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v(t#) T x E In the case T = (IN,+), Act ( T , X ) ~- C ( X , X ) via the correspondence noted in (1).
85 (3) What is a contractive semiflow? Define the ratio r 0 of a semiflow 0 in X to be the least r e [0,oo] such that Yt E T
Vx, y e X
d(x t ,yt) < r t d(x,y), i.e. Vt E T rot<_ f t . Then 0is contractive if r 0 < 1 . Note that if T = ~ with 0 generated by f , r O= r f , so 0 is contractive iff f is a
contraction. (4) Is there an analogue of Banach's contraction mapping theorem for continuous semiflows? Yes. Every contractive continuous semiflow 0 in a nonempty complete metric space X has a metrically attractive f x e d point, i.e. a point p such that Vt Pt = p and Vx E X
d(x t ,p) strictly decreases to 0 as t ~ ~ . of each Ot for t > 0.
This p is necessarily the unique fixed point
The result can be proved nonstandardly using transfer of
Banach's theorem as follows. Take infinite n e IN*, let T = 1In and let f = 07. Then f is a ,-contraction, so let its fixed point be p say. As f n = 01, p = fix f n = fix 01 E X. p is fixed under 0 since Vt > 0, where m r E #t then p = f m p = Pmr E # Pt so p = Pt" Lastly, Vz E X, d(z t ,p) strictly decreases to 0 as t -* c¢, using that d(z t ,p) = d(z t 'Pt )
< r o t d(z,p) and, if xt ~ p and s > O, d(xt+ s ,p ) < r 0 s d(xt ,p) < d(xt ,p). Assume now that our metric space X is locally compact, and suppose ® is a nonempty compact set of contractive continuous semiflows in X (with respect to the topology described in (2)), w i t h ~
®r 0 < 1. We now sketch the nonstandard method
of obtaining a continuous semiflow i n ~ X
(the 'union semiflow' of O).
Taking
infinitesimal r > 0, let F = {07 I 0 E ® *}, which belongs to (Admis X ) *, and has invariant set K say, the fixed point of the *-contraction [J F of ~ X *. Then if all is well (note; we may need to impose a further 'taming' condition on O for this, and perhaps also assume X is boundedly compact;
the details have not yet been fully
checked), we should then be able to obtain a continuous semiflow ¢ i n ~ X such that for t E [0,oo) and nTe #t, ([J F) n e # ¢t ; intuitively U F serves as an 'infinitesimal generator' for the semiflow. And this semiflow will be contractive, with fixed point
st K (the 'attractor' or 'invariant set' of O, as it would be called). As mentioned above, some details remain to be checked; but the work should go through. As a simple example, if O consists of three continuous semiflows contracting steadily about the vertices of a triangle, the invariant set will be the triangle in question.
Chapter 4 Views and Fractal Notions
Section 0. Introduction 1. Views and Similarities 2. 3. 4. 5. 6.
Relative Strength of View Structures and Similarity View Structures View Self-Similarity Self-Similarity of Some w-Extensions of Invariant Sets The View Topology A Definition of 'Visually Fractal'
7. Notes, Questions, and Suggestions for Further Work
Page 87 89 95 98 102 107 116 125
87 O. Introduction
This chapter finds its origins in what was perceived to be something of an imbalance in the study of 'fractals', a term which in the absence of any universally agreed definition is being used here informally to refer to sets having 'detail beyond detail' in some sense, perhaps also having some form of self-similarity. Namely, whilst a considerable amount of work seems to be under way on the dimension-theoretic side (studies involving Hansdorff dimension for example), there seems to be httle being done on the more vmual side, and the present chapter represents an attempt to begin redressing the balance. Specifically, the simple idea of a 'view' is introduced as the basic constituent in a framework designed for use in studying the structure of sets within a given space X. Particularly in the case of X = ~2, the type of structure studied can be thought of as Idsugl structure, as the term 'view' already suggests. Imagining looking down on a subset A of the plane X = ~2, what we'd see would be a region D of X along with the part A n D of A lying adthin that region. Such a pair
(D,A N D) is an example of what we'll formally be defining as a 'view of A', D being the 'domain' of the view, and it should be apparent that this definition can be genera~zed not only to subsets A of any topological space X but, in [ullest generality, to subsets A of arbitrary sets X. This then is the basic idea of views, which can be used to express a variety of concepts to investigate. Indeed, the formal notion of a view was originally designed in pursuit of formulating one of these concepts, namely a certain type of self-similarity of closed subsets A of ~2. The rough idea was that wherever one looked at A one should see the same sort of structure. Or, rephrased somewhat more definitely, that every nonempty view of A should be embedded in every other nonempty view of A, in a sense suggested by the picture below;
a view of A
another view of A
This definition of embedding implicitly involves a group of transformations of X, namely the group of direct similitudes, which also induces an obvious and closely related notion of when two views are similar (namely when some direct similitude maps the one view to the other). However, in a like way any group of transformations of X gives rise to a notion of similarity and embeddings of views so it will be natural to generalize the framework accordingly. In the general setting we'd abstractly refer to the elements of the group C involved as 'similarities', modelling things on the primary
88 example in which X = ~n and the similarity group G consists of the direct similitudes. Another natural similarity group to consider in the case of ~n would be the group of affine bijections. Most of the view-related definitions we'll introduce, particularly that of a 'view class', will involve the presence of a similarity group. We now outhne the work to follow. In Section 1 we give in full generality the basic definitions concerned with views and similarities, with remarks towards the end on the case of topological spaces which will form the setting of subsequent apphcations. Section 2 is concerned with what might be thought of as the relative 'visual power' of the view structures described in Section 1, and also of the augmented structures in which a similarity group is present.
In Section 3 we formally introduce the above-
mentioned notion of self-similarity along with some related concepts, proving a few properties of self-similar closed subsets of a Hausdorff space, and in Section 4 we show that certain 'w-extensions' of Hutchinson's invariant sets are self-similar. In Section 5 we study three topologies arising in the context of views of closed subsets of a Hausdorff space; foremost is the 'view topology' on the set of views, and this induces a topology on the closed sets and a topology on the 'view classes'. Section 6 uses the view topology in defining when a subset A of ~n is 'visually fractal' at a point x E A, namely expressing that as we zoom in on x, what we see never settles down, so that in this sense there is detail beyond detail. If this holds at all points of A we say that A is 'visually fractal', and it is subsequently shown that many invariant sets have this property.
Finally we have Section 7, providing a list of notes, questions, and
suggestions for further work.
89
1_=. Views and Similarities In this section we give in full generality the basic definitions concerned with views of subsets of a set X. We start with the framework in which views are defined.. A view space is a pair (X,.~) where X is a set and.~ is a view structure on X, namely a set of nonempty subsets of X. X is called the domain of the space and.~ the view structure of the space, its elements being called the view domains, thought of as the 'observable bubbles of space', more accurately as the regions of X one can see at a single glance. The primary examples we have in mind are where X is ~n and.~ is the set of open balls (the usual view structure on Rn as we'll call it), in particular the case of 9~2, which the pictures in this and later sections illustrate. A key point here is that the view domains are all bounded, although they also cover arbitrarily large regions ; such a view structure represents the property of being able to see only bounded regions at a glance, albeit arbitrarily large ones, which can be considered an idealization of reality. Returning to the general situation, with (X,.~) in mind along with a subset
Ob (the object set) of 2 X whose elements we shall call objects (thought of as the 'objects of study') we make the following definitions concerned with 'views' of objects. For D E .~ and A E Ob, the D-view of A is DA = (D,A N ]9). An entity v of this form is called a view, this particular one being a view of A. D is the domain of v, denoted by
d o m e , and A N D is the object part of v, denoted by ob ~, thought of as the part of an object (which you might like to think of as being coloured black in an otherwise white space) visible in D. If the object part of v is a//of A we may say v is a whole view of A. The set of views will normally be denoted by Y. However, for $ C.~ and ./gc Ob we define g . / g = { E A [ E E $ and A E ~ } , giving us the more explicit notation .~ Ob for the set of views, should it be desired. We generally abbreviate ${A} to gA, so in particular the set of views of A is denoted by .~A. The set { A N D I AE Ob and D E . ~ } of obiect Darts will be denoted by ObParts. In the following, the letters D and E will be assumed always to denote view domains whilst A, B, C will denote objects and u , v , w will denote views. It is trivial but central to realise that an object A is not necessarily recoverable from a view of A since a view only reveals the part lying in its domain. This prompts the following terminology.
We say an object A is consistent with a view v (or v is
consistent with A ) if v is a view of A, i.e. if ob v = A N dora v. This expresses that we
90 could be looking at A. Generalising, we say A is consistent with a set ~ of views if it
is consistent with every element of ~, i.e. ~ c .~A. If there exists an object consistent with ~g we say ~ is consistent. D is said to be a sub view domain of E if D C E. We say u is a subview of v , written u _ < v , if d o m u C d o m v a n d ob u = d o m u O ob v . This gives
a partial ordering of ~, and synonymous to saying that u i s a subview of v we may say that v is a superview of u. We say v is empty if it has empty object part, expressing that no part of an object is visible in v. At the other extreme we shall say v is full if ob v = dora v , expressing that nothing but object is visible in v. For D E.~ we say objects A and B are D-indistinguishable if D A = D B , i.e. A N D = B N D. More generally, for $ C .~ we say A and B are g-indistinguishable if A and B are D-indistinguishable for all D E $, equivalently if A N [J $ = B N LJ $- In the special case $ = .~ we may just say A and B are view-indistinguishable. The observable space is [J ~ , the union of the view domains, and if this is all of X we say .~ is covering.
Another important property relating to what might be
thought of as the encompassing power of.~ is the following. We say.~ is ideal if it is an ideal basis, i.e. for all
D 1,D 2E.~ there's
D 3E.~ such that
D t U D 2CD~.
Intuitively this represents the property that given any two view domains, you can always take a step back to get a view domain encompassing both the former. A subset of X is said to be.~-bounded if it is covered by some element of.~. In the case where .~ is ideal, the.~-bounded subsets of X form the ideal generated b y . ~ .
The usual
view structure on ~n is of course covering and ideal, and ' . ~ - b o u n d e d ' just means 'bounded'. So far the framework described admits no notion of s i m i l a r i t y between views and is consequently rather static in nature, suggesting little in the way of interesting concepts to investigate.
It is the presence of a notion of similarity that will make
things far more interesting.
To this end we'll now assume we have a group G of
permutations of X, under which ~ is closed.
The triple (X, G,.~) will be called a
similarity view space, (G, .~) being a similarity view structure on X . The elements of G (the similarity group) are called the similarities, denoted usually by f , g , h .
The
primary examples we have in mind are where X is ~ n G is the group of direct similitudes (see Appendix 9) and.~ is the usual view structure on ~n;
(G,.~) will be
referred to as the usual similarity view structure on R~. Returning to the general case, with ( X , G , . . ~ ) in mind along with an object set Ob which is closed under G, (for
91
example, the set of closed subsets of 8n in the examples just mentioned) we make the following definitions. We say objects A and B are similar, written A ... B , if B is the image of A under some similarity, in other words if B is in the orbit of A under the natural group action of G on Ob.
The equivalence classes with respect to ~ will be called object classes,
the object class of A being denoted by A~. Sometimes we may just say B is a ~ p y of A if B is similar to A ; and if g E G maps A to B we may write A ~ B . Identical definitions to the above go for view domains ;
D and E are similar,
written D ... E , if E is the image of D under some similarity, i.e. they're in the same orbit, and so on. This time the equivalence classes are called view domain classes. In addition we say D is embedded in E , written D -4 E , if there is a similarity mapping D to a sub view domain of E , in other words if D is similar to a sub view domain of E. If g maps D into E we'll say g embeds D in E and write D-9* E .
Naturally,
D -.q, E h F :~ D h o g F , and ~ is a preordering of.~. Since.~ and Ob are both closed under G, so is the set of object parts, as
g (A fl 1:)) = gA N gD. We therefore have a natural group action of G on V defined by gv=(gdorav,gobv), in other words comprising the actions of G on.~ and Ob Parts working in parallel. The action is equivalently described by g DA = gD gA, since g DA = g(D, A fl D) = (gD, g ( A fl D)) = (gD, gA n gD) = gD gA . The action of G on V gives us the definition of similarity between views ; u and v are similar, written
u,.~ v,
if
there's
a
similarity taking u to v, i.e. if they're in the same orbit. We may synonymously say v is a co~v of u, and if g maps u to v we may write u ~ v. The equivalence classes will be called view classes, the view class of u being denoted by u ~. In terms of similarities we can now define the notion of one view being embedded in another as mentioned and illustrated in the introduction.
Namely, we say u is
embeddecl in v, written u ~ v, if there is a similarity taking u to a subview of v, in other words if u is similar to a subview of v. If g maps u to a subview of v we may say g embeds u i n r a n d write u - ~ v .
Natura~y, u ~ v h w
~ uh°gw,
and
preorders the set of views. Of course, if g embeds u in v it in particular embeds dora u in dorn v. For a set ?gof views we define ~ ~" = {u ~ I u E ?g}. In particular the set of view classes is thus denoted by Y'~, and the capitals [Jr, V, W will be assumed to denote view classes in the following. The elements of a view class may suggestively be
92 called its realizations.
As with views we can think of view classes as being visual
information about an object highlighted in black in an otherwise white space;
but
whereas a view gives you absolute information in that you know exactly what domain you're looking at and what's visible in it, a view class only gives information modulo similarity.
It's as if you're receiving an image which has been taken by a remote
camera whose bearings are unknown ; more precisely the view domain the camera is looking at is only known modulo similarity. cloaked in a coat of uncertainty ;
The transmitted information then is
the original view v has become v ~, which any
realization of v ~ could have given. This then is the significance of the concept of a view class, and informally you may like to think of them as 'views transmitted by remote camera'. A number of definitions regarding views have natural counterparts for view classes. A view class of A is a view class of a view of A ; the set of view classes of A is thus ( . ~ A ) ~. Now just as A is not necessarily recoverable from a view of A, neither of course is it necessarily recoverable from a view class of A, in fact even less so, and we may say A is consistent with V (or vice versa) if V is a view class of A, intuitively expressing that we could be looking at A (with our remote camera).
This definition
formally conflicts with an earlier one in that V is also a set of views, and consistency with A in the above sense does not equate with consistency in the earlier sense of a set of views, but under the sensible assumption that the above definition is the one involved when dealing specifically with a view class, no confusion should arise. Generalizing the definition, A is consistent with a set ~ of view classes if it is consistent with every element of ?Z, in other words if ~ C_( . ~ A ) ~ . consistent with a view class V, so is every copy of A.
Of course, if A is
If there ezists an object
consistent with ?g, in other words if ~ is a bundle of information we could obtain were we looking at a suitable object, we may say ~ is consistent. We say Uis embedded in V, written U---~ IF, if some element of Uis embedded in some element of V, equivalently if a// elements of U are embedded in all elements of V. Thus u "~ ~ v ~ ~
u --* v. The view classes are preordered by ~ .
A view class containing an empty view contains only empty views ; such a view class is said to be empty. So in general, v "~ is empty ¢~ v is empty. Likewise, a view class is full if its elements are full; equivalently, v ~ is full ~
v is full.
The idea of view embedding leads naturally to two basic relations on the set of objects. We'll say a view u is embedded in B , written u --* B , if u is similar to a view of B . Moreover if g maps u to a view of B we'll say g embeds u in B , written u _9. B . Note that g embeds DA in B iff g DA = gD B, equivalently g (A n D) = B N g D , i.e. gA fl gD = B n gD ; this will be frequently used in subsequent work. We now say A is view-embedded in B ,
written A--* B , if every view of A is embedded in B ,
93 equivalently if every view class of A is a view class of B , i.e. ( . ~ A ) ~ c ( , ~ B ) ~ .
This
can be equivalently phrased as the fact that whenever A is consistent with a set of view classes, so is B ; or in short, the possibility of A (i.e. the possibility that we're looking at A) implies the possibility of B .
The relation ~ preorders the set of objects. We
say A and B are view-similar (or view class indistinguishable), written A ~ B , if A ~ B and B ~ A, i.e. (.~A) "~ = ( . ~ B ) ~ , i.e. A and B have identical view classes, intuitively expressing that A and B cannot be distinguished by images from a remote camera. This is of course an equivalence relation, and it's easily seen t h a t . . . 1.1 Note
Similar objects are view-similar.
Proof: Suppose A ~ B .
Then VD E *~, g embeds DA in B since
g DA = g D g A = g D B .
And since g gives a permutation of.~, we thus have A ~ B .
0
Significantly however the converse is false, even in the case of the usual similarity view structure on ~n ; see note 7.10. In the case where.~ is covering but not ideal, counterexamples are more easy to produce. For example, consider the modification to the usual similarity view structure on ~n in which the view domains are instead only the open balls with radius < 1.
Let A be a singleton and B consist of two points
which are distance >_ 2 apart. Then because all the view domains have diameter _< 2, A and B will be view similar ; but they are not similar.
More generally, let A be a
nonempty closed set and B be the union of a disjoint family of copies A i of A each produced from A by an isometry, such that points in distinct copies are always at least distance > 2 apart. Then A and B will be view-similar, but they need not be similar of course. So far we've introduced view structures and similarity view structures on a set X along with the main basic definitions concerning views of subsets of X .
We conclude
this section by describing the natural upgrading of the two types of structure to the topological setting.
We can lead into this by considering a condition on a view
structure.~ on a set X concerned with what could be thought of as the 'resolution power' of .~. Namely, if .~ covers X, and for all x E X and D 1 ,D 2 E-~ containing x there's D 3 E .~ with x E D 3 c D 1 N D r , we'll say .~ is topological. The reason for this is that the definition in other words says that .~ covers X and the intersection of any two elements of.~ is a union of elements of.~; i . e . . ~ forms a basis for a topology on X. The idea of views has thus led us to the notion of a basis for a topology on a set, which in turn leads to topological notions of course. In a different world then, topology could conceivably have developed from the idea of views, but we shall not pursue this speculation here.
Instead we now point out that topologicaJ view structures can
equivalently be considered as the 'natural' view structures on topological spaces, as follows.
94 For a topological space
(X, 0),
a view structure on (X,O) is a view structure on X
which is a basis for 0 (i.e. it's topological and gives the topology of the space); in other words it's a basis for 0 not containing ~.
The pair ((X,O),.~) will be called a
topological view space. Now where for a topological view structure .~ on a set X 0 ~ denotes the topology for which.~ is a basis ,.~ is thus a view structure on (X,O~). So to consider a topological view structure on a set is essentially to consider a view structure on a topological space.
We'll favour the latter viewpoint since in practice
we'll usually be starting off with a particular topological space in mind.
All the
definitions regarding view spaces naturally also go for topological view spaces (except that this time the domain of the space is defined as the topological space involved). Note by the way that for an idea/view structure .~ on a topological space, every nonempty compact set C is covered by a view domain, i.e. there's a whole view of C
(Proof: ~
covers C s o there's a finite subcover, and in turn some element of.~ covers
all the elements of the finite subcover, hence covers C ).
(I(,0) is a pair (G,.~) homeomorphis~ of (X,O) and.~ is a view structure on (X,O) triple ((X,O),G,~) is called a topological similarity view space,
Regarding similarities, a similarity view structure on where G is a group of closed under G. The
and all the definitions applying to similarity view spaces naturally apply here too. Note that the usual similarity view structure on the view structure on the space ~n.
set ~n is
moreover a similarity
95
2:. Relative Strength of View Structures and Similarity View Structures Until further notice let X be a set. say.~ 1 is weaker than or equivalent t o . ~ ,
For view structures.~l and-~2 on X we written -~1 ~ - ~ 2 , if
(1) Every element of.~ i is covered by an element of.~ 2 , and (2) Every element of.~l is a union of elements of.~ 2 . We synonymously say that.~ 2 is stronger th~n or equivalent to.~ 1 . Note that the relation ~ is really the conjunction of the weaker relations given by (1) and (2), which respectively express that the 'encompassing power' and 'resolution power' of.~ 2 is at least as good as that of.~ 1 (noting that (2) can be reformulated as ffor all x E X and D 1 E.~ 1 with zE D 1 , there's D 2 E ~ 2 with xE D 2 c D l ,). In short then, the definition expresses that.~ 2 represents a power of vision at least as good as that of.~ 1 . In particular, if objects .4 and B are view-distinguishable with respect t o . ~ l (i.e. for some D E . ~ 1 D`4 ~ D B ) they're also view-distinguishable with respect t o . ~ 2 , simply by condition (1). We say.~ 1 and.~ 2 are equivalent, written -~1 N - ~ 2 , if . ~ ~-~2 and -~2 ~-~1" The relation ~ is a preordering of the set of view structures on X (since each of (1) and (2) define preorderings), so ~ is an equivalence relation. Trivially,
-~l C.~2 # - ~ 1 ~ 2 -
Equivalent view structures are thought of as representing
equal powers of vision. In particular note that if -~1 "~-~2, objects .4 and B are view-distinguishable with respect to.~ 1 iff they're view-distinguishable with respect t o . ~ . ; or put in the contrapositive, they're view-indistinguishable with respect to-~l iff they're view-indistinguishable with respect to.~ 2 . View-indistinguishability is thus ~/~uivalence invariant ; in general we apply this term to any view-related concept which remains unchanged if we to switch to an equivalent view structure. 2._!1 Proposition
For any view structure.~ on X there's a largest which is equivalent
to .~, namely M ~ = { U I U is a nonempty union of elements of.~ covered by an element of.~}. Proof:
.~CM~ gives .~ < M ~ , whilst by the definition of M ~ , M ~ < . ~ ; M ~ ~ . ~ . And for any view structure $ ~ . ~ , $ < . ~ says $ c M ~ .
so o
Calling.~ maximal if it equals M ~ , i.e. if it has no proper expansion to an equivalent view structure, then 2._22 Corollary
A view structure.~ on X is maximal iff it is closed under.~-bounded
unions, i.e. for any nonempty $ c_.~ such that [J g is.~-bounded, [.J g e .~ . o Also note that
.~<.~
¢~M~CM~,
so . ~ . ~
¢~M~=M~.
Restricted to the maximal view structures then, < is the partial ordering c . generally note that for maximal~2 and any .~ ~ , .~ ~ ~ ~ ~ ~ ,~ ~ c_.. ~ ~. .
More
96 In the case where X is a metric space and.~ is the set of open balls, M ~ is the set of nonempty bounded open sets. Of course, M ~ is topological and gives the same topology as .~, namely that of X. This could have been anticipated by the following ; 2.__33 proposition
For equivalent view structures.~l a n d . ~ on X, i f . ~ is topological
then so is .~ ~, and .~ ~ gives the same topology as.~ ~.
Proof: Firstly we s h o w . ~ is topological. Given any D~, E~ e - ~ we must show there is F ~ E . ~ with x E F ~ C D ~ f l E ~ . As D~ and E~ are unions of elements o f . ~ let D ~ , E ~ e . ~ l w i t h x ~ D ~ C _ D ~ a n d x ~ E ~ C E ~ . Thus x ~ D ~ f l E ~ C _ D ~ f l E ~ , a n d a s.~ is topological let F~ e ~ ~ with x ~ F~ c D~ ~ E t . As F1 is a union of elements o f . ~ let F~ e . ~ with x ~ F~ c_ F~. Then x e F~ C D~ ~ E~ as required. Secondly, since each element o f - ~ is a union of elements o f . ~ and vice versa, .~1 and.~ ~ give the same topology,
o
For any topological view structures.~ and .~2 on X with . ~ < . ~ , the topology given b y . ~ is a refinement of (i.e. is stronger than or equivalent to) that given b y . ~ of course. In short then, the stronger the topological view structure, the stronger the topology. Also, it's worth pointing out t h a t . . . 2.4 Note For view structures-~1 and-~2 on a topological space (X, 0), - ~ < . ~ ¢~ Every element o f . ~ is covered by an element o f . ~ .
Proof: ¢ : In addition we already know that every element o f . ~ is a union of elements of.~ 2 because it belongs to d for w h i c h . ~ is a basis, o We now turn to the matter of comparing similarity view structures. For similarity view structures ( G 1 , ~ 1 ) and ( G 2 , . ~ 2 ) on X we say ( G 1 , . ~ t ) is weaker than or equivalent to (G 2 ,-~2) , written ( G t , . ~ l ) < (G~,..~), if G2 is a subgroup of G 1 and.~ 1 ~<-~ ~ • Synonymously (G 2 , .~ ~ ) is stronger than or equivalent to (G 1 , .~ 1 ) • The relation G 2 C G i implies that under the similarity group G~ there's no more 'spatial similarity' within X than there is under G 1 ; hence view classes with respect to (G 2 ,-~2 ) will be no more ambiguous about what their realizations could be than those with respect to (G~ , . ~ ), and we can expect the ability of (G 2 ,-~2 ) to distinguish objects by view classes to be at least as good as that of (G~, .~ 1 )- Of course, in the case G 1 = G~, ( G1, .~ 1 ) ~< (G2, -~ 2 ) ¢¢ -~ 1 <~-~ ~. We say (G I ,-~I ) and (G 2 ,-~2 ) are equivalent, written (G l , - ~ l ) ~ (G2 ,-~2), if ( G l , . ~ l ) < ( G 2 , . ~ 2 ) and ( G 2 , . ~ 2 ) <~ ( G 1 , . ~ 1 ) , equivalently if G I = G ~ and -~1 ~ - ~ 2 ; in other words we get an equivalent similarity view structure just by replacing the v/ew structure with an equivalent one. The relation < is a preordering and ~ is the induced equivalence relation.
97 2._55 Proposition Let ( G t , . ~ t ) ~ (G2,-~2 ) be similarity view structures on X, and Ob be an object set closed under G 1 (hence also G2 ). Then for A,B E Ob, (1) A -~ B w.r.t. (a2 , ~ 2 ) • A --, B w.r.t. ( a l , ~ l ) (2) A ~ B
w.r.t. CG2,.~2) , A ~ B
w.r.t. ( G , , . ~ , )
Proof: (1) Then for any D t E . ~ t , where D 2 E.~ 2 covers D t and g E G2 embeds D2A in B, gE G t also embeds DtA in B. Thus, with respect to (G 1 ,-~t ), every view of A is embedded in B , i.e. A ~ B . (2) By (1). 0 Putting (2) in the contrapositive, if A and B are view class distinguishable with respect to ( G t , - ~ l ), they're also view class distinguishable with respect to (G 2 ,-~2 ), thus illustrating the remark made earlier about the relative distinguishing powers of < -comparable similarity view structures. It also follows from the above result that the relations --* and ~ are equivalence invariant, i.e. if (G 1 ,*~1 ) and (G 2 ,"q)2 ) are equivalent then (1) A--*B w . r . t . ( G t , . ~ l ) ¢¢ A - . * B w.r.t. (G2,.~2) (2) A ~ B w.r.t. (G1 , - ~ t ) ** A ~-, B w.r.t. (G 2 , . ~ ).
98 3~ View Self-Similarity This section is concerned with the visual notion of self-similarity for the expression of which the formal idea of a 'view' was originally developed.
After the
relevant definitions and a few remarks we'll concentrate on properties of 'view self-similar' dosed subsets of a topological space. The definitions which follow are made with respect to a similarity view space (X,G,.~) and object set Ob, for nonempty objects A and B .
At least in the case
where.~ is covering and ideal, the definitions u ~ B , A -~ B and A ~ B below are powerful strengthenings of the earlier definitions u --* B , A ~ B and A ~ B , as the terminology will reflect. We say a view u is universally embedded in B , written u --~ B , if u is embedded in every nonempty view of B . Intuitively this means that wherever you look at B ( ' a t ' implying a nonempty view) you can see a copy of u within your view. As long as.~ is covering (so there exists a nonempty view of B ), this trivially implies that u ~ B . We now say A is universally view-embedded in B , written A --~ B , if every nonempty view of A is universally embedded in B .
If.~ is covering and ideal, every
view of A can be expanded to a nonempty view of A, hence the above condition A ~ B moreover implies that every view of A is universally embedded in B , which in turn implies t h a t A ~ B .
As long as .~ is covering, the relation
~
is transitive.
However, it certainly need not be reflexive. If A ~ A we say A is view self-similar (abbreviated hereafter tO 'self-similar'). This then is a formal expression of the idea that 'wherever you look at A you can see the same structure', saying that every two nonempty views of A are embedded in one another. A simple example is that of a line A in ~2, the latter having the usual similarity view structure. A and B are universally view-similar, written A ~ B , if A ~ B and B ~ A. Note that by transitivity of ~
this implies A and B are both self-similar. As long
as.~ is coveting and ideal it also implies A ~ B . Two further definitions are of relevance.
We say A is view-embedded in v,
written A --* v, if every view of A is embedded in v. If also v is a view of A we'll say v is a generative view of A, since (.~A) ~ = {u ~ I u _< v }, expressing that the view classes of A are 'generated' from the subviews of v.
Such a view precisely embodies
the visual structure of A modulo similarity. The relevance to self-similarity is t h a t . . 3.1 Note
If (G,.~) is a similarity view structure on X with.~ covering and ideal,
then for a nonempty object A, A is self-similar ¢:~ Every nonempty view of A is a generative view of A.
Proof:
99
Since.@ is covering and ideal we know that A is self-similar iff every view of A is embedded in every nonempty view of A, i.e. every nonempty view of A is a generative view of A. o 3.__22Proposition
Let ( G t , . @ t ) < (G2,.@2) be similarity view structures on X, and
Ob be an object set closed under G t (hence also G 2 ). Then for A,B E Ob,
(1) A X
w.r.t. (C2 ,.@2 ) * A X B w.r.t. (Cl ,.@1 ).
(2) A ~ B w.r.t. (G 2 ,.@2 ) * A ~ B w.r.t. ( G I , . ~ l ).
Proof: (1) Let DtA and EtB be nonempty.@t-views of A and B ; we must show some element of Gt embeds DtA in EtB. Since .@t < .@2 let D 2 E.@~ with D t C D 2 , and let E 2 E.@2 with E2C_E t and E2Bnonempty. As A ~ B with respect to (G 2,.@2) let #E G2 embed D2A in E2B. Then g E G t also embeds DtA in EtB. (2) By (1). o 3.__33 Corollary The relations ~ and ~ are equivalence invariant, as is the property of being self-similar, o Regarding the possible self-similarity of X itself, 3.4 Note X is self-similar ¢* Every view domain is embedded in every other.
Proof: Since every view of X is nonempty, X is self-similar iff every view of X is embedded in everyother. Andfor D, EE.@, DX ~ EX ¢:~ D ---* E . o For the rest of this section let (X,O) be a Hausdorff topological space on which we have a similarity view structure (G, .@) with.@ ideal, and take the object set to be ~'X, the set of closed subsets of X. We'll now consider some properties of self-similar dosed sets. The first result gives a prime characteristic of such sets, namely that they're residual, i.e. they have empty interior. Bearing in mind that.@ is covering and ideal, recall that an (implicitly nonempty) object is self-similar iff every view of A is embedded in every nonempty view of A. 3._55 Proposition Every self-similar proper closed subset A of X is residual.
Proof: As A ~ X there's a non-full view ~ of A, and since ~ is embedded in every nonempty view of A, no view of A can be full, i.e. no view domain is a subset of A. As the view domains form a basis for the topology, this just says that A has empty interior,
o
X is thus the only closed set which could possibly be self-similar with nonempty interior. A second characteristic of most self-similar closed sets is that of being 'perfect', i.e. each point of the set is a cluster point of the set ;
100
3._.66 Proposition Every self-similar closed set A with at least two points is perfect.
Proof: As.~ is covering and ideal and t A t > 2, there's a view u of A whose domain contains at least two points of A. Now take any z E A . Then for all D E . ~ with z E D , as u ~ DA (and similarities are injective) then D contains at least two points of A, hence at least one point of A - {z} ; so as.~ is a basis for the topology, x is a duster point of A as required, o The next result concerns self-similar compact sets, showing that they're generally somewhat disconnected. You might see why this could be expected by considering the usual similarity view structure on i n. There, any self-similar compact set A is bounded and can thus be covered by a view domain, which gives rise to a view of A having a 'moat' of background space surrounding A ; and since this view is embedded in every nonempty view of A we can expect A to contain 'islands surrounded by moats' everywhere you look, and A should thus be very disconnected (as long as A is non-trivial). This 'island embedding' idea forms the basis of the proof ; 3..77 Proposition If X is regular, every self-similar compact set A with I A [ >_ 2 has at least continuum-many components (moreover, continuum-many components in any open set intersecting A).
Proof: As A is compact with.~ ideal, A is covered by some D E . ~ , and as IAI > 2 and Xis Hausdorff there are disjoint view domains Et and E2 intersecting A with E i C D. By self-similarity of A let g t and g 2 be similarities g~ f~'~,E1A respectively embedding DA in EtA and E2A. In DA ~ ~ ' - ~ - ( f ' ~ particular they embed DA in DA, and gi A c A. For any finite sequence g in {g 1 ,g 2 } we have the composition og which likewise embeds DA in DA (by transitivity of embedding) so g A = A N g D , and since g D is an open neighbourhood of the compact set 9 A and X is regular, g A is a union of components of A. Now for g E {g t ,g ~.}w, ( g t n A) is a decreasing sequence of nonempty compact sets hence has nonempty compact intersection which we'll denote by g A. For distinct f , g E {g 1 ,g 2 }w, where n is least such that I n ~ g n (so f t n = g in) we have f A C f~n+l A C f t n + l D = f i n ( f u n ) C f ~ n E 1 and gA c_ g~n+l A c g~n+l D = f [ n ( g n D ) c f [ n E 2 with f t n E l and f t n E 2 disjoint as E l and E 2 are, so f ~ n + l A and g ~n+l A are disjoint; and remember they're each unions of components of A. It follows that any component of A intersecting ] A (and there is one of course) is disjoint from any
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component of A intersecting g A. Hence A has at least as many components as elements of {g 1 ,g 2 }w i.e. at least 2 ~0 components. Moreover, consider any open set V intersecting A. There's E E .~ intersecting A with E c_ V, and since DA (above) is embedded in EA by some similarity g, the components of A map via g to components of A in E , hence in V. So at least 2 ~0 components of A lie in V. o The existence of non-trivial self-similar compact sets in the case of ~n with the usual similarity view structure, perhaps surprising at first (you may like to try and think of an e x a m p l e . . . ), is demonstrated in the next section after 4.7. Trivially every singleton is self-similar in the case of ~n. However, as the last result shows, to find connected non-trivial self-similar closed sets we must took in the realm of unbounded closed sets. In the next section we'll provide a class of examples obtained by extending certain of Hutchinson's invariant sets.
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4~ Self-Similarity of Some a~Extensions of Invariant Sets Throughout let X be a nonempty locally compact (hence complete) metric space. Since X is locally compact, the body-interiors (see 'Preliminaries') form a bas/s for the topology of X (using that each point of X has a neighbourhood basis of compact sets, and the interior of any compact set is a subcompact regular open set). So let .~ be the view structure on the space X consisting of the body-interiors, and note that .~ is ideal. Using 2.4, note that .~ is equivalent to the (maximal) view structure consisting of the nonempty subcompact open sets (which, if X is boundedly compact, is also equivalent to the view structure consisting of the open balls). In the terminology of Chapter 3, let F be an admissible set of contractions of X with invariant set K, and let hF = {re F [ fis a homeomorphism of X } and GF be the group of homeomorphisms of X generated by hF (this is the trivial group if hF = ~). Then with respect to the similarity view structure ( G F , . ~ ) on X we'll give sufficient conditions for the self-similarity of certain 'w-extensions' of K. Such w-extensions will then also be self-similar with respect to any weaker similarity view structure of course; in particular, if X = ~n and every dement of hF is a direct similitude, the sets in question will be self-similar with respect to the usual similarity view structure on ~n. First we'll describe what 'w-extensions' of K are, and why in suitable cases they look like being good prospects for self-similarity. Consider the example of a Sierpinski Gasket K i n i~ produced by F = { f l , f 2 , f3 } as described in Section 1 of Chapter 3. By visual inspection one can soon convince oneself that any two views of K lying fully within the triangle involved can be embedded in one another, moreover that the embeddings can be done by elements of the monoid generated by F. Roughly speaking then, the 'inside' of Kis self-similar, and it is only due to the presence of an outer edge of K that K itself is not self-similar (for example, no whole view of K is embedded in the inside of K ; indeed we know by 3.7 that K is not self-similar, since K is compact and connected). If we could in some way 'grow' or 'extend' K outwards indefinitely so that there was no edge left then, we might well end up with a genuine self-similar set. This idea of extending K outwards, hinted at in [Man], is the basic idea behind forming an 'w-extension', which, returning to the general case, we now formally describe. For fE hF, as K is dosed under f it's expanded by f - 1 i.e. K C f ' I K . Consequently, for any infinite sequence i of inverses of elements of hF, (i~n K ) is an increasing sequence of compact sets (the notation here being the same as that used in Chapter 3, namely i[n K = o( i~n) K = ( i o o • . • o in_l) K , with i[O K = K ) and we define i K = Lj i ~ n K . A set of this form is called an ~-extension of K with n
respect to F, abbreviated to 'w-extension of K ' with F in mind. We similarly define i A = U i[n A for any A c X dosed under all fE hF (so (i[n A) is increasing). n
103
The sketch on the right illustrates the first few stages in forming an w-extension of a Sierpinski Gasket, which should be of
f l - l f 2 "tK A
help in visualizing the way in which the extending takes place. Note for example that
ft'lf2-1f3-1K
~
t "1K
is to K a s K i s to
fsf2ft g. Letting (hF) -I denote the set of inverses of elements of hF, we'll denote the set of infinite sequences in (hF)-1 by (hF) -w, the set of finite sequences in
f l-I f2 -I f3 -IK
(hF)-1 by ( h F ) - < ~ , and the set of sequences of length n in (hF)-1 by ( h F ) - n . A set of the form i A where i e (hF) -n may be called a (-n}th-level image of A with respect to hF ; it is obtained from A by applying a sequence of n elements taken from (hF)'1 To prove self-similarity of an w-extension i K we'll require two conditions on F, namely the homeomorphism condition and the compact set condition introduced respectively in Sections 6 and 7 of Chapter 3, and one condition jointly dependent on i and F which relates i to the way in which the compact set condition can hold, making sure that the w-extension in question extends over the whole space in a certain sense, thus ensuring that it has no 'outside'. In the following, D will always be assumed to be an element of .~. Bear in mind that if every element of F is a homeomorphism then the recurring phrase 'If the compact set condition holds with D ' in the work below can be replaced by 'If the open set condition holds with D ', by 7.5 of Chapter 3. 4.._!1 Proposition If F satisfies the compact set condition with D then (1) Every fE Monoid hF embeds DKin DK. (2) Vie (hE)-<w, D i g = OK. (3) Vi E (hF)-w, D i g = DK. (4) Vie (hF)-w Vn, o(itn)embeds DKin i K.
Proof: (1) Let H= D. By transitivity of embedding it suffices to show that each fE hF embeds DK in DK. As His closed under the homeomorphism f so is its interior D. It remains to show that f ( g fl D) = g fl f D so that f ( Dg ) = (f D)K. For g e F - { f } , g His disjoint from f_H_= f H = f D so (as K £ Hgives g g ¢ g H ) g Kis disjoint from
f D too. So as Kis the union of its first-level images, Kn f D = f K N f D = f ( K f i D) as required. (2) Let g = oi. Then g -1 e monoid hF so by (1) g -1 (DK) = (g -1D)K, so applying g to both sides, DK = g ((g -ID)K ) = (g g "ID) g K = D g K = D i K.
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(3) By (2), Vn Dfl i t n K = DN K. Hence DN i K = D n
[J
irnK=
n
= LJ(OflK)=DNK.
[j(Dn
irnK)
n
So D i g = O K .
n
(4) Letting i = i~n j and using that D i g = (itnD)(itnjg) =(itnD) ig.
D g by (3), itn ( D g ) = i~n ( o j g ) = o
Note incidentally that (1) above gives an alternative proof of 7.10 of Chapter 3, which essentially said that if F = hF and F satisfies the compact set condition with D , then for all fE Monoid F , f embeds DK in K. Now combining the homeomorphism condition and compact set condition we have the following ; 4.__22 Proposition If F satisfies the homeomorphism condition and satisfies the compact set condition with D , then (1) DKis embedded in every nonempty view of K. (2) For any nonempty subviews u and v of DK, u ~ v.
Pro of: (1) Let E K b e a nonempty view of K. Taking z E E N K, by 6.1(2) of Chapter 3 there's f e Semigroup hF with f D c E ; and we know f embeds DK in DK by 4.1(1), so f also embeds it in EK, f ( D K ) = ( f D ) K being a subview of EK. (2) By (1) DKis embedded in v ; hence so is the subview u of OK. o 4.3 Lemrna For D E.~ dosed under all fE hF, and i E (hF) -w, i D = X ¢~ V E E ~ 3 n E w EC i~nD.
Proof: ~: For E E .~, E is compact and covered by the open sets irn D since their union is i D = X . Hence there's a finite subcover, and this posesses a largest member since ( it n D) is increasing. ¢ : Trivial, as-~ covers X and Vn E w i t n
D C i D.
o
We now have the main result ; 4A Proposition (Self-Similarity of Some w-Extensions i K ). If F satisfies the homeomorphism condition and satisfies the compact set condition with D , then for any i E ( h F ) - W such that i D = X , i K is a self-similar closed set (and DK is a generative view of i K ). Proof: First we show i K i s dosed. Take any x E i K , and let n E w w i t h x E i ~ n D . By 4.1(4), i K N i ~ n D = i [ n ( K N D ) = i ~ n K NilnD,soasi~nDisopen, zEi~nK= itn KC_ i K , giving x E i g a s required. Every view of i K is embedded in DK (Proof: for E E.~, by 4.3 there's n e w with E C_itn D, hence E i K is a subview of (itn D)i K which by 4.1(4) is itn (OK) which is similar to DK ; so E i K is embedded in DK ) so every two nonempty views u and v
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of i K are similar to nonempty subviews u' and v' of DK, and by 4.2(2) we know that u' --* v' , hence u ~ v.
So i K is self-similar. And lastly, since DK is a view of i K
(see 4.1(3)) the above also showed that DK is a generative view of i K.
o
For example, let X = ~2 and K be a Sierpinski Gasket produced by the usual F. Let D be the interior of the triangle involved and let i E F -w such that every element of F -1 occurs infinitely often in i. It is easily seen that i D = X in such a case (indeed, these are the only i giving i D = X ), so by 4.4 i K is a self-similar dosed set.
In
addition, by the following result, every two such w-extensions are universally view-similar (hence also view-class indistinguishable) ; 4.__55 Corollary (Universal View-Similarity Among w-Extensions of K ) . If F satisfies the homeomorphism condition and satisfies the compact set condition with D ,
then for any i,jE(hF) -w such that
iD=X=jD,
ig~jg. Proof: By the last result DK is a generative view of both i K and j K. So for any nonempty view u of i K and nonempty view v of j K , u and v are similar to nonempty subviews u' and v' of DK, and by 4.2(2) u' --* v' , hence u --* v.
o
Regarding 4.4, the following result describes a class of periodic sequences i having the desired property that i D = X in the case where X = ~n and the elements of F are similitudes. We define first that for fE (hF )n, f-1 =
(hE)-n, so o(f-I ) = ( o f ) ' 1 and for m < w we define f - m = (f-1)m. 4.6 Note
Suppose X = ~n, every element of F is a similitude, and F satisfies the
compact set condition with A = D .
Then for any fE F <w such that f A C A,
letting i = f - W w e have i D = X , so i K is a self-similar closed set.
Proof: Since f A ¢ D with f A compact, there's e > 0 with [fA]e c_ D . Letting )~ be the _
scale factor of o(f -1 ), then for all m _> 1 we have (noting A c f-(m-1)A = f - m f A )
[A](~me) c [f-m fA](~m ) = f - m [fA] e c_f - m D C i D . Since ~me converges to oo with m , it follows that i D = X, and by 4.4 i K is a self-similar closed set.
o
Finally we give sufficient conditions for K itself to be self-similar. As well as the homeomorphism condition and compact set condition for some D , we stipulate a condition ensuring that arbitrarily large views of K are embedded in DK, which by 4.2(1) will imply the self-similarity of K. Regarding the term 'arbitrarily large' used in (2) below, when we say 'arbitrarily large' elements of-~ obey a certain condition, we mean that for all D E .~ there's E E .~ with D c E such that E obeys the condition.
106
4._!? Proposition (Self-Similarity of Some K ) . If F satisfies the homeomorphism condition and satisfies the compact set condition with D such that, for arbitrarily large E E.~ there's fE hF with
f E C D and f E disjoint from g D for all g E F - { f } , (and DK is a generative view of K ). Proof:
then K is self-similar
By 4.2(1) and transitivity of embedding, it suffices to show that every view of K is embedded in DK (i.e. that DK is a generative view of K ). Taking any C E .~ then, we wish to show that CK is embedded in DK. For some E E .~ with C c E , there's fE hF with f E C D and f E disjoint from g D for all g E F - {f}. Taking such E, it now suffices to show that EK is embedded in DK, for then CK will be too. As f E c D, it remains to prove f ( K N E) = K N f E , i.e. f K f i f E = K O f E . And this holds as K is the union of its first-level images and g D is disjoint from f E for all g E F - {f}. o For example, consider the case in ~ with F = { f n ] n _< w }, where for n < w f n is the direct similitude mapping A = [0,1] to [an, bn ] = [1/2 n _ I/4 n + l 1/2 n ] c A (i.e. f n contracts about 1 by factor r n = 1]4 n + l then translates by -1 + 1/2 n ) , whilst f w is constant value 0. Note that f w = lim ( f n ) , so F is compact and satisfies the homeomorphism condition. The first-level images of A are shown in the diagram. Letting cn = 2 n + l _ 1, a simple calculation shows that for n > 1, the dosed cn r n fringe of f n A (namely [an - cn rn, bn + cn rn] , which is f([A ]cn) ) is
nonoverlapping
gEF-{f}.
with
gA
for
all
Since (c n) tends to infinity, it's easily seen that the conditions stipulated in 4.7 hold with D= A = (0,1) as we can
0 . . . . .
f 2A
1
f IA
f oA
take the 'arbitrarily large E E-~' to be the interiors of the sets [A ]c " So K is n self-similar with respect to the similarity group generated by hF, and is thus also self-similar with respect to the usual similarity view structure on ~. We thus have an example of a bounded dosed set which is self-similar with respect to the usual view structure on ~ (and examples in ~ can now easily be produced using a similar scheme to the one underlying the above). Establishing the existence of such a set was the main reason we introduced the homeomorphism condition (in Chapter 3) rather than keeping to the simpler condition that every element of F be a homeomorphism. For, in the case where X = ~n, if F satisfies the conditions stipulated in 4.7 then F cannot be a set of similitudes. Indeed, if every element of hF is a similitude then F must contain a constant map. We can prove this nonstandardly as follows. By transfer of the condition on F (and bearing in mind that as .~ is an ideal basis, some element of .~ * expands v.~= U{D* I DE~}=bdX*) for some E E . ~ * expanding bdX* there's fE hF * with f E C D, which implies that the scale factor of f is infinitesimal. It follows that the element of F to which f is near must be a constant map.
107
5. The View Topolog~ Throughout let X be a locally compact Hausdofff space and .~ be a view structure on the space X consisting of b o d y - i n t e r i o r s . . ~ could consist of a//such sets (since they indeed form a basis for X as X is locally compact), but this is not required. The main example we have in mind is where X is ~n and g is the usual view structure on ~n, consisting of the open balls. Our object set will be ~'X, the set of closed subsets ofX. The set of views then is Y = . ~ $ ' X = { D A [ DE.~andAEffX}={(D,I) I D E -~ a n d / i s a closed-in-D subset of D } (beating in mind that each I in the latter is an intersection of D with some A E ~'X ). In this section we describe and investigate three topologies. First of all we'll consider the 'view topology' on V, which will then give rise to the 'view-induced' topology on ~X. Then we'll add in the idea of similarities so we moreover have a similarity view structure on X , thus bringing the concept of view classes into the fray, and we'll show how the view topology naturally induces a topology on the set Y ~ of view classes. We start then by considering how to define a natural topology on Y. The rough idea is that, where (D,I) E 7/and ( E , J ) E Y*, ( E , J ) should be near (D,I) iff E is near D and J looks hke I*. The first of these requirements suggests we need a suitable topology on the set .~ of view domains. To this end then, consider the following nonstandard anatomy of a view domain D. As D is compact with interior D, OD = D - D = 8 D , which is compact as D is. Note then that D is the disjoint union of D and 0]9, and therefore D # is the disjoint union of D # /** \\ and (aD) #, which are each partitioned into monads of course. This is illustrated on the right. Note also that D # c D * C D #, i.e. D * is sandwiched
between
D#
and
D # U (019)#.
Regarding the question of when E E .~ * should be near D, it seems natural to ask that E be sandwiched between D # and D # like D *, i.e. that D # c_ E c D #, so that E differs from D * only within the 'infinitesimally thin' boundary region (0D) #. Such a topology on .~ indeed exists; namely the body-interior topology described in Appendix 6. We shall assume from now on then that .~ has this topology, so that for D E .~ and E E .~*,
EE#D ¢~ D # C E C-D # ¢$ EAD*C(OD) #. The second part of the problem, returning to (D,I) E Y and ( E i J ) E Y* and assuming now that E E #D, is to specify when J 'looks like' I*. Intuitively, the only part of D * we can properly see (assuming we cannot resolve beyond the monadic level,
108
i.e. that points in the same monad cannot be distinguished) is the main body D it ; the
remaining points belong to (OD)it = Lj {itx I x E X and #z contains a point of 0 (D *) = (0D)* } and are thus too near the boundary of D * to be resolved from it. So our task reduces to specifying when J 'looks like' I * within D it.
Since we cannot resolve
monads, J N D it blurs to U {itx ] z E D and itx contains an element of J }, shown below, and likewise I * N D it blurs to U {itz t z E D and #x contains an element of I * }; J n D # shaded blurring
D # partitioned into monads Intuitively then, J looks like I * in D it iff these blurred images agree, i.e. Vz E D, J intersects itz iff I * intersects #x.
Since J intersects itx iff x E st J , and I * intersects
itz iff z E I , the condition is equivalently that Vz E 19, x E st J ¢=~ z E I ; equivalently
(st J ) o D = I. So, we now ask whether there /s a topology on 7/ in which (E,J) E # ( D , I ) ¢~ E e #D and (st J ) n D = I. The answer is yes, and we now give a standard description. The view topolo{~ on Yis the topology generated by the sets of the form w
[g, V] = {(DJ) I K C D a n d D C V}, int (K,U) = {(D,I) [ KC D and I intersects UC D }, or disj K = {(D,I) t K _c D and I is disjoint from K }, where K is compact and U and V are open.
From now on assume Y has the view
topology. The following shows that we have the monads desired ; 5._..! Proposition
For (D,I) E ~ a n d (E,J) E Y*, ( E , J ) E # ( D , I ) ¢~ E E # D and ( s t J ) n D = I ¢~ E E #D and st ( J n D # ) = I .
Proof: The two conditions on the right are equivalent since (st J ) n D = st ( J o D # ). We now show that ( E,J ) E ft ( D,I ) ¢~ E E ItD and ( st J ) n D = I . : By using the sets of the form [K, V] we have that E E #D (recalling a note in Appendix 6 on a basis for the body-interior topology). It remains to show that for
xED,
xEstJ,~
xEI.
First suppose x e I .
compact neighbourhood of x with K C D .
By local compactness let K b e a
For any open neighbourhood U of x with
UC K , (D,I) E int (K,U) so (E,J) E int (K,U)* so J intersects U*. By saturation
109
then, J intersects #z, hence z E st J as required. On the other hand suppose x 1~I . Then as I is closed in D there's a compact neighbourhood K c D of x disjoint from I , so (D,I) E disj K so (E,J) E (disj K)* so J i s disjoint from K*, hence from #z, so x ~ st J as required. ¢ : ( 1 ) If(D,I)E[K,V] then (E,J)E[K,V]* asEE#D. (2) If (D,I)'E int (K,U) then (E,J) E int (K,U)* since firstly K* c E (as K* £ K # C D # c E), and secondly, where /E I N Uand (as ie st J) jE J r # x , we have j E J n U* so J intersects U* C D * (3) If (D,I) E disj g then (noting K* C E as in (2)) (E,J) E (disj g ) * , otherwise J would intersec~ K * and hence I = (st J ) N D would intersect K = st K *. o 5.__.22Proposition
~'is Hausdofff.
Proof: If ( E , J ) E # ( D t , I t ) N # ( D 2,I 2 ) , t h e n f i r s t l y E E g D t N # D 2 s o D t = D 2 as . ~ i s Hausdorff, andthen I t = ( s t J ) f l D t = ( s t J ) N D 2 = I 2; so(O t,I t ) = ( 3 2,I 2). o For D E .~, the D--topology on ~ X i s the topology induced by the map A ~ DA. Denoting monads with respect to this topology by #D then, BE #DA ¢*
D ' B E #DA;
or intuitively, B is near A iff B looks like A* in D*. The view-induced topology on~'X is the conjunction of the D-topologies for D E.~, and we'll denote monads with respect to it by #s, for reasons shortly to become apparent. Thus, B E #s A ¢=~ VD E .~ B E #D A. Equivalently ; 5.___33Proposition For A E ~'X and B E ~ X *,
(1) F o r D E . ~ , BE gDA ¢* ( s t B ) N D = A N D ¢:~ s t ( B N D # ) = AN D. (2) B E # s A <=~ s t B = A .
Proof: (1)
BE#DA ¢~ D ' B E #DA (D, BN D*) E #(D, A N D) st((BND*)ND#)= AflD byh.1 st ( B N D #) = A N D noting D # C D* ¢~ ( s t B ) n D = A n D as s t ( B N D # ) = ( s t B ) N D .
(2) By (1), since.~ covers X .
O
Thus the view-induced topology on ~'X turns out to be the 'S-compact' topology described separately in Appendix 7, which is why we've used #s for the monads. Furthermore, for D e.~, the 'D-topology' we've defined here is the 'D-topology' defined independently in that appendix. This shows then how the S-compact topology on ~qX arises as the natural topology induced by views, and how in particular each D-topology (for D E .~) models 'likeness in D '. Assume g X has the view-induced topology for the rest of the section.
110
We could have given alternative view-related justification for the desired monads of a view-induced topology on S X without needing the view topology, as follows. Imagine looking at X * with vision that is the 'limit' of our standard abilities. Intuitively then, the 'observable part' of X * is the union monad of .~, namely
U{D* ] D E . ~ } = n s X * , so for B O X * the observable part of B i s B N n s X * . But we cannot resolve beyond the monadic level, i.e. points in the same monad cannot be distinguished, so B N ns X * 'blurs' to U {pz ] z E X and pz contains an element of B } = U {p~ I z E st B } = (st B )#, which we shall call the monadic imag.e of B , denoted by B # (which is B ~' if X is metric and B C_ns X * ). The natural question arises then of whether there's a topology on ~ X i n which, for A E ~ X and B E ~ X *, B is near A i f / B and A * have the same monadic image. Now since st A * = A , the monadic image of A* is just the monadic cover of A, hence
B # = A* #
( s t B ) # = A # ¢~ st B= A, so the answer to the above question is 'yes', the topology being (again) the S-compact topology on ~ X . Note that the idea of monadic images was partially used in developing the view topology, since we essentially decided in considering when (E,J) should be near (D,I), that J looks like I * in D # iff the intersection with D # of the monadic images of J and I * were the same (since for any L c X * , L # N D # = U{#x] x E D a n d #zcontains an element of L } ) . And note that in the path we took to the view-induced topology on ~¢X, the idea that the 'observable part' of X * is U {D * ] D E -~ } is embedded in our forming the topology as the conjunction of the D-topologies for D E.~. Developmental paths aside, these remarks give further perspectives on the topologies defined so far, and from Appendix 7 we know that under the view-induced topology, ~ X is a compact Hansdorf/space.
5.4 Note
A ,-view is nearstandard if/ its domain is.
In other words, for
(E,J) E ~*, (E,J) is nearstandard ~=~ E is nearstandard. Proof: : Trivial. ¢:LetEE#Dsay.
Then(E,J)E#(D,(stJ)ND).
In the next lemma we consider the .-views of a ,-closed set B ; 5.5 Lemma F o r B E ~ X * , (1) F o r E E n s . ~ * , ° ( E B ) = ° E ° B . (2) st (.~*B ) = ~ ° S .
Proof: D°B=DstS=(D,(stS)ND), by 5.1 we need to show that s t ( B f l E ) n D = ( s t B ) n D . This holds since i f z E D with x E st B, then moreover x E st (E fl B ) using that ~z c E . (2) By (1) we have st ( .~*B ) C .~ °B. And the reverse inclusion holds since for all D E .~, by (1) D °B = °(D *B ) E st ( .~*S ). o (1) Let D = ° E .
Since EB=(E, B N E )
and
111 5._..66 Proposition For D E ~ , A E ~fX, E E .~ * and B E ~'X *,
EBE/~DA ¢~ E E # D and B E # D A . Proof: Both sides imply E E/~D, so assuming now that E E #D it remains to show that
E B E # D A {=} B E # D A .
By 5.5(1),
EBE#DA ~ DA=DstB
~ AnD=
(st B ) fl D {=} B E/z D A , as required,
o
The view map is the map .~ x ~ X ~ 7 / i n which (D,A) ~ DA = the D-view of A , and we shall denote it explicitly by u when the need arises. Since .~, ~'X, and 7/ are all now equipped with topologies, we may consider the topological aspect of u. As mentioned in 'Preliminaries', a map f : Z ~ Y between topological spaces is said to be
perfect if it is continuous and closed with compact fibres f-ly, the nonstandard formulation being that VyE Y, f-l#y= (f-ly)# (the inclusion '2' expressing continuity, the inclusion 'C' expressing the rest). And if f is also surjective, f is an identification (or 'quotient') map (i.e Y has the finest topology such that f is continuous), and Yis locally compact iff Zis. 5.___77Prouosition The view map is a perfect surjection.
Proof: We know the map is surjective, and it's continuous by 5.5(1). It now only remains to show that for all (D,I)~ V, u - I # ( D , I ) C_(u-l(D,I))#. Suppose then that ( E,B ) E u -1 ~ ( O,I ) , i.e. ( E,B ) E.~* x ~ X * with EB ~ # ( D,I ) . Then by 5.5(1), ( D , I ) = D ° B = u(D,°B) with (E,B)E/~(D,°B), so ( E , B ) E ( u ' I ( D , I ) ) # as required, o In particular, continuity of the view map gives that (1) For A E ~X, the map .~ ~ 7," in which D ~ DA is continuous, expressing that "as D varies smoothly so does the D-view of A". (2) For D E .~, the map ~ X ~ 7 / i n which A ~-* DA is continuous, expressing that "as A varies smoothly so does the D-view of A". Furthermore we have the following characterization of the view topology ; 5.__.88Corollary The view topology is the finest topology on which the view map is continuous,
7/ with respect to o
And noting that (as ~ X is locally compact, being compact) .~ x ~ X compact iff .~ is locally compact, we h a v e . . .
5.__.99Corollary
7/is locally compact iff .~ is.
is locally
o
In particular, as noted in Appendix 8 (just before A8.3), the usual view structure .~ on ~n is locally compact; thus 71 is too.
112
5.10 Corollaxy For A E ~'X, the set .~A of views of A is closed in Y.
Proof: .~A is the image of the dosed set .~ . {A} under the (closed) view map. For A E ~ X then, .~A E ~ ~
O
Now if .~ is locally compact, so 7/is too, we can
give ~ 7/the S-compact topology, and we then obtain a topology on ~ X induced by the map A ~--~.~A, in other words giving B E #A ¢* .~*B E # s . ~ A . The following shows that this is in fact just the view-induced topology again ; 5.11 Proposition
I f . ~ is locally compact and ~' ~ has the S-compact topology,
then the view-induced topology on ~ X is the topology induced by the map A ~-*.~A of ~ X into ~¢
Proof: We must show that for A E ~ X a n d
st(.~*B)=~
5.5(2)
°B.
BE~¢X*,
So s t ( . ~ * B ) = H A
B E # s A ¢=~ s t ( . ~ * B ) =
¢~ . ~ ° B = . ~ A
.~A.
¢~ ° B = A
B E #sA, as required,
By
<=~ o
Looked at another way, the above essentially says that the map A ~--,.~A is an embedding of ~ X in ~" 5.12 Proposition For D E.~, the set D~fX = {(D,I) I / i s a closed-in-D subset of D} of D-views is homeomorphic to ~ D (the set of closed-in-D subsets of D, under the S-compact topology) via the natural map (D,I) ~ I , and is compact.
Proof: The map is of course a bijection, and it's moreover a homeomorphism since for I E ~ D and J E ~ D * ,
(D*,J)E#(D,I)
¢:~ s t ( J N D # ) = I
¢* J E # s I s i n c e s t ( J n D # )
is the standard part of J in D *. Since g D is compact so is D~fX (alternatively note that D ~ X is the image of the compact set { D } . ~¢X under the view map).
o
The next result shows that the subview relation _< is preserved by o ; 5.13 Proposition
For nearstandard u 1 ,u 2 E 7/* with u 1 < u 2 , °u 1 < °u 2 .
Proof: Let u i = (E i,Ji ) E # vi ; say v i = (D i , ~ ). Then D~= subst E t c subst E~. = 02 , so to show that v l < v 2 it remains to show that for z E D t ,
x E I t ¢~ x E I 2,
i.e.
z E D 1fist J1 ¢* x E D 2fist J2, i.e. z E s t Jr <=~ z E s t J2. And this holds since, as Jl = J2 N El and lzx C E1, Jt f) lzx = ( J2 n E1) fl #x = J2 N ( Zt fl #x ) = J2 fl /zx . o Assume now that G is a group of homeomorphisms of X such that, giving G the compact-open topology, G is a topological group, i.e. (since we automatically have continuity of composition) such that the inverse map -i is continuous, equivalently (as is easily shown) such that whenever g E G * and f E G with g E #f, Vx E X g #x = f # x .
113
As ~'X is closed under the elements of G , (G, .~) is a similarity view structure on the space X, and we have the following result ; 5.14 Proposition The natural group actions of G on .~ and 7/are topological, i.e. the evaluation maps G x .~ -~ .~ and G x 7 / ~ 7 / a r e continuous.
Proof: L e t g E G ' a n d r E GwithgE#f. So V z E X gl~x=f#z=f#z. (1) Supposing EE#D, we need to show g E E # f D . Since D#CE,
(fD)#=
f(D # ) = g ( D #)c gE. Since E C D # , gEC g(D#)= f ( D # ) = ( f D ) # = f - D # So (f D)# C gEe-f-D#, i.e. gEE # f D as required. (2) Supposing (S,J) E # (D,I), we need to show that g (E,J) E # f(D,I), i.e. (g E,g J ) E # (fD, fI). We have g E E # fD by (1), so it remains to show Yy E fD, y E s t g J ¢ ~ y E f I , i.e. VzED, f z E s t g J ¢ ~ f z E f I . Now for z E D , f z E f I ¢ * z E I C $ z E s t J , so the above amounts to showing that VzED, f z E s t g J ¢~ z E s t J . So, let z E D . I f x E s t J t h e r e ' s j E J w i t h j E # x , and then gjegJwithgje#fz,sofzEstgd. Conversely, if f z e s t g J there'sjEJwith gjE # f z , and as g'l e #f-l, j= g qgjE # f q f z = #z so zE st J . o For g E G then, let g@ denote the homeomorphism of.~ induced by g, and let G@ = { g ~ ] g E G }. Likewise, let g 7/denote the homeomorphism of 7/induced by g and let G y = { g T / I gE G}. I f . ~ is locally compact, so is Y, and we give the groups G ~ and G 7/of homeomorphisms of .~ and 7/the compact-open topology, in which case 5.14 says that the natural group actions of G on ~ and 7/(which are formally the group homomorphisms in which g ~ g ~ and g ,-~ g 7/) are continuous. We now give the set 7/~ of view classes the quotient topology induced from the view topology on 7/ by the relation of similarity, calling this the view class topology. Since the relation ..~ of similarity of views is the equivalence relation induced by the topological action of G on Y, it follows from the general result noted in 'Preliminaries' under 'Open Identification Maps', t h a t . . . 5.15 Corollary is open.
The quotient map ~ : 7/~ 7/~ sending each view to its view class o
5.16 Note If G acts transitively on .~ (i.e. if every two view domains are similar), then 7/~ is compact.
Proof: Taking D E . ~ , every view is similar to a D-view, so 7/~ is the image of the set of D-views, which is compact by 5.12, so 7/"~ is compact, o
114
5.17 Prouosition
Let X = ~n, .~ be the usual view structure on X, and G be a group
of similitudes acting transitively on .~ with G closed in the group of similitudes of X . Then Y ~ is a compact Hausdofff space. Proof:
7 / ~ is compact by 5.16. To show V ~ is Hausdorff is equivalent to showing that the relation
... is closed in
y x V (see the section on 'Open Identification Maps' in
'Preliminaries'). So, suppose u 1 ,u 2 e Y* and v~ ,v2 E Y with u I E # v1 and u 2 E # v2 , and u 1 ..~ u 2 ; let g E G * map u 1 to u s . It suffices now to show that g is nearstandard in G * , since if g E # f
say, then f w i l l map v1 to v2 (for, using 5.14, we'd have
u s = g u 1 E # f v 1, hence since also u s E # v 2 and
Y i s ttansdorff, f v l = v 2).
Let
= dora u~ and D i = dora v i . Since E i E # D i , it easily follows that E 1 and E 2 are both of finite noninfinitesimal diameter (use that diam :.~ ~ (0,oo) is continuous ),
hence the ratio of g (being
diam E 2 / diam E 1 ) is finite and noainfinitesimal.
addition, there's x E X such that g D 1C_gE l = E 2cD--~ # ) .
g x is nearstandard
(note that
In
as D l c E l ,
From these facts it follows from A9.2 that g is n e a r a
similitude f . And as Gis closed in Sire X, f e G .
o
Finally we show that if X = ~n and every view domain is convex then the view topology is given by a natural metric as follows. Firstly we define the silhouette of a view v = ( D , I ) to be Si/~J = 0/9 U I , which, being also 0D U I (as I - I_C OD ), is compact (as 8D = 0 D is compact and I is compact, being closed in compact D ) and nonempty (as 0D ~ 0).
We thus have what we shall call the silhouette map
Sil: 7/--. J d X , and giving J~X the Vietoris topology (which recall is given by the
tIausdorff metric) we find t h a t . . . 5.18 Prooosition
For X = ~n with every view domain convex, the silhouette map is
an embedding. Proof:
We must show that for ( D,I ) E I/ and ( E,J ) E 7/ *, (E,J) E #(D,I)
¢# S i I ( E , J ) E # S i l ( D , I )
i.e. OEU JE # ( 0 D U I ) . Note that injectivity of Sil will automatically follow due to 3 [ X being Hausdorff. : As E E #D then E E # D so by continuity of O (proved in A6.4), O E E # 0 D , i.e. OE E # OD , so OD = st OE. We now have ODOI=ODU(DflstJ)
as I = D f l s t J
=ODUstJ
as s t J - O D
= st OE U st J
as OD = st OE
C D-OD
= D
= st (OEU J ) .
Since also OE U J _c E c_ D # C_ns X * we have 8E U J E # (OD U I ) as required.
115
¢ : By continuity of cony : J ~ X -~ J ~ X (see A8.5), cony (OE U J ) E # cony (0/9 U I ) , i.e. E ~ # D , hence E ~ #D (and also 019 = st OE as proved above in '~'). It now remains to show that I = (st J ) 0 D . We know that OD U I = st (OE U J ) , so I = st (OE u J ) - OD
= (st OE U st J ) - 019 = (ODU st J ) - O D = st J - 01) =(stJ)f~D
as s t J C s t E = D ,
o
Thus, defining the view metric to be the metric d on Y induced from the ttansdorff metric h by the silhouette map, i.e. given by d(u ,v ) = h(Sil u , Sil v ), we have the following result; 5.19 Corollary
For X = ~n with every view domain convex, the view topology is
given by the view metric,
o
Note that if we'd only asked that the view domains be connected, the above work would break down because the silhouette map would not necessarily be injective. For example, let X = ~2, D~ be the open unit ball (0)t , 11 = [01112, D 2 = D t - I1, and I 2 = ~ . Then (D 1,I 1 ) ~ ( D 2,I 2) yet their silhouettes are both aD 1UaI I. The view metric can also be used to naturally metrize the compact Hausdorff space Y ~ of view classes in the case of the usual similarity view structure on ~n; see note 7.11 for details.
116
6. A Definition of 'Visually Fractal' Throughout let
X =
~n, .~ be the usual view structure on X, and Y = .~ ~ X .
G will denote the group of rotation-free direct similitudes, namely generated by the translations and dilations, and the relation of similarity among views with respect to G will be denoted by - .
We shall call the resulting view classes oriented view classes,
these being (for n > 2) distinct from the more usual type of view class arising from the whole group of direct similitudes.
Intuitively, oriented view classes model the idea of
images coming from a remote camera whose orientation is c o n s t a n t . For all D 1 ,D 2 E .~ there's a unique g E G mapping D 1 to D 2 , so in particular G acts transitively on .~, and since G is also closed in Sire X we know by 5.17 that the space
Y-
of oriented
view classes is a compact Hausdofff space. For x E X and r > 0 let (Z)r denote the open ball (i.e. view domain) with centre x and radius r . For D E .~ the unique g E G mapping D to the unit open ball (0) 1 is said to normalize D .
A (0)vview is called a normal view, and for a view v the map g
normalizing dora v is said to normalize v, being the unique g E G mapping v to a normal view; we shall call g v the normalization of v, denoted by norm v. The fact that each view is similar to a unique normal view says that each oriented view class V contains a unique normal view, i.e. has a unique normal realization, and we'll denote it by norm V. Denoting the set of normal views by norm V, t h e n . . . 6.__! Proposition
~-
-~ n o r m 7r
via the map V~-* n o r m V .
Proof:
The inverse of the bijection in question is - : n o r m ~---* Y = - , which we already know is continuous.
Since n o r m V is (by 5.12) a compact Hausdorff space, the map --- is
hence a homeomorphism, hence so is its inverse,
o
We thus have the option of using normal views in place of oriented view classes, replacing use of v - by that of n o r m v , which for some purposes is more convenient. The quotient map norm: ~-*
norm ~
-:7f--,
~-
is replaced by the view normalization map
noting that in general n o r m v = n o r m ( v - ).
The latter map
thus has the properties of the former, and in p a r t i c u l a r . . . 6.__22 C.orollary The view normalization map is an open identification map.
o
Bear in mind that by 5.12 with D = (0)1, n o r m Vis naturally homeomorphic to ~'(0)1 with the S-compact topology, v E n o r m Y corresponding to its object part ob v. In addition we know from 5.19 that the topology involved is given by the view metric. We can give an independent short proof of this as follows.
~ ,~2 ~ (,~orm r )*, ~ and v E n o r m
First note that for
,~2 ) ~ o ** St(ohOb ~ = St(o) Ob ~ . So for ~ (norm r ) *
117
,v* )
0 ¢,
t(0) ob = t(0) ob v*
¢~ St(O)lOb_.
u = ob v
¢:~ u e # v , as required. This metric on the normal views makes an implicit appearance in some work of Tan Lei on the Mandelbrot set; see note 7.16 for further details. For A e Y X and x e X, an z -view of A is a view of A whose domain is a ball centred at z , i.e. a view of the form (Z)r A.
We now come to the key definitions in
this section. Letting x e A, suppose that as we zoom in on z , what we see settles down towards a hmit;
normi/
the formal expression of this is that
n o r m (x)r A
as r decreases to 0, equivalently that ( ( x ) r A ) -
converges in
converges in
I/-= as r
decreases to 0. If this occurs we'll say A is visually convergent at z ; otherwise A is visually fractal at z , which is one interpretation of the idea of A having 'detail at all levels of magnification' at z .
A is thus divided into two parts which we'll call its
convergent part and fTactal part, consisting respectively of the points at which A is visually convergent or fractal, and denoted by cv A and ~r A. Bearing in mind that
norm I/is a compact ttausdorff space, the nonstandard formulation that A be visually convergent at z is that o n o r m ((X)r A *) be the same for all infinitesimal r .
If this
holds, the limit in norm I/will be called the limit view of A at z , denoted by lira A z . We'll say A is visually convergent if it is visually convergent everywhere, equivalently if it is visually convergent at its boundary points (since trivially it's visually convergent at any interior point). Likewise A is visually fractal if it is visually fractal everywhere, which imphes that A is residual. To avoid constant repetition in what is now an estabhshed context, in the following work we shall generally drop the adjective 'visually' on the above definitions, except in the formal statement of results.
Some
examples of convergent sets in X = ~2 are illustrated below ;
Note that in the case of the first example, the limit view at any boundary point is a view of a half-plane just touching the origin. This is illustrated in the sketch on the next page, which shows the normalizations of sucessive views of A at a boundary point z , converging to the limit view as we zoom in on z ;
118
These pictures also suggest the proof of the following result; 6.__33Proposition Every convex dosed set is visually convergent.
Proof: Let A be a convex closed set, and take any z e A . For r > 0 let I r = ob norm (Z)r A. As A is convex about z , / r increases as r decreases. So using A7.4(2) , as r decreases to 0, / r converges in ~'(0)l (namely to the closure of Ur Ir = ~n I 1In in (o)l), in other words (recalling the way in which norm 7z is homeomorphic to
norm (Z)r A converges in norm ~ as required,
~(0)t ) o
Indeed, by A7.4(2) a set A is convergent at any point z for which ob norm (X)r A is eventually increasing or eventually decreasing as r decreases to 0. Later we shall provide a sufficient condition for certain of Hutchinson's invariant sets KF to be fractal. In the meanwhile we note one simple example of a set being fractal at a point. Namely, let X = ~ and A = { 0 } O { 1 / 2 n l n > 0 } ; then A is fractal at 0. Indeed, A is 'visually periodic' at 0 in a sense made precise later on, and this will imply fractality at the point in question. Note in the present example that A is convergent at all points 1/2 n (a set being convergent at any isolated point), showing that the convergent part of a set A need not be dosed in A. On the other hand, the example on the right in ~2 shows that the fractaI part of a set A need not be closed in A either. Here, A is fractal at all points a n = (1/2n,0) (the set of points above a n being a vertically aligned copy of {am I m >_ 0 } ), but A is convergent at the a 3 a2 at ao origin. The upper curve involved is tangential to the lower line at the origin, and it should be noted that A would have been convergent at the origin regardless of how we had 'filled in' the space between this curve and the lower line. Despite the example in X = ~ of A = {0} U {1/2 n I n >__0 } being fractal at 0, note that the set B = {0} U {1/n I n > 1 } is actually convergent at 0 (despite the existence of a homeomorphism of X mapping A to B ), the object part of the limit view
119
being [0,1). We prove this as follows, letting bn = 1In. Take any infinitesimal r > 0, and let n e ~ * with bn < r <_ bn_ 1 . Letting g be multiplication by l / r , g normalizes u=(0)rB*,and
o b n o v m u = { O } U { g b m[ m>_ n } .
gbm+ l ~ g b m noting that
Now for all m > n we have
gb m - g b m + 1 = ( b m - b m + l ) / r < n(b m - b i n + l ) =
n](m (re+l)) < n/(n ~ ) = 1In ~ 0. Also, g bn ~. 1. It follows that the standard part of ob norm u in (0)l is [0,i) as required. This example also shows that we can have a set B which is residual, yet having a limit view whose object part is not residual in (0)t ; indeed, it can have a limit view which is fuU. A limit view is a view of the form liraA z for some x e A E ~X, and we'll provide a complete description of these shortly, facilitating later work. First, for A e (0,1) we define a _zoomoperation zA on norm ~ by ((0), ,x) = norm
,xn
It is easily seen that for A e ~X, zA (norm (Z)r A ) = norm (X)Ar A . In a d d i t i o n , . . . 6.4 Lemma VA e (0,1) z~ is continuous. Proof: Let v e n o r m T z and u E ( n o r m Y ) * with u E # v , a n d l e t
I=obv
Then for ze(O)A , z e s t ( J n ( O ) A * ) ¢~ z e s t g
so st(Jn(O)A* )n(O)A
¢=~ x e I ;
and J = o b u .
= I n (0)A , which says that ((0)A*,Jfl (0)A*) E # ((0)A ,In (0)A), so by continuity of the normalization map (part of 6.2) norm ((0)A*,J n (0)A*) E # norm ((0)A ,I fl (0)A), i.e. z~u ~ # zAv.
o
We'll say a view v = ((X)r,I) is radiant if I = {z} U{ Ix,y) [ y e Y} for some YC O (X)r. In this case, Y is unique, easily shown to be I - I , and we'll call Y the radiant set of ~, denoted by tad v, this being a closed subset of the sphere O (Z)r ( the unit sphere S n-1 in the case v is normal). Some examples of radiant views in ~2 are illustrated below;
For z e A we'll say A is radiant at z if some z-view of A is radiant; trivially this implies that A is convergent at x, but not conversely. However, at least the limit view of A at x is radiant; for as we now show, the limit views are precisely the radiant normal views. For closed YC S n-1 let I Y = {0} U { [0,y) ] Y E Y}.
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6...55 Proposition For a normal view v the following are equivalent; (1) v is a limit view. (2) v is radiant. (3) v is nonempty and invariant under every zoom operation z)~. Proof:
(1) $ (3) : Say v = l i m A x , and for r > 0 let u r = n o r m ( x ) r A . Naturally v is nonempty since 0 E ob v . And for )~ E (0,1), since v is the limit of u r as r decreases to 0 then by continuity of z ~ , z~v is the limit of z~u r = u)~r as r decreases to 0 ; but this is of course just the same limit, so z~v = v, i.e. v is invariant under z)~. (3)~(2):LetI=obv. If I - { 0 } = 0 t h e n I = { z } so v is radiant. Now assume I - {0} ~ ~. From (3) it easily follows that for all p E I - {0}, (0,y) C_I where y is the element of S n-1 such that p E (0,y) ; moreover [0,y) C I a s / i s closed in (0)l. So /is a nonempty union of sets of the form [0,y) with y E S n-1 hence v is radiant. (2) ~ (1) : Then letting Y = tad v and A = I Y (which is a closed subset of [011 ), A is radiant at 0, and v = (O)IA = lira A 0 , so v i s a limit view. o Using this, we thus have the following homeomorphism between the space lira of limit views and ~(S n-l), defining for YE ~(S n-l) that lira Y = ((0)t , I Y ) ; 6.6 Proposition Giving ~'(S n-l) the Vietoris topology, lira Y ~- ~ ( S n-l) mutually inverse maps v H rad v and Y~--* lira Y .
via the
Proof:
The fact that these maps are mutually inverse bijections is straightforward. We now show that the second is a homeomorphism, which (recalling how norm Y ~- ~(0)1, and denoting monads in the latter space by #s, given by B E #sA (=~ St(o)lB = A) reduces to showing that for Y E ~(S n-l) and Z E f¢(S n-l)., Z E # Y ¢¢ IZ E #s I Y . This is proved as follows, first of all noting that for y E S n-1 and z E (S n - l ) ,
z
[0,z)
[0,y).
: ~ : Z E # Y m e a n s that VyE Y 3 z E g z E # y and VzEZ 3yE Y z E # y . above note it easily follows that St(o ) IlZ = I Y , i.e. IZ E #s I Y .
By the
¢ : We need to prove that YyE Y 3 z E Z z E # y and VzEZ 3yE Y z E # y . Taking y E Y, y/2 E [0,y) C_I Y = St(o)llZ so for some z E Z and a E [0,z) we have a E # (y/2), which easily gives z E #y. A similar argument shows that for z E Z there's yE Y w i t h z E # y . o As a corollary, lira Vis compact as g ( S n-l) = {0} U~?~(Sn-l) with ~ ( S n-l) compact. Note that O, isolated in ~(S n-l), corresponds to the limit view with object part {0}, which is the limit view of a set at any of its isolated points (indeed, for x E A,
121
x is isolated in A iff A is convergent at z with scale, S n-1 corresponds to the
full
rad limAx =
0). At the other end of the
limit view, which is the limit view of a set at any
of its interior points. We now give a sufficient condition that a residual invariant set K = K F be fractal ; 6.77 Proposition Let F be a finite set of contractive direct similitudes which satisfies the compact set condition with some A such that A K is not embedded (by a direct similitude) in any limit view, and I K] _> 2. Then K is visually fractal.
Proof: Let z e K .
Taking infinite n , l e t
fEF nwith
zEf(K*),
which is infinitesimal. Note that f ( A * ) c_ [z]e as z e rf diam A *
Where g e C * normalizes
and let
f(A * )
we have
e=rfdiamA,
with diam f ( A * ) = = 1/,
so
of =
rg rf = 1/diam
A , which in particular is a finite noninfinitesimal, and since g o f also maps a bounded point to a bounded point, by A9.2 it's near some similitude h . Letting D = A , we now show that h embeds DK in °(norm (x)e K * ) ; for it will then follow by our suppositions that the latter is not a limit view, so K will be fractal at z as required.
By transfer of 4.1(1), and the fact that
embeds
(DK )* in
i.e. (go
f)(Dg)* <_norm(z)e g*,
(x)e g * , i.e. f
(OK)* < (x)e g*,
g f (DK )* < g (x)e g*,
f)(Dg)* ) <_°(norm(x)e K* ). O((gof)(og).)=h(Dg). So h ( D g ) <
so by 5.13 °((go
And since ( g o f ) e # h , then by 5.14
°(norm (x)e K* )
hence
f(D * ) c (x)e, f
as required,
o
One easily seen sufficient condition that A K not be embedded in any limit view is that K n ,4 (the object part of A K ) contain at least two points and expand no line segment [x,y ] ; and since this condition in particular holds if K is totally disconnected and K N A ~ 0 (hence moreover I K N A I _> 2 using 10.7 of Chapter 3), note t h a t . . . 6.__88 Corollary Let F be a finite set of contractive direct similitudes which satisfies the compact set condition with some A such that K fl A ~ 0 , totally disconnected. Then Kis visually fractal,
and suppose K is o
In particular the Cantor set in ~ is fractal. In practice, with X = ~2 a computergenerated picture of K will quickly suggest whether or not A K can be embedded in a limit view, and more often than not the answer is likely to be 'no' as required, since views embeddable in a limit view are rather special. For example, the Sierpinski Gasket is clearly fractal, by looking at A K where A is the triangle involved. A formal proof is provided by the following strong result,
affK
denoting the affine span of K ;
122
6.J P r o ~ s i t i o n
Let X = ~ and F be a finite set of contractive direct similitudes
which satisfies the compact set condition with some A ~ K such that K n A ¢ O and
affK= X. Then K i s visually fractal.
Proof: Firstly note that K i s residual by 7.13 of Chapter 3. Letting D = A we show that
DK
is not embedded in any limit view, which by 6.7 will imply that K is fractal. Suppose for a contradiction that there is a direct similitude h embedding This is illustrated on the right.
Using
DKin a limit view v.
h
that K is residual and v is radiant, it follows that
for
all
z E obDK-h-10
L z through x such L x is a Now o b D K = K N D
there's a unique line
that some neighbourhood of z in subset of
obDK.
is not a subset of a line (otherwise, as
affK¢ X , lob DK t > 2) let z E ob D K - h -10, and then let y E ob DK with y t~ L z • Note that Ly is defined since
K n D is dense in K b y 6.3 of Chapter 3, Kwould be a subset of a line, i.e. contradicting one of our assumptions) so (since in particular
h -10 E L x . Taking a .-neighbourhood N of x with NO_ #x, by transfer of result 6.1(2) of Chapter 3 there exists some
I E (Semigroup F ) * and
by
4.1(1)
with
I embeds
I(D * ) c_N, (DK)* in
I(DK ) * ~ ~ ¢ -
~z
(DK)*. As I is a .-similitude the angle I ( L x * ) and I ( Ly * ) is the same as that between Lx * and Ly * (namely that
between between
L x and Ly ). But by transfer of a standard fact, I(L x* ) = L *Ix and
I(Ly * ) = L *IY' the angle between which is infinitesimal using that Ix~ IY (so
h Ix ~ h IY ). We thus have a contradiction as required,
o
Changing direction, we now ask whether for A E ~ X and x E OA it is generally true that A is fractal at z iff
OA is fractal at x. Of course, this is only of interest for
non-residual sets. The answer is that the forward implication holds whilst the reverse
an= 1/2 n for n > 0 , and let A = { 0 } U U { [ a n+ en,an_ 1] I n > l } where en = ( 1 / n ) d(an,an_ 1 ) = ( 1 / n ) a n. Then OA={O}U{a nl n>.O}U{a n + e nt n_> I } is fractal at 0 b u t A i s n o t , all due to the fact that en / a n = 1/n converges to 0 as n tends to (x) . We now however give the proof of the forward implication above ;
fails.
As an example of this failure in ~ , let
6.10 Proposition
For A e fOX and x E OA, if A is visually fractal at x so is
Proof: We prove the contrapositive; assuming that
OA.
OA is convergent at z we now show that
123
A is convergent at x too. Let Y = rad limoA z . For z E s n - l - y ,
theset
Mz = # x
N U{(z,w] [ wElZ(Sn-1)z}
subset of A * or is disjoint from A * . For, suppose a E M z N A * other / ~ E M z we have f l E A * ;
is either a
Then taking any
for if
fill A * then by transfer of a standard
....... !!i:
fact the line segment [a ,fl] must cross the boundary of A * at some point 7 , but this would give the contradiction that z E Y (for where r = 2 d(x,7) and g E G *
normalizes (X)r , then z / 2 = o (g 7) E o ( ob norm ( X)r OA * ) = ob lira OA x
giving
z E Y as lira OA x is radiant). Letting Z = { z E S n - I - Y I M z C _ A * }
then, it is easily seen that for any
infinitesimal r , o (norm (X)r A * ) is radiant, with radiant set Y U Z (Ycoming from the fact that OA c_ A ). So A is convergent at z as required,
o
We continue with a few further results concerning boundaries. For a view v = ( D , I ) we define Ov = (D,a D I ), where 0 D denotes 'boundary in D ' (note that 0D I = D N aD since D is open), av is a view with the same domain as v, and the following result applies to any topological view space ; For D E ~ and A E ~ X , 0 DA = D OA .
6.11 Proposition
Proof: We must show that
01)(DNA)=DNO"4'
i.e. D N 0 ( D N A ) = D N 0 . 4 ,
i.e. for
x E D, x E 0 (D N A) ¢:~ x E OA. Using t h a t / ~ C D *, this is proved as follows; E O ( D N A) ¢~ 3a, flE #~: with a E ( D N A ) *
and f l ~ ( D N A ) *
¢~ 3a,/~ E #x with a E D * N A * and f l ~ D * f l A * ¢~ 3a,/~E#x with a E A *
and f l l ~ A *
xE OA. Note that the map
vEnormY
0: ~ " ~ Y is not continuous.
uE(normV)*
and
o
with v full and
For example, in X = ~{, let
obu=(O)l*-(O)e
for some
infinitesimal e > 0. Then u E #v but 0u ~ # 0v ; indeed, 0u is near the normal view with object part {0}. The next result describes 0v in the case where v is a non-full limit view ; 6.12 Proposition
For Y E ~ ( S n - l ) with Y ~ S n-1 , Olim Y = l i m O ( s n - 1 ) r .
Pro of: Let Z = O(S n - l ) r .
We must show O(o) I Y = I Z , i.e. Vx E (0)1, x E 0 I r t=} x E I Z .
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Both sides hold for x = 0 (using Y ¢ S n-1 to show 0 E 0 I Y ) , so now take x ¢ 0. f l E # x with f l ~ I Y * ,
$:Let
and let a E ( S n - 1 ) * with fie (O,a).
a ~ Y*. Now x E ( 0 ) I N O I Y = O ( 0 ) I I Y C I Y
Since ~ I Y * ,
so let yE Y w i t h x E ( 0 , y ) .
ThenaE#y
so y E Z , hence z E IZ as required. ¢ : Let z E Z with z E (0,z).
Where
a E (S n - l ) . _
y * with a E # z , (0,a) is disjoint
from I Y * , hence since some/3 E Co,a) is near x we have x E 0 I Y . For Y = S n-l,
Olim Y is the empty normal view.
o
Although we have not
formally included this view under the term 'limit view', note that it does arise as such, in the sense that if we zoom in on a point x ¢ A we eventually see only an empty view;
norm (X)r A is the empty normal view for all sufficiently small r . By 6.12, the set consisting of the limit views and the empty normal view is closed under O. We'll say a view v = ( D , I ) is residual i f / i s residual in D, equivalently 0/9 I = I , equivalently ~v = v. T h e n . . . 6.13 Note
For a limit view v, v is residual ¢:~ rad v is residual in S n-1
Proof: lira Yis residual ¢=~ Yis residual
Equivalently we must show that for YE ~¢(S n - l ) ,
in S n - l . It's true for Y = S n-1 as both sides fail. For Y~ S n-l, using 6.12, lira Yis residual ¢=~ Olim Y = lira Y ¢~ l i m O ( s n - 1 ) Y =
Iim Y ¢~ O ( s n - 1 ) Y =
Y ¢:~ Y is
residual in S n-1.
o
Finally, we describe the notion of visual periodicity mentioned in passing earlier on. For A E ~ X and x E A we'll say A is visually periodic at x if for some r t > r 2 > 0 ,
(X)r2A-(x)rlA
and V ~ E ( r 2,r l )
(x)~A~(X)rlA.
If this holds then r t / r 2 is the
same for a//such pairs (r 1 ,r 2 ) (Proof: take any such pair (~t ,~2 ) and assume ~ <_ r 1 . Now where ~1 = A r l , ( A r I ,A r 2 ) is clearly such a pair too, hence we must have 62 = A r2, thus
~1/62 = rl/r2 ), and we define the period of A at x to be In (r 1 / r 2 )
= In r 1 - In r 2 . The reason for this definition of the period (rather than r i / r 2 say) is as follows. Imagine zooming in on x at a constant rate compared to the existing size of the view domain.
More precisely, suppose that where at time t our view domain is
( Z ) r ( t ) , we have r i 0 ) = 1 and 7"(t) = - r ( t ) , so r ( t ) = e - t and r 2 = r (t2) we have In r I - In r 2
Then taking r l = r(tl)
t 2 - t 1 , the time taken between the appearance
of successive identical normalized views (once we've zoomed in far enough).
Visual
periodicity of A at z thus equivalently states that the map [0,c~) ~ norm ~" in which
t ~-* norm (Z)r (t)A is eventually periodic (the period then being that of A at z ), and it is immediately apparent that in such a case norm (Z)r (t)A does not converge as t increases to ec (i.e. as r (t) decreases to 0 ) , so A is fractal at x ; 6.14 Note
A is visually periodic at x # A is visually fractal at z .
o
125
7. Notes, Questions. and Suggestions for Farther Work The notes below are grouped under the sections to which they are most relevant, except for the final 'miscellaneous' group.
Several of the notes simply point out
various natural view-related definitions worth bearing in mind for further work. Throughout, assume unless otherwise implied that we're dealing with X = ~n with the usual similarity view structure, A and B denoting elements of the 'object set' ~ ' X , and the similarity relation being denoted by ~ .
The symbol =
will be reserved for
denoting the relation of similarity arising from the rotation-free direct similitudes, as described in Section 6.
Despite the convention of X = ~n, note that several of the
points below (or in some cases, suitable generalizations) make sense for any topological similarity view space, or, even more generally, any similarity view space. Section 1 : Views and Similarities 7.__! Although the term 'view' we have used is quite apt in the case of ~2 (thinking of looking down on a plane and obtaining views of a set), its meaning differs significantly from everyday usage in the case of ~3 (not that we intended to model the latter usage). Here, our 'views' are as if obtained not from a normal camera but from one which can 'see through' structure, taking a whole 'sample' of an area and the object-structure within it. However, if we 'looked down' on ~3 from within ~4, the term 'view' would be appropriate in an analogous way to the earlier ~2 case. The main point here as regards interpretations is that one should be aware that there are two things involved;
the
space one is studying and the vantage point one is studying it f r o m . 7._.22 Recall that an x-view is a view whose domain is a ball centred at z.
For A we
define the equivalence relation ~ in A by x A Y ¢~ some z -view of A is similar to a y - v i e w of A, saying A is similar at z and y.
The equivalence classes are called the
similarity classes of A, and should be worth investigating. For example, consider the similarity classes of the Sierpinski Gasket, or other invariant sets K F . Generalizing A , for a E A and b E B we may say A and B are similar at a and b if some a -view of A is similar to a b -view of B . For a weakening of this relation see 7.16. 7.._.33 Does there exist a set A consistent with every view class, i.e. ( . ~ A ) " = Y ~ ? (If so, there's a compact example since, as can be proved constructively in a few hnes, for any A there's compact Cwith ( . ~ A ) ~ C ( . ~ C ) ~ ) .
Since I ~X] = I Y I = 2 ~0,
an answer in the negative cannot immediately be given on cardinality grounds alone. In connection with the question, preorder * X by A < B ¢~ (.~A) ~ C (.~B) ~. The question then asks whether a greatest element exists.
If not, consider maximal
elements. Minimal elements may also be of interest; for example, X and 0 are both minimal, having only one view class each. Note that every countable subset of ~ X has
126
an upper bound with respect to < .
Consider replacing ~ X in the above matters by
smaller classes of sets, for example the connected closed sets, or the residual closed sets. In general, say A is universal for ~ C ~ X if A is consistent with all view classes of elements of 2 ; investigate such universality. 7.4 Note that a view space (X,.~) can be identified with the similarity view space (X,{Id X }, . ~ ) , the similarity group being trivial. Section 3 : View Self-Similarity 7.5 Call a view generative if it is a generative view of some set (the latter concept being defined prior to 3.1). Then it's quite easily proved that a view v is generative iff it is properly embedded in itself, in the sense that some similarity g embeds v in v such that g dora v c dora v. 7.6 Look for examples (if there are a n y . . . ) of self-similar sets which are significantly different from w-extensions of invariant sets, examples which are not even linked to the matter of invariant sets. At present I know of none. Is there perhaps some iterative method for producing examples by forcing the self-similarity in some way? Is there a result which merely asserts the existence of self-similar sets with one property or another (perhaps by use of Zorn's Lemma)? 7.__Z7For self-similar A, is it necessarily true that for all r > 0 there's a ball of radius r which is disjoint from A (i.e. that "A contains arbitrarily large holes")? Section 4 : Self-Similarity of Some w-Extensions of Invariant Sets 7.88 The notion of w-extensions can be used to give explicit examples of tilings of X by isometric copies of an invariant set K satisfying the conditions in 7.15 of Chapter 3. For suppose i E F-W with i K = X , and for n >_ 0 let T n = {i~n ( f K ) [ fE F n }, the image under iln of the set of nth-level images of K, which is a tiling of iln K by isometric copies of K . Noting that in general Tn c T n + l , ~ T is thus a tiling of U i~n K = X by isometric copies of K . Note also that there ex/sts i E F -oJ such that n i K = X , by using 4.6 of the present chapter with A = K ; for we can namely take i = f - w for any fE F <w with f K C_K (and such f exist in abundance). 7.9 Concerning 4.4 and 4.6, investigate further the question of which i E ( h F ) - w give i D = X. 7.10 In connection with 4.5, although i K ~ j K (and in particular, although i K is view-similar to j K ) i K need not be similar to j K . It can be shown for example that in the case where K is the Cantor set generated by the usual two contractions,
127
i K ~ j K iff some tail-end of i equals some tail-end of j (a tail-end of a sequence i being a sequence of the form (i n ,in+ 1 ,in+ 2 , . . . of 1.1 is false.
) for some n ). Thus the converse
Section 5 : The View Topology 7.11 Using the view metric d, or more specifically its restriction to the normal views (see Section 6 for the definition of normal views and normalization),
we can
metrize ~ ~ as follows. Define p on Y by
p (v I ,v 2 ) = min { d ( f l v 1 ,f2v2 ) [ each f i is a similarity mapping dora v i to (0)1 } = min {d(u 1 ,u 2 ) ] each u i is a normal view similar to v i } , the minimum existing by compactness of the group of rotations about 0.
This is a
pseudometric in which p (v 1 ,v 2 ) = 0 ¢=~ v 1 ~ v2 , hence we obtain what we shall call the view class metric d c on 7/"" (the subscript c standing for 'class') by
d c (vl~,v2 ~ ) = p (v 1 ,% ), or equivalently, d c ( V t ,V 2 ) = min {d(u 1 ,% ) I each u i is a normal realization of V/}. The following proves that d c gives the view class topology. We must show that for UE ~"* and VE Y, U E # V ¢:~ d c ( U , V * ) , ~ O . Suppose UE # V . ~:Y~
Y~
is open).
Taking normal v E V, there's u E U with u E #v (as the map Where g normalizes u , then as
domuE#(O)l
it easily
follows that g is near the identity map f , so by 5.14 g u E # f v = #v. d(gu,v)~0,henceasguE
U andre
We thus have VC V*, d c ( U , V * ) <_ d(g u ,v) -- 0.
Conversely, suppose d c ( U , V * ) ,~ 0 .
Then where d c ( U , V * ) = d(u,v)
u E UN (norm Y ) * and v E V * N (norm Y)*, since norm
with
is a compact Hausdorff
space with topology given by d, and d(u ,v) -- 0 , there's w E norm Y with u ,v E #w. Thus
V*=v~E#w~
giving V = w ~ , a n d
U=u~E#
w~=#V.
7.12 It is quite easily proved (using continuity of the view map) that
~" is
path connected. Hence so is Y ~ . 7.13 Call A view class compact if its set ( . ~ A ) ~ of view classes is compact (i.e. closed) in the compact Hausdorff space ~ ' ~ (equivalently, if every . - v i e w of A is similar to a nearstandard . - v i e w of A). Which sets have this property? Note that no compact connected set does; indeed, every compact A which is view class compact has an isolated point, noting that (for any x E X ) the limit in Y ~ of ((z)r A ) ~ as r ~ oo is ((0)1 ,{0}) ~. Any affine subspace of X is view class compact;
for example a line.
Less trivial, we have the following. Firstly, say ~'A is compactly generated if for some
128
compact $ C.~, YA = ( $ A ) ~ (every view of A is similar to a view of A with domain in $). This implies A is view class compact (but not conversely), and may be the more significant condition of the two to study.
Certain (perhaps all?) w-extensions i K
satisfying the hypothesis in 4.4 should obey this condition (stemming partly from the fact that they have generative views) and would thus be view class compact. Is every self-similar set view class compact? Is there any interesting special theory to be done in connection with view class compact sets? Consider also the stronger condition that the set of oriented view classes of A be compact in Y-=, equivalently that the set
{norm v I v is a view of A } of normalized views of A be compact in norm 7.14 Regarding the view topology on Y, if we add in the extra generating sets [K] = {(D,I) I KC I } where K is compact, we get a finer topology on Y, in which
(E,J) is near (D,I) iff (E,J) is near (D,I) in the original view topology and I # c J . This topology is a natural one to consider in connection with views of regular closed sets; roughly speaking, it compares with the original view topology in the same sort of way that the body topology compares with the Vietoris topology. Further exploration is needed on its possible uses and benefits. Section 6 : A Definition of 'Visually Fractal' 7.15 For x,y e A we say A is limit-similar at z and y if A is visually convergent at x and y to the same limit. Note that this condition is independent of the condition that A be similar at x and y ; the latter doesn't even imply visual convergence at x and y . The equivalence classes are called the limit-similarity classes of A. Generalizing, for a E A and b e B
we say A and B are limit-similar at a and b if A and B are
respectively visually convergent at a and b to the same limit. 7.16 The view metric on the normal views is implicitly used in the phrasing of a result of Tan Lei described in [DK] (see page 102 of Bodil Branner's contribution), concerning the visual similarity between the Mandelbrot Set M and the Juha Set Jc around a Misiurewicz point c.
The result in question prompts the following definition.
Suppose a e A E ~ X and b e B e ~ X , and for some direct similitude h with fixed point a (i.e. h is the composition of a dilation about a and a rotation about a ),
d(norm ( a ) r h A, norm ( b ) r B ) converges to 0 as r decreases to 0. Then we'll say A and B are alike at a and b (which in particular holds if A and B are similar at a and b, or limit-similar at a and b ). In this terminology the result of Tan Lei implies that M and J c are alike at c and c. 7.17 The notion of limit views can be used to give formal justification to certain informal statements about gradients in the subject of calculus. Consider for example the graph Gf in ~ of a continuous map f : ~ - ~ .
Gf is closed in ~ since f is
129
continuous, and we can therefore talk of views of G f . Then where x E ~ and p = (x, fx), f is differentiable at x with derivative A iff Gf visually converges at p to the normal view of the line through 0 with gradient ~ ; this then is a formal counterpart to saying that as we look closer and closer at p, the part of Gf we see looks more and more like the line through p with gradient A. Note also the close connection with the idea of the 'infinitesimal microscope' described in [Ke] and attributed to K. D. Stroyan. 7.18 Limit views can be used to formulate classes of sets which may be interesting to investigate. For example, in ~ consider the visually convergent sets A such that for
all x E 0.4, lim A x is a view of a half-plane whose boundary line passes through 0. 7.19 As a weakening of the concept of visual convergence, we'll say that A is weakly convergent at z E A if ((Z)r A) ~ converges in 7 / ~ as r decreases to 0. This is implied by convergence at z , but the concepts are not equivalent. For example, if A is a spiral in ~2 with centre x, then ((Z)r A) ~ is the same for a// r , so A is weakly convergent at 0 ; but A is not convergent at 0. 7.20 Generalize 6.9 to ~n. 7.21 Is it true that a view self-similar set A ~ X is fractal iff it is fractal somewhere ? 7.22 Characterize the boundary points at which the Mandelbrot set is fractal. Conjecture:there are only countably many boundary points at which the set is
convergent. Miscellaneous 7.23 In matters concerned with ~n we concentrated on having the direct similitudes form our similarity group.
Consider generalising as much of the work as possible to
the case where the similarity group consists instead of all affine bijections. 7.24 The following view-related properties should be worth considering.
We say A
has indeterminable scale if every view class of A has A-realizations (realizations which are views of A) of different radius.
This expresses that the scaJe of A cannot be
determined by view classes alone. Stronger, A has completely indeterminable scale if every view class of A has A-realizations of arbitrarily small or large radius ;
for
example, any w-extension i K in 4.4 has this property. For self-similar A, every view class of A at least has A-realizations of arbitrarily small radius. Another condition of the same ilk is that every view class of A have at least two A-realizations. All these conditions and others are concerned with how well a set A serves as a landmark by which one is trying to ascertain ones bearings, given view classes (views from a remote camera) alone.
130
7.25 So far we have talked only of views of sets, but it is possible to sensibly define views of other types of object; for example, views of continuously coloured closed sets. For a space X, and a space Y thought of as a space of colours, we can define a
continuously coloured closed set as a continuous map f : A --* Y for some ,4 E ~ ' X . Taking a view structure.~ on X, the D-view of f is then defined as (D,f tD ), f tD being the 'object part' of the view, our 'object set' comprising the continuously c01oured closed sets. The basic definitions and theory regarding views of sets largely carry over to the new case (including the matter of topologizing the view set jr in a natural way). Note incidentally that a closed set can be thought of as a continuously coloured closed set which is constant ; we thus have a generalization of closed sets. A significantly different area to explore and develop is that of views of Borel measures m on closed subsets A of a space X (more precisely, on the Borel algebra 2 A of A) such that the support of m is the whole of A. For such a measure, the D-view of rn would be defined as (D, m [ D ) where ra tD is short for rn ~~ ( A fl D ). In view of these remarks, the question arises of whether or not we can usefully
axioraatize the framework of 'views of objects' (along with the later additional structure of having a 'similarity group' G involved). It seems that one can go a significant way along these lines, but on balance I believe the matter should be placed on 'hold', pending further exploration in the various separate instances described so far (namely views of sets, coloured sets, and measures), along with any other significantly different instances (of which I presently know none).
Appendices
Appendix 1. Topological Monoid Actions
P age 132
2. Finite and Infinite Sequences in a Hausdorff Space
134
3. Continuity of fix : Contrac X --*X 4. Reductions of a Metric Space 5. Nonoverlapping Sets, and Tilings
137 139 144 149
6. 7. 8. 9.
The Body Topology The S-Compact Topology The Hyperspace of Convex Bodies Similitudes
152 155 158
132
Appendix I : Topological Monoid Actions Recall that a monoid consists of a set M together with an associative binary operation on M (usually just denoted by juxtaposition) and an identity element e with respect to the operation.
If M is further equipped with a topology making the
operation continuous (as a map M ~ M ~ M, giving M ~ M the product topology) we have a topological monoid. Suppose M is one such. Then for a topological space X, a topological monoid action of M on X is a monoid homomorphism O: M ~ C (X,X) (the latter being the monoid of continuous operations on X under composition o ) whose corresponding 'evaluation map' the action of p ,
M ~ X ~ X is continuous.
For p E M, Op is
and (0p)z is usually just written as pz; the homomorphism
requirement thus asks that in general
(pq)z = p (qz) and ex = x.
map sends each (p,x) to pz of course.
The evaluation
Continuity of this map (i.e. q E #iv and
a E #z # qa E # px ) actually implies that the action of each p E M is continuous. Note that if X is a locally compact Hausdorff space, a monoid homomorphism
O:M-* C (X,X)
is a topological monoid action iff 8 is continuous, where C ( X , X )
has the compact-open topology. A1.1 Proposition
For a Hausdorff monoid M, ~ M forms a Hausdorff monoid under
P Q = { p q [ p E P and qE Q}, and the map
the operation
p~{p}
is a
topological monoid embedding of M in J ~ M .
Proof: Each PQ is i n ~ ' M being the continuous image of compact P ~ Q under the monoid operation on M. Associativity of the new operation follows from that of the original, and the identity element is { e}. Likewise with continuity, for if P ' E # P and Q' E #Q then P ' ~ Q' E # ( P x
Q) i n J ~ ( M x M) so by continuity of the original operation
M x M ---* M and hence of the induced operation J ~ ( M x M) ~ ~ M
P' Q' E # PQ as required.
Chapter 1) we get
(see 8.1 of
Lastly, we know the map
p ~-* {p}
embeds the space M in the spaceJ~M, and it's also a monoid embedding since in general {pq} = {p}{q} and {e} = the identity element o f ~ M .
o
A1.2 Proposition If a Hansdofff monoid M acts on a Hausdorff space X, then
pA = {pal aE A}. (2) ~ ' M a c t s o n ~ X b y P A = {pa I p E P a n d a E A } = (1) Macts o n ~ X b y
LJ pA.
pEP Proof: (2) Each PA is inJ~'M being the continuous image of P x A under the evaluation map oftheoriginalaction.
aEA}=P(QA)
(PQ)A = {(pq)a [ pE P, qE Q, aE A} = {p(qa) [ P E P , qE Q,
and { e } A = { e a ] a E A } = A
continuity of the evaluation map ~ M x ~ X ~ ~ X
so we h a v e a m o n o i d a c t i o n .
And
follows from that of the original.
For if P' E #P and A' E #A then P' x A' E # ( P ~ A) i n J ~ ( M ~ X ) so by continuity
133
of the evaluation map M x X ~ X and hence of the induced map J~'(M x X ) ~ ~ X we obtain P' A ' E # PA as required. (1) By (2) and the embedding of M i n i M .
o
Note that if we take X to be a locally compact Hausdorff space and M to be C ( X , X ) under composition and the compact-open topology, with M acting on X in the natural way (each fE M being its own action on X ), then the action of J ~ M on ~"X is the map
U : J~C ( X , X ) --. C ( J d X , ~ X )
defined at the end of section 9 of
Chapter 1, and the continuity of this map, noted there in 9.4, is now alternatively proved by (2) above, which gives a fuller picture of what's going on.
134
Appendix 2 : Finite and Infinite Sequences in a Hansdorff Space Throughout this section let F be a Hausdorff space. A finite sequence in F is a function n ~ F where n E w = {0, 1, 2 . . .
} ; here we are treating naturals in their
set-theoretic sense so n consists of all lesser m .
n is the ~
of the above finite
sequence ; the set of finite sequences in F of length n is thus F n In distinction to the generic symbols f and g for elements of F we may denote finite sequences in F by f and g in bold type.
The length of f will be denoted by
I l l • The set of finite
sequences in F is denoted by F <w, being U F n , and this forms a monoid under the n natural operation of concatenation wherein fg is ' f followed by g ', a finite sequence of length I l l + [g [, the first If l terms being those of f and the subsequent terms being those of g. Algebraically this is the free monoid generated by F. The identity element is the finite sequence of length 0, namely 0 (the 'empty sequence'). We also have a partial order < wherein f _< g ~
g extends f, the least element being ~. A finite sequence f of length n may sometimes be written as ( f 0 , • • • , f n _ l ) or even just f0 • • - "fn-1 " In particular a sequence ( f ) of length i is often denoted by (or even identified with) its term f . By a sequence in F (without the term 'finite' appended or implied) we mean a function w -* F . The set of sequences in F is thus F w. As with finite sequences we usua~y use bold type f and g for sequences in F, though a sequence f may also be written as ( f n ) • For fE F w and n E w, f In denotes the restriction of .f to n , a finite sequence of length n . The product topology on F <w is the unique topology on F <w in which each set of the form F n is open and has its natural product topology. It is easily seen t h a t . . . A2.1 Proposition
The product topology on F <w makes F <w into a Hansdofff
topological monoid in which F i s embedded by f ~ ( f ) .
o
Under the binary operation and topology described above then, F <w can be thought of as the 'free topological monoid generated by F '. We now consider the set F _<w = F <w U F w of finite or infinite sequences in F. Extending the operation of concatenation to the case of forming the infinite sequence
fg
from finite f and infinite g in the obvious way, it is clear that F <w forms a
'partial monoid' in the sense that concatenation is associative wherever the definition makes sense (namely, f(g h ) = (fg) h whenever f and g are finite), and there is an 'identity element' having no effect under the operation (namely 0). F <w can be extended to one on F <w in a natural way too.
The topology on To facilitate the
description let 0 denote some element not in F , and let F 0 = FU {0} have the topology in which F forms an open subspace and {0} is open (i.e. 0 is isolated). Now identify each f e F <w with f 0 w E F0 w where f 0 w is f followed by repeated 'O's ; you
135
can think of the 'O's as indicating places in which no element of F is present. Under this correspondence F _w can be considered as a subspace of Fow, the latter naturally being given the product topology, and in this way we have described a topology on F _<w which we'll call the product topolok-y on F -<~ The salient facts are as follows, where for ~fE F <w i w denotes the element of F w in which I is endlessly repeated ; A2.2 Proposition Under the product topology, F _<wis Hausdorff a n d . . . (1) Each product space F n forms an open and closed subspace of F _<w. (2) F <w (with the product topology) forms an open subspace of F <w and the product space F w forms a closed subspace of F < w (3) F o r I E F w, M={gE(F<-W)* [ wCdomg a n d V n E w
gn E # I n } .
(4) f o r I E f w, f = hm (Itn) • (Hence r < W i s densein f <-w). (5) For I e f <w, i w = lim (I n ) .
Proof: As F 0 is Hansdorff so is Fow hence so is the 'subspace' F <_w. By the formulation of the monads in Fowthat in general #h = { g E ( F 0 w)* I V m E w g m E # h m
}' and
that in F 0 # 0 = {0) and (for fE F ) /~f = the monad of f in F , it's easily seen that (3) holds, hence F w receives its natural product topology, and that for IE F n, #~={gE(Fn)*
I Vm< n g m E # h m
}=the
monad of I
in F n ,
hence F n
receives its product topology and is open. The union F <w hence also receives its product topology and is open, the complement F w therefore being closed. And for IE F <w, if #f intersects (F n ), then IE F n, showing that F n is closed. (4) follows from (3) since for infinite n I In E • . subsequence of {Iwt n) .
And (5) is a corollary to (4), ( I n ) being a
o
From the description of the monads it's easily verified that the partial binary operation of concatenation is continuous wherever defined, making F _<w in this sense a 'partial topological monoid'. Also note that the map F <w ~ F w sending each f to I w is continuous. A2.3 Proposition F-<W is compact ¢=~ Fis compact.
Proof: : F ~ F 1, and the latter is closed in F _w, hence compact. ¢ : Then F 0 is compact hence so is Fow. And it's easily seen that F _<w is closed in F0w, hence is compact too. We could alternatively have proved the above by u s i n g . . .
o
136
A2.4 Note
For g E ( F _<w )*,
g is nearstaudard ~
for all finite n E dom g, g n is nearstandard.
Pro of: :Let g e # f .
IffEFWthen
VnEw,gnE#fn
sognisnearstandard.
IffEFm
then dora g = m and Vn < m , g n E # f n so 9 n is nearstandard. ¢ : S u p p o s e w C _ d o m 9 . Then l e t t i n g f E F W w i t h V n ~ w g n E # f n , w e h a v e g E # f so g is nearstandard. On the other hand suppose dom g = m E w. Then letting fE F m with Vn < w g n E # f n ' we have g E # f so 9 is again nearstandard,
o
Lastly we show that for compact F, analogous to the fact that each infinite sequence is the limit of its finite restrictions, F w and F <w are each the limit (with respect to the Vietoris topology on 3g" F <w ) of their 'finite restrictions' ; A2.5 Proposition For compact F, with respect to the Vietoris topology on ,Y~ F <w (t) F W = l i m ( F n ) . (2) F - < W = l i m ( f E n )
wheref_
t m < n }.
Proof: First note that as F i s compact so are F w, F n F -<w, and F -< n . (1) We must show that for infinite n , where S n is the set of internal sequences in F * of length n , S n E # F w. By compactness of F, every element of S n is near an element of F w. Conversely, for all fE F w, f . tn belongs to S n and is near f . So S n E # F w as required. (2) Similar sort of proof to the above,
o
137
Avvendix 3 : Continuity of fix : Contrac X --,X In this section we consider the possible continuity of the fixed point map
fix:ContracX--*X with respect to the compact-uniform and bounded-uniform topologies on the set Contrac X of contractions of a nonempty complete metric space X.
If X is locally compact the compact-uniform topology is sufficient for
continuity to hold, and this can be proved by purely topological means as follows. For a Hausdorff space X and fE C(X,X ), we say z is a tooologically attractive fixed point of f if for all y E X, (fay) --, z , which indeed implies z is fixed under f as f is continuous. Such a point will be the unique fixed point of f , and denoted by fix f . A3.1 Proposition Let X be a locally compact Hausdorff space and F = {rE C (X,X) ] f has a topologically attractive fixed point}. Then the natural fixed point map fix : F ~ X is continuous with respect to the compact-open topology on F.
Proof: Let fE F and g E F * with g E #f. Let x = fix f and suppose for a contradiction that fix g ¢ #x. Then let C be a compact neighbourhood of x with fix g ~ C *. Now -1C * is a .-neighbourhood of fix g, so as (g nx)n E w * converges to fix g in X * , there's n e w * such that V m > n
gnx t~ C*. Let n be the least such. Note that n > l
as
g x E # f x = # x C C*. By leastness of n, gn-lxE C* so by compactness of C let cE Cwith gn-lxE #c. As (fmc) --, x let m_> 1 with fmcE C. By continuity of composition in C (X,X ), g m E # f m
so g m(g n-lx) E # f m c , i.e. g n+m-lx e # fmc
giving (as # f m c C C*) g n + m - l z E C*, a contradiction as n+m-1 > n.
o
Assume from now on that X is a nonempty complete metric space. Since the compact-uniform topology on C (X,X) is the compact-open topology, and since in the above result Contrac X C_F , it follows t h a t . . . A3.2 Coronary If X is locally compact, fix : Contrac X ~ X respect to the compact-uniform topology,
is continuous with o
In Theorem 2 of [Nad], sequential continuity of fix : Contrac X --*X is proved, namely that if (fn) converges to g then (fix f n ) converges to fix g. For countably based X this is equivalent to continuity since Contrac X (moreover C (X,X)) is then also countably based (see [Du], page 265, 5.2). In his result Nadler uses the topology of pointwise convergence, but on Contrac X this coincides with the compact-uniform topology, as pointed out in 'Preliminaries'.
Without local compactness, continuity with respect to the compact-uniform topology may fail, one example being given in [Li,1] along with another proof of A3.2.
138
However, by placing an upper bound u on the Lipschitz ratios allowed, we do at least have a restricted version. For u E [0,1) let
ContraruX = {fE Contrac X [ rf <_u }.
ContracX*withrg~l,
VxEX,
: Then by transfer of a standard inequality,
d(z, fix
A3.3 ~ m m a
Forge
gxux
¢:~ z ~ f i x g .
Proof: g ) < d(x,g x)/(1-rg ) u 0
as
1-rg is finite and noninfinitesimal and d(z,g z) u 0. ¢:Then
gzufixg
too, so g x ~ z .
A3.4 P r o l ~ i t i o n
o
For any u < 1, fix :
ContracuX ~ X
is continuous with respect to
the compact-uniform topology.
Proof: fEContracuX
Let
and
gEContracuX*
with
geM.
Then where
x=fixf,
g x ~ f z = z so by the last lemma z ~ fix g as required,
o
The above is also proved in [Li,1]. As an alternative to reducing the domain of the fixed point map in order to obtain continuity, we could instead weaken the topology on
Contrac X.
Weakening to the bounded-uniform topology proves to be
sufficient, which follows from the next result along with another proof of A3.2. Recall from Section 6 of Chapter 1 that an ideal J on X is said to be 'topological' if every element of X has a neighbourhood belonging to J . A3.5 Proposition
For any topological ideal J on X,
continuous with respect to the J-uniform topology on
Proof: fE ContracX
Let
and
ge ContracX*
with gE #f.
fix: Contrac X ~ X Contrac X.
Letting
z=
is
fix f , since z has a
neighbourhood N belonging to J then for all sufficiently small e > 0, letting A = [x ]e we have A C N so A * c u J , and we find that A * is closed under g ;
gA*C(f*A*)~ = ((fA)*)
~
c ([Z]rfe
*)~
cA*
as
VaeA* ga~f*a
as
fA C [X]rfe
as
rfe<e.
So as A * is also .-dosed, transfer of a standard fact gives fix g E A *, hence d(fix g ,z) _< e. So fix g ~ z as required, If X is locally compact,
o
subCp X is topological so the above gives an alternative Bd X of bounded sets is always topological, s o . . .
proof of A3.2. Meanwhile the ideal
A3.6 Corollary fix : Contrac X --*X is continuous with respect to the bounded-uniform topology.
139
Appendix 4 : Reductions of a Metric Space
Throughout let X be a nonempty metric space. For the next paragraph Y denotes a metric space too, though we're mainly interested in the case Y --- X. A control of f : X--* Y is an increasing operation 3 on [0,oo) such that Vz, y E X d(fz, fy) < 3 d(z,y), equivalently V~ _ 0 Vz, y E X (d(z,y) < E * d(fz, fy) < 3e ). The idea is that 3 gives a distance-wise upper bound on the expansiveness of f . For example, for a map f with finite Lipschitz ratio rf , multiplication by rf is a control of f . If f has a control we'll say f is controlled, the nonstandard formulation being that f * is 'macrocontinuons', i.e. that Vz,y E X *, if z is finitely distant from y then ] z is finitely distant from f y . If f is controlled it has a least control, as follows. For f : X-+ Y define 6 / : [0,oo) ~ [0,oo] by 3 r e = ~ / { d(fz, fy) ] d(x,y) _< e }. This is clearly the least increasing function 3 : [0,oo) ~ [0,oo] such that Ve > 0 Vx,y E X (d(z,y) < e * d(fz,fy) < 3e ), i.e. ¥z,y e X d(fz, fy) < 3 d(z,y), and it follows that f is controlled iff 3f is finite-valued, in which case gf is the least control of f . ~f is known in the literature as the 'modulus of continuity' of f . Note incidentally that f is controlled iff Ve > 0 3g > 0 Vz, y E X (d(z,y) _< e :) d(fz, fy) < /~) ; compare with the definition of uniform continuity (and compare the nonstandard formulations, macrocontinuity and microcontinuity). A technical result we'll be using later is the following, where an operation 3 on [0,~) is said to be lower continuous (also known as right continuous) if whenever (an) decreases to b then ( ~f an) converges to ~f b. A4.1 Note For continuous f : X ~ Ywith X compact, (1) Ve > 0, 6re = max {d(fx, fy) I d(z,y) < e }. (2) 3f is lower continuous.
Proof: 0
(1) By a nonstandard result on 1.u.b.s in [0,oo], 3re = d(fx, fy) for some x, yE X * O
O
with d(z,y) < e. But then d( z, y) =
°d(fz, fy)
O
d(z,y) < e and d(f °z,f Oy) = d(ofz, Ofy) =
3fe.
=
(2) Let e E [0,oo) and 3' ~e with 7 > e ; we must show 3f 7 ~ 3 f e , i.e. (as ~fe < 3f 7 ) °3f7<_ ~fe . 0
0
By transfer of (1) let z, yE X* with d(x,y) _< 7 and 3 f 7 = d(fx,fy).
Then d( x, y) =
0
0
d(x,y) < 7 = e
so d(f
0
x,f Oy) < 3fe,i.e. d(°fx,°fy) < 3fe,i.e.
°d(fz, fy) < 3fe ,i.e. ° 3 f 7 < 3fe as required,
o
Note that for any control ~ of f : X - + X , for nonempty bounded A c X we have
diamfA < 3diam A , giving by induction that diamfnA <_ 6ndiam A for all n. Hence if (3 n diam A) converged to 0, so would (diam f nA). An interest in this condition leads us to consider the following definition.
140
We'll say an operation b on [0,oo) has attractor 0 if Ve > 0 also increasing (i.e. in general
e < 7 ~ be_< 67)
(bne) ~ O. If 6 is
note that
60 = 0, and the
formulations below hold. A4.2 Proposition
For an increasing operation b on [0,oo),
(1) b h a s a t t r a c t o r 0
(=~ V e > 0 ( b e <
(2) If b is lower continuous,
e and 3 7 > e 67_< e )
6 has attractor 0 ¢~ Ve > 0
be < e.
Proof: (2) is a simple corollary to (1), which is proved as follows.
bne >_ e, (bne) ~ O. So e < be. Secondly, if V7 > e 67 > e , then by induction,
# : Let e > 0. If 6e >_ e then as 6 is increasing, by induction we'd have Vn contradicting
for 7 > e we'd have Vn 6n7 > e contradicting (6n7) -~ 0. So 37 > e b7 _< e . ¢ : First note that
b 0 = 0, since
Ve>O bO<_ be< e (using that b is increasing). It
remains to show that for e > 0, (bne) ~ O. Suppose (bne) does not converge to 0. Then no 6he can be 0, so (bne) is strictly decreasing since Vr > 0 6r < r . So let r=lim(bne) =/kbne. As r > 0 there's 7 > r with 67 _< r , and since for some n n
6he-< 7 t h e n a l s o 6n+le=66ne _< 67_< r giving 6n+le _< r , a contradiction. So ( bne) ~ 0 as required. o We'll say f : X ---X is a reduction of X if it has a control with attractor 0, in which case 6f is the
least such control (being sandwiched as it is underneath a//
controls of f ) . By the last result note t h a t . . . A4.3 Corollary
For f : X --*X,
f is a reduction ** Ve > 0 ( 6re < e and
>e
bfT_<e)
Proof: : By the last result, putting 6 = 6 f . ¢ : Then bf is finite-valued, hence is a control of f , and by the last result bf has attractor O. o A4.4 Note
Any composition of reductions is a reduction.
Proof: If f and g are reductions, so is
fog since 6fog <- bfobg < 6g.
o
The next result compares the property of being a reduction with various other contractivity conditions, for which some definitions are in order. Encountered in [Ha] is the definition that f : X ~ X
is a weak contraction if
Ve>0
3 7 > e 6 f 7 < e.
Hata notes that every contraction is a weak contraction but not conversely, f : X - * X is said (in the terminology of some of the literature of fixed point theory) to be contractive if for all distinct
x,y E X, d(f x,f y) < d(x,y).
141
A4.5 Proposition Below, for f: X ~ X with equivalence if X is compact. (1) f is a weak contraction. (2) f is a reduction.
(3) v,>0
the implications (1) ~ (2) ~ (3) $ (4) hold,
by,<e.
(4) f is contractive.
Proof: (1) $ (2) : Then also, V e > 0
~fe<e
(for where 7 > e
with ~ f T < e ,
we have
e < ~f7 < e. So by A4.3, f is a reduction. 2) ~ (3): By A4.3. (3) $ (4): Then for all distinct z,y e X, d(f z, f y) <_ ~fd(z,y) < d(x,y) . Assuming now that X is compact and (4) holds, we prove (1). Let e > 0. By contractivity of f and A4.1(1), we have ~fe < e . And since by A4.1(2) ~f is lower-continuous, it follows that 37 > e 6f7 < e as required, o
~f
As a corollary, note that reductions are continuous since contractive maps are. The following example shows that reductions need not be weak contractions. Let X = { x n[ n>_ 1}U{Yn[ n>_ 1}U{p} where the symbols involved denote distinct points. Let X have the post-office metric about p in which d(xn ,p) = 1 + en and
d(y n ,p) = 1, where en = 1/2 n . Being a 'post-office metric about p' means that for a # b, d(a,b) = d(a,p) -t- d(p,b). Note that X is bounded and also complete since distinct points are distance >__1 apart. This fact also implies that every point is isolated, so in particular X is locally compact. Define f : X--,X by fXn = yn, f Y n = p ' and f p = p. By checking the various cases carefully it can be shown that 6fis0on[0,1],lon(1,2],and2on(2,oo).
Thus for e > 2 , S f 3 e = 6 f ~ 2 = ~ f l = O ,
so (~fne) is eventually 0 hence converges to 0. So f i s a reduction. However, f is not a weak contraction since there is no 7 > 2 with ~f7 < 2. We now show that Banach's contraction mapping theorem generalizes to reductions. Note that we say a sequence (an) in [0,oo) strictly decreases to 0 if it's a decreasing sequence converging to 0 and for all nonzero an , an+ 1 < an . A4.6 Lemma For a reduction f of X, (1) For all nonempty bounded A c X, (diam f n A ) strictly decreases to 0. (2) For all x,y e X, (d(f~x, fny)) strictly decreases to 0.
Proof: (1) In general diam f n A <_ ~fn diam A, so (diam f n A ) ---* 0 as (6f n diam A) --, O. As f is contractive, (diana f n A ) is decreasing ; and for any nonzero term diana fnA, diam f n + l A = diam f ( f n A ) < 6f diam f n A < diana fnA. (2) By (1) with A = {z,y}, noting diana f n A = diana {fnz,fny} =
d(fnz, fny).
o
142
A4.7 Proposition For complete X, every reduction f o f X has a metrically attractive fixed point, i.e. a fixed point p (necessarily unique) such that for all z E X (d~f nz, p)) strictly decreases to 0.
Proof: Let z E X. Letting x n = f n z , we first show that (xn) is a Cauchy sequence. Putting y = f x in (2) of the last lemma, (d(x n ,Xn+l) ) decreases to 0, so for all infinite m we have z m ~ Zm+ 1 . Suppose (xn) is not Cauchy. Then let e > 0 such that Vm 3n > m d(x m ,Xn) > e. n > m with
>
Taking infinite m E w * and letting n be the least
then
so
< 6f
, so
&re > ° d ( f x m ,fxn_l) = °d(Zm+l ,x n ) = °d(z m ,z n ) > e (using along the way that x m ~. Xm÷ 1 ) contradicting that ~fe < e . So, (xn) is Cauchy. So by completeness of X, (Xn) converges, say to p , which by continuity of f is therefore a fixed point of f .
And for all y e X, (d(fny,p)) = ( d ( f n y , f n p ) ) which by
(2) of the last lemma strictly decreases to 0, showing that p is metrically attractive,
o
Note that for compact X the above result says that every contractive map on X has a metrically attractive fixed point ; this result appears in [Ed]. It is worth pointing out a dynamic difference between contractions and reductions. For f : X---*X we define the ~ of the path ( f n z ) of x under f to be ~ d ( f n x , f n ÷ l x ) , which if f is a contraction, is finite (hence ( f n x ) is Cauchy, n_>0 which then leads to Banach's contraction mapping theorem). However, under a reduction, or even a 'weak contraction', paths may have infinite length, so that intuitively, although as a point traces out its path it converges to a limit (assuming X is complete), it may travel an infinite distance along the way. Here is an example. Let X = {zn ] n >_ 1} U {p} where the symbols involved denote distinct points, and give X the post-office metric about p in which d(x n ,p) = I / n . Define f : X ~ X by f p = p and f ~ n = ~ + 1 " Then f is contractive hence (as X is compact) a weak contraction, but the path (xn) n _> 1 of z 1has infinite length. For the remainder of the section assume X is complete and let Reduc X denote the set of reductions of X. We now consider the possible continuity of the fixed point map fix : Reduc X ~ X . Basically all we have to do is modify some of the proofs of the corresponding results for contractions given in Appendix 3. For a start, with the same proof as that of A3.2, A4.8 Proposition If X is locally compact, f i x : R e d u c X ~ X respect to the compact-uniform topology,
is continuous with o
143
The next two results parallel A3.3 and A3.4. For any increasing operation u on [0,c~) with attractor 0, let ReducuX = {re Reduc X I uis a control off, i.e. ~f _< u }. A4.9 Lemma For any g E Reduc X * such that for all real e > 0 VzEX,
gz=z
~g e ~ e ,
~=~ z ~ f i x g .
Proof: :Suppose x ~ f i x g ;
let e > 0 b e r e a l w i t h
fixg ~[x]e*.
I n X * the ,-sequence
(g nz)n E u~* converges to fix g, so for all sufficiently large n E w *, g nx ~ [x ]e * Let n be the least such. n > 1 since g z u x.
d(gnz, g z ) < 6 g e
so
By leastness of n, d(g n-lx, x) <_ e so
°d(gnz, gz)_<°~ge<e.
But as g z u x
then
°d(gnx, z ) =
°d(g nz, g z) < e which contradicts d(g nx, z) > e. So z ~ fix g after all. ¢:Then gz~fixg
too, so g z ~ x .
o
A4.10 Proposition For any increasing operation u on [0,oo) with attractor 0, fix : ReducuX ~ X is continuous with respect to the compact-uniform topology.
Proof: Let /e ReducuX and ge ReducuX* with gE #f. Then where z = f i x / , gz~ f z = z so by the last lemma (noting that for all real e > 0, ~g e _< ue < e giving 8g e ~ e), x ~ fix g as required,
o
Finally, in the same way as in Appendix 3, the following provides an alternative proof of continuity with respect to the compact-uniform topology, and gives continuity with respect to the bounded-uniform topology ; A4.11 Proposition For any topological ideal J on X, fix:Reduc X--*X continuous with respect to the J-uniform topology on Reduc X.
is
Pro o/: Identical to the proof of A3.5, except replacing rf e by ~f e.
o
A4.12 Corollary fix : Reduc X ~ X is continuous with respect to the bounded-uniform topology,
o
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Appendix 5 : Nonoverlapping Sets, and ~ n g s Until further notice let X be a Hausdorff space. Recall that the boundary of a closed set A is OA = A - `4. We'll say closed sets A and B are nonoverlapping in X if they can intersect only at their boundaries, i.e. if A N B C OA N OB, equivalently if A fl B = 0 and B fl A = 0.
This implies that A and B are disjoint, whilst being
equivalent to the latter if A and B are regular. Note also that if A and B are nonoverlapping, so are A' and B ' for any closed A' c A and B' C B . If Uand Vare
Residual dosed sets are trivially nonoverlapping. We'll say a family (Ai] iE I ) of closed subsets of X is nonoverlapping in X if i $ j ~ A i and Aj are nonoverlapping. disjoint open sets, U and V are nonoverlapping.
A5.1 Proposition
For A,B C C C X with A and B closed in X,
A and B are nonoverlapping in C :~ A and B are nonoverlapping in X, with the converse holding if A and B are regular.
Proof: : Where - c denotes 'interior in C ' we have ,4 c A_c , so where 0c denotes 'boundary i n C ' , O c A C OA. Likewise0cBC ~B. Hence A N B C 0cANOcB C OANOB. ¢ : A is disjoint from B c since if it intersected B_c (which is open in C ) so would A (as A is dense in A) hence _A would intersect B , a contradiction. Likewise B is disjoint from AC, so A and B are nonoverlapping in C. We say a family ( A i l i E I )
o
of subsets of X is a tiling of X if it's a
nonoverlapping cover of X by nonempty regular closed sets. This can be reformulated as the property (2) below of 'strong nonoverlapping' which automatically implies the regularity condition (that each A i be regular) ; A5.2 Note For a cover (A i I i E I ) of X by nonempty closed sets, the following are equivalent ; (1) (Ai[ i E I ) is a tiling of X. (2) V i E I A i a n d ~!Aj arenonoverlapping.
J (3) V i E I
A i is disjoint from the interior of ~iAj .
J
Proof: (1) a (3) : Let i E I and B =
U A..
j$i
If A i intersected the interior of B then so would
)
the interior V of A i (being dense in A i as A i is regular), thus B would intersect V, hence so would ~iAj (as this is dense in B), hence some Aj (j $ i ) would intersect V,
J
contradicting that Aj and A i do not overlap. (3) a (2) : Take i and B as above. We need to show that B is disjoint from the interior V of A i . If B intersected V, so would ~iAj , hence so would some Aj (j $ i ), hence
J
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Aj would intersect the interior o f UjA k (as this expands V), contradicting (3). (2) =~(1) : Since in general A j C j ~ i A j , we see that (A i I i E I ) is nonoverlapping, so it remains to show that each A i is regular. Let B = ~iAj and Vbe the interior of A i .
J
The complement -~B of B is an open subset of A i so -~B c V ; and as A i is disjoint from _Bwe have A i c_ -~ B = ~B C V, so A i is regular as required,
o
AS,~ Corollary If A 1 and A 2 are nonoverlapping nonempty closed sets with union X , then A 1 and A 2 are regular.
Proof: (A i I i e I ) obeys (2) above, hence obeys (1), so each A i is regular,
o
So far we've been talking of tilings as families, but for some purposes it is more natural to consider tilings in the form of sets. For the remainder of this appendix a tiling of X will mean a set ~ of mutually nonoverlapping regular closed sets which covers X. (Of course, ~ C~ X i s a tiling of X iff its corresponding family (A I A e ~ ) is a tiling of X in the earlier definition.) The following, used in 7.1 of Chapter 3, says that if ~ is compact in the Vietoris topology on ~ X then ~ must be finite ; A5.4 Proposition With respect to the Vietoris topology on ~X, every compact tiling of X is finite.
Proof: Suppose ~
is infinite.
compactness of ~ BE ~ * - { A * }
Then let B be a nonstandard element of ~ *, and by
let A E~f with B e #A. we have
Letting C= U ( ~ - { A } ) ,
B C U(~ *-{A *})= U((~-{A})*)=
then since C * , hence
A c st B c st C * = - C , contradicting that A and C are nonoverlapping (essentially given by (2) of A5.2).
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We conclude this appendix with a nonstandard proof of an existence theorem for tilings, bringing various hyperspace work into play. Throughout let X = ~n, ~BX be the set of bodies in X under the body topology (see Appendix 6), and ~" E 2~BX. We shall be concerned with the existence of a tiling of X by copies of elements of 3", 'copy' being used in the following variable sense. Let G be any group of isometries of X which is closed in the group Sire X of similitudes of X under the compact-open topology; for example G could consist of aU isometries, or just the direct isometries, or just the translations. Then letting ~ be the equivalence relation in ~BX induced by the action of G on ~BX, we'll say B is a copy of A if B ,.. A. Roughly speaking, the tiling existence theorem states that if we can do an arbitrarily good job of tiling over bounded subsets of X with copies of elements of ~',
146
there exists a tiling of X by copies of elements of 22. To state this more precisely we have our last few definitions.
A uacking of X is a set of mutually nonoverlapping
nonempty regular dosed subsets of X ; thus a tiling of X is just a packing of X which covers X. If the sets involved are all copies of elements of 2" we shall respectively refer to a 2"-packing and 22-tiling of X. As a weakening of the notion of covering, we'll say a 3"-packing Jg of X covers B C X towithin toleran¢.e .e if Be_ [[J Jg]e" T h e n o u r tiring existence theorem states that if every bounded subset of X can be covered by 22-packings to within arbitrarily small tolerances, there exists a 22-tiling of X.
In
particular we'll have the trivial corollary that if every bounded subset of X can be covered by a 3"-packing then there exists a 22-tiling of X ; a restricted version of this is given in [GS] for X = ~2 and 22 being a finite set of topological discs. The basic idea of the nonstandard proof is similar to that used for example in the nonstandard proof of the graph-colouring theorem that if every finite subgraph of a graph G is n-colourable, the whole graph G is n-colourable.
There, one takes a
hyperfinite intermediary H of G, for which by transfer there's an n-colouring, and one then simply restricts this colouring to G to obtain an n-colouring of G. The situation for tilings is more involved but has the same flavour. Assuming every bounded subset of X can be covered to arbitrary tolerances by 2"-packings, then taking a *-bounded set B expanding bd X *, by transfer there's a . - ( 22-packing of X) ~ covering B to within some infinitesimal tolerance, and by taking the 'standard part' of ~
we'll
obtain a 22-tiling of X. The work comes in specifying just what this 'standard part' is, and verifying that it is a 3"-tiling of X. A5.5 Note The group action of Sim X on ~BX is topological.
Proof: In other words we must show that the evaluation map
S i m X x ?BX~ ?BX is
continuous. Suppose g E #f and B E #A (recall that the latter means A # C B c A # ). We need to show g B E 9 f A , i.e. [_A# c g B C_(fA) #. As _A # c_ B then (using A9.1 for the third equality) f__A_# = (fA) # = f ( A # ) = g (A # ) c g B . And since B C A #,
g S C g(A #) = f ( A # ) = (f A) #.
o
Let 2 " " denote the set of copies of elements of 2". For use in the next result, note that since any g e G * has finite noninfinitesimai scale factor 1, g E G * either maps bdX*
entirely outside bdX* or leaves bdX* invariant, in which case g E ns G * (using A9.2 and closure of G in the group of similitudes). Let , ~ X have the Vietoris topology (given by the Hausdorff metric h recall). A5.6 Proposition
The body topology on J ~ coincides with the Vietoris topology,
and 3" ~ is dosed in £gX.
Pro of:
147
We show that if K E JYX and B E ( 5" ~ )* with B E/~vietoris K, then K E `V ~ (which shows 5" "" is closed in ~ X ) and B E P~ody K (which shows (taking K E 5" "" in the first place) that on 5" ~ the Vietoris topology is a refinement of the body topology hence (as the latter's a refinement of the former) they're equal). As B E (`V ~ )* there's S e `V * and g E G * with g S = B .
As `V is compact in
the body topology, let T E`V with S E g-body T . Since g maps some bounded point (e.g. any element of S ) to a bounded point, by the remarks preceding this proposition g is near some f E G . B E ~ietoris f T
By A5.5 we then have
so as ~ X
B=gSE#bodyfT,
is Hansdofff, K = f T E `V ~,
hence also
and we also have
B E ~body K .
o
A5.7 Note (1) There exists 6 > 0 such that every element of `vexpands a ~-ball. (2) {diam T ] T E `V } is bounded above in (0,oo). Proof: (1) For A E ~BX, using compactness of A there's a largest $ > 0 such that A expands a 6-ball. The map !BX ~ ( 0 , ~ ) sending each A to its largest such ~ is easily shown to be continuous, hence it achieves a minimum on compact `V. (2) Use compactness of `V and continuity of diam : !BX-~ (0,oo). Alternatively note that `V is a compact subset of 3dX, hence bounded, hence U `v is bounded in X.
o
A5.8 Proposition Every `v-packing of X is closed in JgX. Proof: Let ~4 be a `v-packing of X. Then where 6 is as stated in (1) of the last note, it follows that for any distinct A 1 ,A 2 e~4, h(A t ,A s ) > 6 (because where f~ G and T e ` v w i t h f T = A1, and [x]6C T , then [fz]~ = f [ z ] ~ c A1, so as [fz]~ and A s are nonoverlapping, d(f x, A 2 ) >_ ~ ). Hence ~4 is dosed, o Every `v-packing of X thus belongs to ~¢JgX, which we now make into a compact Hansdorff space by giving it the S-compact topology ; so for 2 E ~J~X *, ° 2 = st~T = { ° B I B E b~ with B E ns Jff X * } = { s t S {stBI BE~withBC b d X * }.
I B E ~ with S C ns X * } =
A5.9 Proposition The set of `v-packings of X is closed (i.e. compact) in ~ X . Proof: Let 5~ be a *-(.V-packing of X); we must show that s t ~ is a `v-packing of X. As .~ C (5" ~ )*, s t ~ C s~ (5" "~ )* = 5" ~ as `V ~ is closed. It remains to show that distinct members A 1 and A s of s t ~ are nonoverlapping. Let BiEb~ with BiE #.4 i (with respect to the Vietoris topology, hence (by A5.6) with respect to the body topology). Then A_i # c B / , so as B_t N B~ = 0 (B 1 and B 2 are nonoverlapping, noting B t ~ B 2 as J~X is Hansdofff) we have (as .4i c _Ai # ) A t N A s = O as required,
o
148
AS.10 Lemma For any .-(Y-packing of X ) ~ ,
U s t ~ = st U ~ .
Proof: For any B e ~
with s t B ¢ O ,
Bc b d X * (noting that B has finite diameter, by
A5.7(2) and the fact that the elements of G are isometries).
Lj{stS] B e 2 w i t h B C _ b d X * } =
[J{stB I B e ~ } , i . e .
Thus we have
[Jst~=st[j,~.
o
AS.11 Proposition For any *-( Y packing of X ) 2 , s t ~ is a Y-tiling of X ¢~ st U ~ = x ,
Proof: w e know that s t 2 is a Y-packing of X by A5.9. Thus s t ~ is a Y-tiling of X iff LJstS~ = x , i.e. (by A5.10) st LJ ~ = x.
o
As a corollary we can note that the set of Y-tilings of X is closed (i.e. compact) in g ~ X , the standard part of every .-(Y-tiling of X ) ~ being a Y-tiling of X as st LJ ~ = st x * = x. However, we now come to the main result; A5.12 Tiling Existence Theorem If every bounded subset of X can be covered by Y-packings of X to within arbitrarily small tolerances, there exists a Y-tiling of X.
Proof: Taking ,-bounded B expanding bd X*, and infinitesimal e > 0, by transfer there's a • -(Y-packing of X ) ~ w i t h Y-tiling of X .
BC[U~]e,hence
st U ~ = X , s o b y A S . 1 1 s t ~
isa o
A5.13 Corollary If every bounded subset of X can be covered by a Y-packing of X, there exists a Y-tiling of X. o
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Appendix 6 : The Body Topol0g~ Throughout let X be a regular Hansdofff space, ~BXbe the set of bodies in X, and ~3iX be the set of body-interiors. As noted in 'Preliminaries', the bodies correspond bijectively with the body-interiors via the mutually inverse maps A ~ `4 and D ~-* D . In this appendix we consider a topology on ~BX (equivalently one on ~BiX, via the above correspondence) which takes account of interiors of sets in a way the Vietoris topology does not. Consider a body A.
A is the disjoint union of its interior A and its boundary
0.4 = A - , 4 , hence A # is the disjoint union of A # and (0A) #, which are each partitioned into monads of course. Note that `4 # = U {#x [ x E X and #x c A * } and A # = [J{#z[ z E X and #z intersects A * } (since /,z intersects A * iff z E ~ = A), and
A#
A * is sandwiched in between these two sets; A # c A * c A I Z = A # U ( O A ) #. This is all illustrated on the right. The natural question arises of whether there's a topology on ~3X in which B E # A ¢:~ A # f i B C A # so that B is sandwiched between A # and A # just like A *, so B is substantially the same as A *, differing only within the 'infinitesimally thin' boundary region (0A) #. The 'body topology' will provide an affirmative answer. We point out first that in the Vietoris topology, B E #A does not imply that A # c_ B ; for example, in the case X = [R we could have A = [-1,1] and B = A * - [ i 1 ,i2] for any infinitesimals i t < i2 . Further examples in which B is thoroughly r~ddled with infinitesimal holes within the region A # are similarly easy to produce. The lower body topology_ on ~BX is the topology generated by the sets of the form [K ] = {A E fl3X [ K c A } where K is compact. The upper body topology~ on ~BX is the topology generated by the sets of the form sub V = {A E fl3X 1 A C V} where Vis open. The body touology on ~BX is the conjunction of these, and it's easily seen that the sets of the form [K,V] = {A E ~BX [ KC/land A C_ V} where Kis compact and V is open, form a basis for the topology (noting in particular that [/41, Vl ] N [K2, V2 ] = [K t U K 2 , Vt N V~ ] ), a collection of 'basic sandwiches' if you like.
A6.1 Proposition (1) The monads of the lower body topology on ~BX are given by
#A= {BE~BX* I AP'c B}. (2) The monads of the upper body topology on ~BX are given by
#A= {BE~BX* t BC_A#}.
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(3) The monads of the body topology on ~ X are given by /ZA = { B E ?BX * t A_/zc- B C- A/Z}.
Proof: (1) Let AE~BXand BE~BX*.
Suppose BE/ZA. Then V x E A , there's a compact
neighbourhood K of z with K £ A , so A E [K ] hence B E [K ] * giving K * C-B and in particular /zx C- B .
Thus A/Z ¢ B.
Conversely, suppose A # c B , which moreover
implies A/Z C-B (since for internal B C-X * , p~ C-B ~ /zx C-B ). Then for any K with A E [ K ] w e h a v e K c - A s o K * C K / Z c A / Z c B g i v i n g B E [ K ] * . SoBE/ZA. (2) This just follows from 1.1(2) of Chapter 1 since the topology is the restriction to ~BX of the sub-open topology on ~X, and as A is compact #X A = A #. (3) By (1) and (2), taking the intersection of the monads, o Assume from now on that ~BX has the body topology. Note that if B E/ZA then (where a denotes symmetric difference) B a A * c (aA)/z since both B and A * are sandwiched between A/Z and A/Z, whose difference is A # - A / Z = (0A)/Z. In fact the converse also holds, giving the alternative characterization of the monads that B E/ZA ¢$ B a A * c (aA)/z. The body topology is Hausdorff s i n c e . . . A6.2 Note The body topology on ~ X is a refinement of the Vietoris topology. Proof: Suppose B is near A in the body topology. We must show it's near A in the Vietoris topology, i.e. A c st B and B C-A/Z. We already have B C-A/Z of course, and secondly, as A/Z c B we have A c st B hence (as A is regular and st B is dosed) A c st B . o ~BX should not be expected to be locally compact; it can be shown for example that ~B ~n is nowhere locally compact. However, on the set of convex bodies of ~n the topology agrees with the Vietoris topology and is locally compact; see A8.2. Under the bijective correspondence between ~BX and ~BiX, the body topology on the first induces a homeomorphic topology on the second (likewise with the lower and upper body topologies), which we'll naturally call the body-interior topology on ~BiX (likewise the lower body-interior topology on ~BiX and the Upper body-interior topology on ~BiX ). The monads can be expressed as follows ;
A6.3 Proposition (1) The monads of the lower body-interior topology on ~BiXare given by
# D = {EE ~ i X * I D#C- E}. (2) The monads of the upper body-interior topology on ~BiXare given by
/zo= {Be
iX* l Ec-- /z }.
(3) The monads of the body-interior topology on ~ i X are given by /ZD= {EE~BiX* [ D/ZC EC-D/Z}.
151
Pro of: (1) We must show that for D E ~3iX and E E ~ i X *, D # c E ¢:¢ D # C E (as the left hand side expresses that D E # E in the lower body topology on ~ X ). This is done as follows; D # C E ¢:~ D#C(~-~ ¢:~ D#C_E ( a s ( ~ = E ) . (2) We must show that E ¢_D # ¢~ E C D #. Suppose E C D #, i.e. (as D is compact, so D # = #X ~ ) E C #X ~ "
As X is regular and ~ is compact, D has a neighbourhood
basis of closed sets, and it follows easily that E c #X ~ ' i.e. E c D # (3) By (1) and (2), taking the intersection of the monads,
o
Similar to the case of the body topology on ~X, it can easily be shown that for DE~iXandEE~BiX*,EE#D ¢:~ E a D * C _ ( O D ) * . Also, E e # D :~ D = s u b s t E . And note that the sets of the form [K, V]' = {D E ~ i X [ K c D and D c V} where K is compact and V is open, form a basis for the body-interior topology on ~3iX, namely the natural image of the basis we noted for ~ X . Finally we give a result on the continuity of the boundary map 0 : ~ X ~ Cp X, giving the set Cp X of compact subsets of X the Vietoris topology (note: if X = ~n we can replace Cp X by 2 ; X since boundaries of bodies are nonempty). A6.4 Proposition If X is locally connected, then giving ~ X the body topology and Cp X the Vietoris topology, the boundary map 0 : ?BX --* Cp X is continuous. Proof: Let A E ~ X and B E #A. We must show OB E #vietoris0A, i.e. (bearing in mind that #X OA = ( OA) # as OA is compact) OB C ( OA) # and OA c st OB. As A # c B C B c A # we have OB = B - B
cA #-A
belongs to stOB. As a E A C s t B
#=(OA)#.
It remains to show that every aEOA
let f l E B N # a .
As B E #A then B is near A in the
body-interior topology, so A = subst B , i.e. A = subst B, hence (as a l~ A ) #a is not a subset of B, so let 7 E #a -- B. We now have f1,7 E # a , so as X is locally connected at a (i.e. a has a neighbourhood basis of connected sets) there's .-connected C with f1,7 E C c #a (an elementary use of saturation). By transfer of the easy result that "for any closed B c X and connected C C_X such that C intersects both B and X - B, C intersects OB ", it follows that C intersects cOB,so cOB intersects # a , so a E st cOB as required, o Note that if we instead give ~3X the (coarser) Vietoris topology, cOneed not be continuous. As an example let X = R, A = [-1,1], and B = A * - (-5,5) where 5 is a positive infinitesimal; then B E #vietoris A but cOB~ #vietoriscOA ; indeed, st cOB = {-1,0,1} = OA U {0} $ cOA so cOBis not even near cOAin the S-compact topology.
152
Appendix 7 : The S-Compact Topolog~ Throughout let X be a locally compact Hausdorff space, and ~ X denote the set of closed subsets of X. In [Nar], Narens gave a purely nonstandard description of a topology on ~'X making ~'X a compact Hausdorff space. This topology was later elaborated on by Wattenberg in [Wa], who also gave a standard description and dubbed the topology the 'S-compact' topology. In between the publication of these two papers the topology appeared extensively (and, so it seems, independently) in the book [Mat] of Matheron, though apparently given no name. In this appendix we'll give a brief description of the topology along with the results relevant to its application in Section 5 of Chapter 4. Recall from Chapter 1 that the Vietoris topology on ~ X was defined as the conjunction of the 'open-intersecting' and 'closed-avoiding' topologies. The S-compact topoloL~ on ~'X on the other hand is the conjunction of the open-intersecting and compact-avoiding topologies, the latter being generated by the sets of the form disj K = {A E ~'Z [ A is disjoint from K }. Since every compact set is closed, it follows then that the S-compact topology is a coarsening of the Vietoris topology. This can also be seen from the formulation of the monads in (3) below ; A7.1 Proposition (1) The monads of the open-intersecting topology on ~ X are Oven by
#A= { B E ~ X * I A C stB ) . (2) The monads of the compact-avoiding topology on ~'X are Oven by
#A= {BEffX* I stBC_A }. (3) The monads of the S-compact topology on fiX are given by
#sA= { B e ~ X * I s t B = A }. Proof: (1) Already proved in 1.1 of Chapter 1 ; we've listed it again for convenience. (2) Let B E ~ ' X * . First suppose BE#A. Then s t B C A since for any x E s t B , for every compact neighbourhood K of z, B intersects K * so B 1~(disj K)* so A ¢ disj K , so A intersects K ; so (as x has a neighbourhood basis of compact sets) x E A = A. Conversely suppose st B c_A. Then for any compact K with A E disj K we have B E (disj K )* (for if B intersected K * then A = st B would intersect K) ; so B E #A. (3) By (1) and (2), taking the intersection of the two monads, o As indicated in (3), we may use the notation #s for monads with respect to the S-compact topology. For B E ~'X *, since B is internal then st B is dosed, and by (3) above, B E/~s st B , i.e. °B = st B, showing in particular that every element of ~fX* is nearstandard, i.e. that ~¢X is compact. ~¢X is also Hausdorff since if B E #sA1 n #sA2
153
then A l = s t B = A 2 so A t = A 2. Also note that # s O = { B E ~ X *
I Be_ r m X * }
where rm X * is the set of remote (i.e. non-nearstandard) points of X *. For open D c X , the D--topologg on ~fX is the topology on ~ X induced by the map ~¢X ~ ~ D (= the set of closed-in-D subsets of D) in which A ~-* A N D , where ~ D has the S-compact topology (note that D is a locally compact Hausdorff space so the S-compact topology on ~'D /s defined). In other words, denoting the monad of A E ~ X with respect to the D-topology by #D A , then for B E ~'X *,
B E # D A ~ B N D * E#s(AND ) in ~'D*. This can be expressed as follows ; A7.2 Note For open D C X , and A E ~ X and B E ~ X *,
BE#DA ¢=~ s t ( B N D # ) = A f l D ~=~ ( s t B ) N D = A N D ¢:~ VxED, x E s t B ¢=~ z e A . Pro of: Where stD C denotes the standard part in D of a subset C of D *, we have B E #D A
¢=~ stD(BND*)= A N D .
And as s t D ( B N D * ) = s t ( B N D # ) = ( s t B ) N o ,
result follows,
the o
Intuitively, 'B E #D A' can be thought of as expressing that B looks like A * within D *;
this interpretation is eaaborated upon in Section 5 of Chapter 4.
The
S-compact topology on KX can now be formulated in terms D-topologies in the following way ;
A7.3 Proposition For any open cover .~ of X , the S-compact topology on ~ X is the conjunction of the D-topologies on ~ X for D E.~; in other words, for A E ~ X andBE~X*,
BE#sA ¢=~ V D E ~ , B E # D A .
Proof: Since.~ covers X and each D E .~ is open, st B = A ~=~ VD E .~ (st B ) N D = A N D ; i.e. BE#sA ¢~ VDEfO, B E # D A .
o
Also note that in general, if D 1 C D 2 then the D2-topology is a refinement of the Dvtopology ; this can be seen from the last formulation of 'B E #D A' in A7.2. Finally we give a result on limits of monotonic sequences in ~'X, the proof closely following that of 4.4 of Chapter 3 ;
154
A7.4 Proposition
For a sequence ( A n ) in ~ X ,
(1) If ( A n ) is decreasing, ( A n ) ~ ~] A n . n
(2)
If ( A n ) is increasing, ( A n ) ~
~n A n .
Proof:
(1) Let B = r] A n . Taking any infinite m , we must show that st A m = B . n finite n , B'CA
A m C_ A n * so
m so B = s t B * C s t A
(2) Let B =
st A m C_st (A n *) = A n ",
so st Am c_ B .
-
B = B . Conversely, for all finite n -
st A m ; so U A n c st A
n
m'
Conversely,
m.
UA n . Taking any infinite m , we show s t A
st A m C_st B *
For all
,
=B
Since A
A n * C A m so A n
so as the latter's closed, B c st A m"
~
st(
CB* A
n * )C o
155
Appendix 8 : The Hyperspace of Convex Bodies Throughout let X = IRn. Let ~ B X denote the set of convex bodies in X, namely the convex regular nonempty compact sets, and give ~ 3 X the Hausdorff metric. The main result in this appendix will show that Lrc2X is loca~y compact and that its topology, namely the Vietoris topology, coincides with the body topology and the S-compact topology (respectively described in Appendices 6 and 7), from which we may conclude that the topology is the natural topology on C23X. As usual we make use of nonstandard analysis, which in fact finds rewarding application to the theory of convex sets (as yet unreported in the literature so far as I ' m aware) due to the attractive interplay between convex subsets of X and convex subsets of X * Generalising notation used in the case of ~, for z,y E X , [z,y] denotes the line segment with endpoints z and y , whilst (x,y ] denotes [x,y ] - {x}. A8.1 Lemma For ,-convex C , st C is a closed convex set with interior subst C.
Proof: As C is internal, st C is closed and subst C is open. And st C is convex since for all z, y E s t C
there exist a, bE C with a E # z a n d
asia, b i t C. We now show s t C = s u b s t C .
b E # y , giving
[x,y]=st[a,b]CstC
Since subst C is an open subset of s t C
it remains to show s t C CsubstC. For z E s t C
there's an open ball V on x with
VC st C, so VvE V #v contains a point of C , so by convexity of C clearly # x c C (indeed, V # c C ), so z E subst C as required,
o
A8.2 Proposition CBX is locally compact, and its topology coincides with the body topology and the S-compact topology. Proof: Give the set J d X of nonempty compact subsets of X the Hausdorff metric. The set ~X of convex elements of JdX is closed in J d X ; for if A E ~'-'Xthere's B E ~X * near A, so A = st B which is convex as B is .-convex. In turn ~ B X is open in ~X ; for if AE~X
and B E ~ X *
with B E # A ,
substB=stB=A~(3
hence B ¢ O
(as
(subst B) # c B ), so B is ,-regular, hence belongs to L~3X *. Now local compactness is closed-hereditary and open-hereditary, hence as JdX is locally compact, so is ~ X , and in turn so is C23X. Next we show that the body topology and S-compact topology coincide with the Vietoris topology on ~ X .
Since the body topology is a refinement of the Vietoris
topology which is a refinement of the S-compact topology, it remains to complete the circle and show that on ~ X , topology.
the S-compact topology is a refinement of the body
So, suppose A E ~ B X and B E ~ 3 X * with B e #sA. We must show B is
156
near A in the body topology, i.e. A # C B c A #. Firstly, B C ns X *; for if this were false, say
bEB-nsX*,
then taking any
cEBNnsX*
(such c exists as
s t B = A ~ ( 3 ) we have [b,c]CB so st[b,e]CstBC_A, but st[b,c] is unbounded, contradicting that A is bounded. We now have B C ns X * and st B = A, i.e. B is near A in the Vietoris topology, which in particular gives B c A #. Lastly, A # c_ B since
A = st B = subst B using A8.1.
o
Note that since the set of closed balls is dosed in Lr*BX (quite easily proved nonstandardly), it too is locally compact. It should be noted incidentally that Lrc2X is not boundedly compact ; for example we can have B E ~ X
* with B { #z for some x,
hence i n J ~ X * B E # {z} so B is not near any element of Lrc2X. Let ~ i X
denote the set of interiors of convex bodies, equivalently the set of
convex body-interiors.
Lrc2X and Lr~BiX correspond bijectively under the mutually
inverse maps A *--*A and D~--* D (using the facts that interior and closure both preserve convexity), and we naturally give C~iX the topology which makes it homeomorphic to Lr~BX under this correspondence ; this is namely the body-interior topology inherited from ~BiX (see Appendix 6). Lr~BiXis thus locally compact, as is the set of open balls (namely comprising the interiors of the closed balls). A8.3 I,e m m a For .-convex C with subst C# (3, st C is the closure of subst C , so st C and subst C are corresponding nonempty regular convex sets.
If also
CC ns X * , st C and subst C are corresponding elements of Lr~Xand Lr~iX.
Proof: Firstly we show st C = subst C. st C is a closed expansion of subst C , so it remains to show that s i c
CsubstC. Let z E s t C ; say cE C N # z . Taking some y E subst C , we obtain that (z,y ] # c C , because for z E (z,y ] there's b E [c,y ] with b ~ z , and necessarily b q~c, hence b ~" C C (as b ~ is the image of #y c C under the dilation about c taking y to b ), i.e. #z C C . So (x,y ] c subst C , giving z E subst C as required. By AS.1 we now have that st C and subst C are corresponding regular nonempty convex sets. If also C c ns X *, then st C is compact, hence st C E ~ X , and correspondingly subst C E ~ i X . o The following gives a simple formulation of the monads of ~BiX ; A8.4 Proposition
ForDE~iX,
# D = {EE ~ i X *
I substE= D } .
Proof: For E E ¢x~iX * we must show E E #D ¢~ subst E = D . Both sides give subst E ~: 0, so now assume this. Hence by A8.3 st E and subst E are corresponding regular sets, and we can reason as follows ;
157
EE#D
¢~ E E # D
by the homeomorphism of ¢'c~iX with ¢'~BX
<=} s t E = D
since Cr~BXhas the S-compact topology
¢:) s t E = D
as st E = st E by taking interiors,
4=) substE = D
o
Lastly, making use of the following result on continuity of the convex hull operation cony : ~ X --- JgX, we show that ¢'~X is embedded in JdX by O ; A8.5 I~m~ma
The convex hull operation cony : JgX --, ~ X
is uniformly continuous
with respect to the Hausdorff metric. Proof: For A,B E ( ~ X )* with A z B, we must show that cony A ~ cony B. By transfer of a formula
for
convex
hulls,
convA={A la 1+.
• • + A n + 1 an+ 1 [ alla l E A
and
Ai E [ 0 , 1 ] * w i t h A l + . . • + A n + 1 = 1 } . For any c = A l a 1+ . • . + A n + 1 an+ l i n convA, for each i there's biE B with biz ai, and for such bi we then have c z A lb 1 + . • • + A n + 1 bn+ 1 E c o n v B . Likewise every element of convB is infinitesimally distant from an element of cony A. Thus cony A ~ cony B. o A8.6 Proposition
Giving JYX the Vietoris topology, the boundary map
O : ¢'~BX --- ~ X
is an embedding.
Proof: By A6.4 the map O above is continuous.
Now O is also injective since for A E ¢'78X,
A = convS.4. And for AEC'~BX and BE(Cr~BX)*, if 0 B E # 0 A cony OB E # cony ~A, i.e. B E #A.
then by A8.5 o
158
A p ~ e n ~ 9 : Similitudes In this appendix we provide a summary of the main facts about 'similitudes' of X = ~n used in the monograph, including a few nonstandard results. A similtude of X is a bijection f: X ~ X for which there exists rE (0,oo) such that Yz, y E X d(fz, f y ) = r d(z,y). This ris unique and is called the scale factor o f f (it is the 'Lipschitz ratio' rf of f too of course). The similitudes form a group Sire X of homeomorphisms of X, and the scale factor map r : S i m X--, (0,oo) is a group homomorphism. Under the compact-open topology, assumed throughout, Sim X forms a topological group (with the scale factor map becoming continuous). Continuity of composition holds simply from that in C ( X , X ) , whilst continuity of the inverse map f~_, f-1 can easily be proved with the aid of the following nonstandard result A9.1 for example. First we define that a bijection g : X * ~ X * is strongly microcontinuous if Vot,#EX* ~ z # <=> g a z g f l , equivalently if g and g-I are microcontinuous, equivalently if VaE X* g ( a : ) = (g a ) z . Note then that a .-similitude g is strongly microcontinuous iff rg is finite and noninfinitesimal (which in particular holds if g is nearstandard in (Sire X )*, by continuity of the scale factor map). A9.1 Note F o r g E ( S i m X ) * a n d f E S i m X , gE #f ¢~ VzE X g # z = fizz. Proof: : Let z E X. Since g is strongly microcontinuous (being nearstandard) and g z z f z , VaE X*, aEizx ¢:~ a z z ¢:~ g a z g z ¢~ g a ~ f z which is fizz. ¢: Then VzE X gizzC i z f z ; i.e. ge #f.
¢~ g a e i z f z .
Sogizz=izfz, o
A9.2 Proposition For g E (Sire X )*, the following are equivalent ; (1) g is nearstandard in (Sire X) * (2) g b d X * = bdX*. (3) rg is finite and noninfinitesimal, and g maps some bounded point to a bounded point. Proof: (1)#(2):gbdX*=g(X
#)=
U g#z= U f#x= U #f x=x#=bdx* Z
Z
Z
(2) ff (3) : Since g bd X * c_ bd X * , rg is finite (else g could only map at most one element of X t o a bounded point). And since bd X * c_ g bd X *, rg is noninfinitesimal. (3) :~ (1) : Since rg is finite and g maps a bounded point to a bounded point, we have g bd X * c_ bd X *. By the same reasoning applied to g -1 (noting that r(g -1) = (rg)-1 which is thus finite and noninfinitesimal) we have g -lbd X * C bd X *, i.e. bd X * c g bd X *. So g bd X * = bd X *. Since g is also stongly microcontinuous, g maps
159
monads to monads, and letting ] be the unique map X ~ X
such that VxE X
it is easy to show that f is a similitude with scaling ratio ° r a . By the last note we then have g E #f. o g#x = #fx,
The definition of a similitude given above was purely metric, but similitudes can be resolved into certain 'geometric' transformations as follows. Firstly, an orthogonal map of X is a linear isometry of X, and these are in fact the only isometries of X leaving 0 fixed. For p E X and r E (0,oo), the dilation about p by factor ris the map f defined by f x = r ( x - p ) + p (note: if r < 1 we may refer to f as the contraction about p by factor r ), and a map of this form is called a dilation. It can be shown that any similitude f can be expressed uniquely in the form 'orthogonal map followed by dilation about 0 followed by translation'. We say f is direct if the orthogonal map involved is a rotation, i.e. has determinant 1. Letting Orthog X be the group of orthogonal maps, D//oX the group of dilations about O, and Trans X be the group of translations of X, all these being topological subgroups of Sim X , the map Trans X x Di~)X ~ Orthog X ~ Sire X in which (f, g , h ) ~-* f o g oh is a homeomorphism.
And in the natural way, Trans X ~- X
and
DiloX~-(0,oo)
of
course, so Sire X ~- X x ( 0 , ~ ) - Orthog X. It follows that Sim X is locally compact since each of the three factors on the right are (Orthog X is compact moreover). The group of direct similitudes (namely generated by the translations, dilations, and rotations) is dosed in Sim X, hence likewise locally compact.
References [AFHL] [Cu] [Da] [DE]
[Du] [Ed]
[csl [Ha] [Itaus] [HLI [Hu] [uw] [Ju] [Ke] [Ku] [Li,1] [Li,2]
[Li,3] [Man] [Mar] [Mat]
Albeverio, Fenstad, H0egh-Krohn, & Lindstr0m, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, 1986. N. J. Cutland, Nonstandard Measure Theory and its Applications, Bull. London Math. Soc., 15 (1983), 529-589. M. Davis, Applied Nonstandard Analysis, John Wiley & Sons, 1977. R. Devaney & L. Keen (Eds.), Chaos and Fractals : the Mathematics Behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics, Vol 39, A.M.S. J. Dugundji, Topology, Allyn and Bacon Inc., 1966. M. Edelstein, On Fixed and Periodic Points under Contractive Mappings, J. London. Math. Soc. 37 (1962) 74-79. K. J. Falconer, The Geometry of Fractal Sets, C.U.P. 1985. B. Griinbaum & G. Shephazd, Tilings and Patterns; an Introduction, W. H. Freeman, New York, 1989. M. Hata, On the Structure of Self-Similar Sets, Japan J. Appl. Math., 2 (1985), 381-414. F. Hausdorff, Mengenlehre, Dover, New York, 1944. A. E. Hard & P. A. Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, 1985. J. E. Hutchinson, Fractals and Self-Similarity, Indiana Univ. Math. J., 30 (1981), 713-747. W. Hurewicz & H. Wallman, Dimension Theory, Princeton Univ. Press, 1941 I. JuMsz, Non-standard Notes on the Hyperspace, pp. 171-177 of Contributions to Nonstandard Analysis, W. A. J. Luxemburg & A. Robinson (Eds.), North-Holland, 1972. H. J. Keisler, Foundations of Infinitesimal Calculus, Pfindle, Weber & Schmidt, Massachusetts, 1976. K. Kuratowski, Topology Vols 1 & 2, Academic Press, New York, 1966. T. Lindstr0m, A Nonstandard Approach to Iterated Function Systems, preprint. T. Lindstrcm, Brownian Motion on Nested Fractals, Memoirs of the A.M.S., Vol 83, No. 420 (1990). T. Lindstrom, An Invitation to Nonstandard Analysis, in Nonstandard Analysis and its Applications, N. J. Cutland (Ed.), C.U.P. 1988. B. B. Mandelbrot, The Fraetal Geometry of Nature, W. H. Freeman, New York, 1983. G. Martin, Transformation Geometry, Springer, 1982. G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, 1975.
161
[Mi]
E. Michael, Topologies on Spaces of Subsets, Trans. Amer. Math. Soc.,
7x (195,), 152-182. [Mu]
[Nad] [Nad [Ro] [SL] [Vi] [Wa]
[Wicl
[Will
M. G. Murdeshwar, General Topology, Wiley Eastern Limited, 1983. S. Nadler, Sequences of Contractions and Fixed Points, Pacific. J. Math., 27 (1968) No.3, 579-585. L. Narens, Topologies of Closed Subsets, Trans. Amer. Math. Soc., 174 (1972) 55-76. A. Robinson, Non-standard Analysis, North-Holland, 1966. K. D. Stroyan & W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, 1976. L. Vietoris, Bereiche Zweiter Ordnung, Monatschefte fiir Mathematik und Physik, 32 (1922), 258-280. F. Wattenberg, Topologies on the Set of Closed Subsets, Pacific. J. Math., 68 (1977) No.2, 537-551. K. R. Wicks, Spiral-Based Self-Similar Sets, Mathematics Research Reports, University of Hull, Vol 3 (1990) No. 1. Also to be published in Spiral Symmetry (Eds. I. Hargittai & C. A. Pickover), World Scientific Ltd. R. F. Williams, Composition of Contractions, Bol. Soc. Brasil Mat. 2 (1971) 55-59.
Notation Index Preliminaries A*
4
A a
4
#~¢
wig
4
~X
5
~ X
Jfx
5
#x
5
ns X *
5
st B
5
JA
6
rm X *
5
°a
subst B
5
A #
5
~A
6
~x A
~
cp X *
6
a
subCp X Bd X
10
C(X,Y) fix f
6
10
10
Contrac X
11
5 5
Cp x
6
a ~ 13
10
bd X *
10
F(X,Y)
4
11
rf
10
11
LiplX
11
Chapter I • Nonstandard Development of the Vietoris Topology J~
23
~
26
Fz
27
LJF
27, 2s
F~
2s
FA
2s
Chapter 2" Nonstandard Development of the Hausdorff Metric [A ]5
h(A,B)
31
pns X *
Fz Fjd
d(x,A)
31
36
41
[Y ]5
31
A~
32
a~
y c
36
lira d
[J F
41, 42
LiPu(Z , ~ X )
FA
42
31
32 37 42
42
Chapter 3" Hutchinson's Invariant Sets KF , K
47
Admis X
A F
53
xF
Id X
56
Monoid F
fA
56
kf
48
55
59
56
cx
48
of
56,61
fx
56
Semigroup F
63
163
A^ B
hF
64
Admis u X
dim A
Contrac u X
76
Reduc-Admis X
67
79
76
Reduc-Admis X 15
Contu( X, Y )
89
ob v
89
g~,g
u_< v
90
A~ D'-* E
91
89 90
91 91
91
D~E
D~
91
gv
91
u,~ v
91
u~
u'-* V
91
u-~ v
91
?_Z' ~
91
u-.q* B
U~
92
-~t < -~2
95
(a,,~,) 5 (as , ~ ) A~
B
98
GF
102
(hE) -w #/9
103
109
91 91 91
V
92
u--~ B
92
A ---* B
92
A~B
93
"~1 ~ " ~ 96
89
A~B
A,~ B E
89
Ob Parts
u~ v
V~
89
//
(X,G,~)
91
DA
89
.~4
(c~ ,~1
95
(c~ ,.~
) ~
M~
)
96
95
u~ B
98 98
A~B
98
A~
iK, i A
102
(hE)-1
(hE) -<w B#
103
110
u
113
G~
113
gy
G•
113
Sil v
114
-
(Z)r
116
norm v
cv A
117
frA
116 117
v
103
(hF)-n
g~
103
111 113 116
norm T liraA x
116 117
z~
119
tad v
119
S n-1
119
IY
119
lira Y
120
lira Y
120
aff A
121
Ov
123
d
127
x ~
y
125
78
Contrac-Sim ~n
80
Chapter 4 Views and Fractal Notions (x,~) 89 Ob 89 dora v
68
c
6~D
123
81
164
Appendices ill
134
F <w
./tn
134
F -< w
134 134
~f
139
Reduc X
3"
145
S" "~
~iX ~X
149
,us
155
~'23i X
Orihog X
156 159
142 146
j.w
134 135
ReducuX ~X
155
cony
157 159
143
149
~X
152
[~,y] DiloX
f-< 9
Sim X Trans X
155
158 159
Term Index admissible, 47 alike, 128 attractor, 47, 78, 140 map, 76 Blaschke selection theorem, 35 body, 12 body-interior, 12 bounded point, 10 boundedly compact, 10 Cantor set, 60 closed ~-ball, 31 ~-fringe, 31 under compact unions, 41 code map, 59 code space, 59 compact point, 6 compact set condition, 71 compactly generated, 127 composition map, 56 concatenation, 134 consistent, 89, 90, 92 continuously coloured dosed set, 130 contraction, 11 contractive, 140 control, 139 controlled, 139 convergent, see visually convergent convergent part, 117 copy, 91, 145 D-indistinguishable, 90 D-topology, 109 D-view, 89 ..@-bounded, 90 -indistinguishable, 31
dilation, 159 disjoint family, 60 domain of a view, 89 of a view space, 89 S-indistinguishable, 90 embedding of a view in an object, 92 of views, 91 of view classes, 92 of view domains, 91 equivalence-invariant, 95 expands, 54 F-range, 53, 55 finite-level image, 56 fractal, see visually fractal fractal part, 117 Hausdorff distance, 31 Hausdorff metric, 32 homeomorphism condition, 67 hyperspace, 14 ideal induced ~ , 23 topological ~ , 23 view structure, 90 indeterminable scale, 129 completely ~ , 129 infinitesimal fringe, 32 infinitesimally indistinguishable, 32 interlinked, 64 intermediary, 4 invariant set, 47 intersection monad, 4 isometric copy, 75
166
object, 89
length of a finite sequence, 134
class, 91
of a path, 142
part, 89
limit
set, 89
(An), 37 (An>,
point of
observable space, 90
set of 37 -similar, 128
open set condition, 71 orthogonal map, 159
-similarity class, 128 view, 117, 119 linking chain, 64 Lipschitz map, 11 lower continuous, 139
packing, 146 Y- ~ , 146 periodic point, 63 sequence, 63
metric
pre-nearstandard, 36
Hansdorff ~ , 32 view ~ , 115 microcontinuous, 11 strongly ~ , 158 monad
realization, 92 reduces, 54 reduction, 140 -admissible, 78
intersection ~ , 4
regular, 12
of a point in a space, 5
remote, 5
of a subset of a space, 6
reptile, 74
union ~ , 4
residual, 12
monadic
rotation, 159
cover, 5 image, 110 monoid, 132
saturation, 4 scale factor, 158
action, 132
self-similar, see view self-similar
topological ,~, 132
sequence, 134 concatenation, 134
near, 5 nearstandard, 5
finite ~ , 134 length of a finite ..., 134
nonoverlapping, 144
Sierpinski Gasket, 54
normalization, 116
silhouette
nth-level image, 56
map, 114
(-n)th-level image, 103
of a view, 114 similar at z and y, 125 objects, 91 view domains, 91
167
similarity, 90
topology (ctd.)
class of a set, 125
lower body ~ , 149
group, 90
lower body-interior ~ , 150
similarity view space, 90
of pointwise convergence, 11
similarity view structure, 90
open-intersecting ~ , 16
equivalent ~ , 95
product ~ , 11
on a topological space, 94
S-compact ~ , 152
stronger ~ , 95
sub-open .-., 16
weaker ~ , 95
uniform ~ , 11
similitude, 158
upper body ~ , 149
contractive ~., 81
upper body-interior ~ , 150
direct ~ , 159
Vietoris .~, 16
rotation-free ~ , 116
view ~ , 108
standard part of a point in a space, 5
view class ~ , 113 view-induced ~ , 109
of a subset of a space, 5 sub view domain, 90
union
subcompact, 6
function, 27, 28, 41, 42
substandard part, 5
map, 22
subview, 90
monad, 4
superview, 90
universal, 126 universally
tiled, 70
embedded, 98
tiling, 70, 144, 145
view-embedded, 98
existence theorem, 148 3"- ~ , 146
tolerance, 146
view-similar, 98 usual similarity view structure on usual view structure on ~n, 89
topologically attractive, 137 topology body ~ , 149 body-interior ~ , 150
Vietoris topology, 16 view, 89 class, see below
bounded-uniform ~ , 11
D- ..~ , 8 9
closed-avoiding ..., 16
domain, 89
compact-avoiding ~ , 152
domain class, 91
compact-open ~ , 10
empty ~ , 90
compact-uniform ..~, 11
-embedded, 92, 98
D- ... , 109, 153
full ..~, 90
generated by, 9
generative ..~, 98, 126
J - u n i f o r m ~ , 11
-indistinguishable, 90
IRn,
90
168
view (ctd.)
weak contraction, 140
-induced topology, 109
weakly convergent, 129
limit ~ , 117, 119
w-extension, 102
map, 111
wth-level image, 62
normal ~ , 116 normalization map, 116 normalization of a ~ , 116 radiant ~ , 119 radiant set of a ~ , 119 residual ..~, 124 -similar, 93 structure, see below sub..~, 90 s u p e r ~ , 90 topology, 108 whole ~ , 89 x - ~ , 117 view class, 91 compact, 127 e m p t y ~ , 92 full . ~ , 92 indistinguishable, 93 metric, 127 of a set, 92 oriented ~ , 116 topology, 113 view structure, 89 covering ~ , 90 equivalent ~ , 95 ideal ~ , 90 m a x i m a l ~ , 95 on a topological space, 94 stronger ~ , 95 topological ~ , 93 weaker ~ , 95 visually convergent, 117 fractal, 117 periodic, 124
zoom operation, 119
