
also
element
w =
(v,8)
straight
of
z,
of
z.
As
for
(b)
In t h e
in
the
P,
then
that
for
P =
=
is p r o v e d .
assumption
(9.1.)
singular
to
~ ~ 0.
F 8.
(z,0)y
~ i
and
where
w
Y'Yl
sufficiently
the
Zz,
S p ( w , w v)
(9.1.)
v is
to
8iI.
close
line
cases
P.
L(z,@ )
it c u t s
case
to
the
Three
cuts
the
interval
(a)
and
(b),
(c),
the
proof
possibilities straight
(z,z),
the is
proof the
may
line
(c)
£~
it c u t s
is e x a c t l y
same
as
for
the the
occur at
here:
the
Z-z same
at
as
singular
(a)
right the
left
in
D 3.
element
C2 .
F 6 - Here
the
singular
elements
are
exactly
the
same
as
in
F 5.
273
Among
the
cases
sented
here.
As
as
far
the
Proposition
i0.
Let
which
will
the
case
(c)
us be
Fr
centered the
delete
The
difference
in
lisions
with
in Sec.
Secs.
proceed
flows the
four
billiard
in
H
where H
all
with
which
and
(c)
the
are
same
follows
(
of
repre-
as
in
directly
D 4.
from
If
(4,6)
the
reflection,
M
then
crossing
of
opposite
case.
L
M
~(4,6)
MLi
of
=
the
and
(41,81 )
the
in
H
Now
let
proceeded
us
define and
directed
toward
the
linear
elements
for
the
denote
by
of t h e 4 U L duri=l ±
L =
[0,~]
the
is n o t
sections
of
us
col-
horizon
trajectory
let
= L. × i
billiards
as w e
Indeed,
crossing
consider
[Sin] 2.
length.
supplementary
Any
Let
like MF =
in FrX
interior
which
[0,~], of
belong
MLi.
defined
when
3.
and
here
can
in
consecutive
infinite
square.
are
different
is n o w
of
H
by
One
Sinai
two
billiard set
consecutive
where
of
says,
scatterers.
at m o s t
O M F,
in
between
the
unit
identification
@
H
the
two
elements
the
H.
C.
opposite
the 1 0 < r < ~.
r,
by
billiard
~ 2/Z2
bounded
r radius
by Ja.
T2 =
identified
D
boundary
as o n e
in
with
disc of
time
of
the
length
to t w o
mapping
the the
torus
square
and
singular
not of
undergoes
linear
obvious
a
insert are
( O MLi) i=l
simultaneously The
proof
the
studied
in c a s e
sides
period
M =
we
unit
with
bounded;
obtain
that,
L1 - L4
2,
(b)
is e x a c t l y
dimensional
(1/2,1/2)
between
If w e
6, w e w i l l
billiard
Sec.
cases
the
torus
2 - 9 is t h a t is n o t
a time
the
the
extensively
F
To a v o i d
two
manifold
in
main
with
at
obtained H
flat
from
billiard
studies
ing
the
identified
circle
bounded.
only proof
is c o n c e r n e d ,
consider
us
denote
F5 the
REMARKS
Let
the
in (b)
•
sides.
us
case
9.1.
FINAL
i0.i
enumerated
In
exactly
(41,81 )
colision
with
correspond
in t h e
where Fr to
same
(41~8 I)
way
as
in Sec.
correspond
takes
place
before
the
crossing
of
to
the L
in t h e
2.
274
Fig. 43. A t r a j e c t o r y of the plane b i l l i a r d in the d o m a i n by the unit square and the circle F is designed.
~
bounded
r
Fig. 44. A t r a j e c t o r y of the toral b i l l i a r d in H. The intervals of the t r a j e c t o r y between two c o n s e c u t i v e c o l l i s i o n s are m a r k e d by the same letter, the c h a n g e of the index indicates the c r o s s i n g of L.
275
It from
is e a s y
Secs. Let
horizon" out
us
to
remark
is n o t
for m o r e Finally
many
other
when
our
see
that
2 - 9 remain that
us
sections
results
the
specific
general let
for
true
such
word
above
to
the
defined
mapping
~,
all
results
by word. reduction
considered
of
the
case
example
and
of
"unbounded
can
be c a r r i e d
situations.
note
that
which
apply.
instead
also
of
provide
section the
L
reduction
one
can
choose
to
the
situation
APPENDIX
A.I.
2.
The proof
the o r i g i n a l [Led]2
and
of O s e l e d e c
paper
M(m, IR)
entries
and by
THEOREM.
Let
some ing
subsets
theorem
[Ose].
[Rue]2, 3. The
t h e o r e m are p r o v e d By
OSELEDEC MULTIPLICATIVE
in we
Other
denote
the
dimensional
X.
Let
below found
c a n be in
versions
[Rue] 3 • set of all m
× m
the g r o u p
be a p r o b a b i l i t y
of s p a c e
can be
THEOREM
found
[Rag];
in
see also
of the O s e l e d e c
in
GL(m,]IR) c M(m,]19) ~
formulated
proo~
infinite
[Man] 2 and
ERGODIC
measure
T :X ÷ X
matrices
with
of a l l i n v e r t i b l e
defined
real
matrices.
on a J - f i e l d
be a m e a s u r a b l e ,
~
@
of
preserv-
transformation
T.I.
Let
tx
A : X ÷ M(:m, IR)
log+llA(x) iid~(x)
be a m e a s u r a b l e
mapping
such
that
< +~.
+ where
log
a
measurable
subset
(T.I.I) {0}
= max(loga,
For
: L0(x)
Y c
every
and
X
c...c
linear
exists
c y
there
)
,
exist
a
T
~(Y)
=
Ls(x) (x) = IRm,
subspaces
__°f
]19m
and
that
the
following
logl]A(n,x)v]l
where
:
=
for
convergence
A(Tn-lx)
The tic
every in
numbers
exponents.
is c a l l e d
for all
that:
filtration
that
Ll(x)
vectors
limit
exists
(i)
1
• A ( T n - 2 x ) • ... • A(x) ;
-~ < X I ( X ) < X 2 ( x ) <...< Moreover
such
X. (x)
n
A(n,x)
1
where
such
= Li(Tx)
n-~+oo
invariant
a measurable
v ( L i ( x ) \ L i_l(x) , 1 _< i _< s(x),
lim
A(x)Li(x)
there
T(y)
x ( Y
c L2(x)
are
Then
(i.e.
point
c Ll(X)
{Li(X)}l_
0).
(1)
linear
subspace
is u n i f o r m
number
the m u l t i p l i c i t y
ki(x) of
the
< +~.
F c L (X)l , F n Li_ l(x)
for v e c t o r s
{Xi(X)}l~i!s(x) The
Xs(X)(x)
are
v C F,
called
exponent
characteris-
- d i m Li_ l(x)
Xi(X)-
the
!)v!! = i.
the L y a p u n o v
= d i m L.(x)l
= {0},
277
(T.I.2) Im(X)
For e v e r y p o i n t x ( Y, let us d e n o t e all L y a p u n o v
Xi(x)
is r e p e a t e d
with
Xl(X),
k.(x)1
times;
the f o l l o w i n g
let fl(x) ..... fm(X) for e v e r y
characteristic
exponents
i.e.
k2(x)
by
ll(X)
the first kl(X) with
be an a r b i t r a r y
S 12(x)
at x, w h e r e
X2(x),
~ ... Z
any e x p o n e n t
of t h e m c o i n c i d e s
etc.
Now,
for x ( y,
f a m i l y of b a s e s of IR m such t h a t
i,l ~ i ~ m,
lim _in Then,
log
Ild~(fi(x))!I
= I.(x).i
if m ~ 2, for e v e r y
two n o n - e m p t y
disjoint
subsets
p , o c {1,2 ..... m} we have: 1 log y(d@~(Ep) , d@x(EQ)) n --n-
lim
= 0,
n+~
where
Ep and EQ d e n o t e
vectors y
{fi(x) } ieP and
is d e f i n e d
T.2.
in Sec.
If m o r e o v e r
I
the l i n e a r
T
subspaces
of IRm
{fi (x) }isO r e s p e c t i v e l y
2.11
(see Sec.
is i n v e r t i b l e ,
3.1 of
A(x)
spanned
by the
and w h e r e
the a n g l e
[Rue]3).
6 GL(m,IR)
and
log+HA_l(x)lld~ (x) < +~ M
then there
exists
a
T-invariant
T(y I) = Y1 ) , ~ (YI) = i, (T.2.1)
measurable
subset
Y1 c
X
(i.e.
such that
for e v e r y p o i n t
there
x ( Y1
exists
measurable
a
decomposition
s(x) ]I~m
=
such that
@ i=l
H. (x) , i
A(x)H
(x) = H. (Tx), 1
all v e c t o r s
1 <_ i <_ s(x)
and that u n i f o r m l y
for
!
v ( H
(x),
llvll = 1
one has the
finite
limits
1
lim lo@llA(n,x)vll n n+_+~ where
for
n < 0
=
we t a k e
X. (x), 1 A(n,x)
i = 1 ..... s(x)
= A(T-(n-l)x)
• ...
.A(T-Ix)
.A(x) .
278
(T.2.2)
For every
lim n+_+~ where
1 -~-
Ep a n d EQ h a v e
Let
x ( Y1
l o g y ~d~n (Ep) n(EQ) x , dCx ) = 0,
L e t us n o t e A.2.
point
the
same meaning
as
in
(T.I.2).
t h a t n o w k. (x) = d i m H. (x). 1 1
+ IRm be a linear mapping. For every k, 1 Ak l e t us d e n o t e by B the k - t h e x t e r i o r p o w e r of B a n d let A0 m k m take B = Id. L e t us d e f i n e t h e o p e r a t o r BA: • AIR m ÷ ~ k=0 k=0 BA m Ak =
~
B : ]Rm
_< k _< m, us k ]Rm A by
B
k=0 The role
following
in Sec
PROPOSITION. for e v e r y
proposition
12 of P a r t Let
x
II a n d
X, M,
Let
us d e n o t e
characteristic repeated Xl(X)
Prom
U, T, A(x)
[[ : X
by
(T.I.I)
at
times;
following and
i
ll(X)
exponents
k.l(x)
the
Ruelle
([Rue] 1 ) p l a y s III a n d
and
Y
an i m p o r t a n t
IV.
be as
in
(T.I) .
Then
( Y
l i m 1 logI[ ( A ( n , x ) ) A n n÷~
Proof.
of
in P a r t s
[ Xi(x)ki(x) (x) >0
_< }~2(x) ~..._< Im(X)
x,
i.e.
k 2(x)
where the
with
(T.I.2)
(2)
any
first
exponent kl(X)
X 2(x) ,
it e a s i l y
of
all L y a p u n o v Xi(x)
is
them coincide
with
etc.
follows
that
for e v e r y
k,
I _< k <_ m, Ak lira 1 logii(A(n,x)) n+ ~ n
and this
implies
(2).
II
We
=
m ~ ~i (x) i=m-k+l
leave
the details
to t h e
reader.
D
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M. Wojtkowski, Invariant F a m i l i e s of Cones and L y a p u n o v Exponents, Ergodie Th. 8c Dynam. Sys. 5, 1985, 145-161.
[Woj ] 2
M. Wojtkowski, P r i n c i p l e s for the design of billiards with n o n v a n i s h i n g L y a p u n o v E x p o n e n t s (preprint, 1984) Note added in proof R e c e n t l y A. V e t i e r in papers
[Vet] I
A. Vetier, S i n a i - b i l l i a r d in potential field (Construction of stable and u n s t a b l e fibers), in Limit T h e o r e m s in P r o b a b i l i t y and Statistics, Vol. II, 1079-1146, Ed. by P. R4v4sz, Colloq. Mat. Soc. J. Bolyai 38, N o r t h Holland Publ. Comp., 1984.
283
[Vet] 2
A. Vetier, Sinai b i l l i a r d in p o t e n t i a l field (Absolute continuity), in Proc. of 3rd P a n n o n i a n Sympos. on Math. Stat., 341-351, Publ. House of Hung. ~cad. of Sci., Budapest, 1983,
studied d i r e c t l y a class of dynamical systems w i t h singularities, which seems to belong to the class of systems studied in the present book. See also [Kub] 1
I. Kubo, P e r t u r b e d b i l l i a r d systems (1976), 1-57.
I, Nagoya Math. J. 6_!i
[Kub] 2
I. Kubo, H. Murata, P e r t u r b e d b i l l i a r d systems II; B e r n o u l l i properties, Nagoya Math. J. 81 (1981), 1-25.
[Rob]
M. Robnik, M. V. Berry, C l a s s i c a l b i l l i a r d s in m a g n e t i c J. Phys. A: Math. Gen. 18 (1985), 1361-1378.
fields,