Fractals and Hyperspaces

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zfirich E Takens, Groningen

1492

Keith R. Wicks

Fractals and Hyperspaces

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Keith R. Wicks Department of Mathematics and Computer Science University College of Swansea Singleton Park Swansea SA2 8PP, U. K.

The picture on the front cover shows a zoom-in on Fig. 14, page 51

Mathematics Subject Classification (1991): 03H05, 05B45, 5 IN05, 52A20, 52A45, 54A05, 54B20, 54C60, 54E35, 54E40, 54H05, 54H20, 54H25, 54J05

ISBN 3-540-54965-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54965-X Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, t965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper

Foreword The main theme of this monograph is the study of fractals and fractal notions, backed up by a self-contained nonstandard development of relevant hyperspace theory, particularly as regards the Hausdofff metric and Vietoris topology. The fractal study itself is in two parts, the first developing and making contributions to the theory of J. E. Hutchinson's invariant sets, sets which are self-similar in the sense of being composed of smaller images of themselves. The second part explores newer territory, introducing the formal notion of a 'view' as part of a general framework concerned with studying the structure and perception of sets within a given space, and in particular we use views to express and investigate new concepts of self-similarity and fractality which are then considered in connection with invariant sets, a large class of which are shown to be 'visually fractal' in a certain precise sense. Complete with many figures and suggestions for further work, the monograph should be of relevance to those interested in fractals, hyperspaces, fixed-point theory, tilings, or nonstandard analysis. The work was undertaken at the University of Hull during the period 1987-90, financed for two years by an SERC research grant which I gratefully acknowledge. My grateful thanks go also to Professor Nigel Cutland for assistance and advice throughout, and to Dr. Dona Strauss for help with many topological and other queries. Keith R. Wicks Hull, August 1991.

Contents Page

Introduction

1

Preliminaries Chapter 1- Nonstandard Development of the Vietoris Topology 0. Introduction

14

1. The Vietoris topology on ~¢X

16

2. The Vietoris topology on J g X

19

3. Compactness of J g X

20

4. Subspaces o f ~ X

21

5. The Union Map [J : ~ g ~ X ~ X

22

6. Induced Ideals on ~ X

23

7. Local Compactness of J~X

25

8. The Map C ( X , Y ) ~ C( J~X,J(Y)

26

9. Union Functions

27

Chapter 2- Nonstandard Development of the Hausdorff Metric 0. Introduction

30

1. The Hausdorff Metric on ~ X

31

2. Boundedness and Bounded Compactness o f ~ X

34

3. Completeness of.)YX

36

4. Limits of Sequences i n J ~ X

37

5. The Map C ( X , Y ) ~ C( J(X,,2gY)

40

6. Union Functions

41

Chapter 3 : Hutchinsons's Invaxiant Sets 0. Introduction

45

1. Existence of Invariant Sets

47

2. nth-Level Images

56

3. The Code Map

59

4. Periodic Points

63

5. Connectedness of K

64

6. Regularity and Residuality of K

67

7. Tiling of K

70

8. Continuity of the Attractor Map

76

9. The Generalization from Contractions to Reductions

78

10. Notes, Questions, and Suggestions for Further Work

81

Chapter 4 : Views and Fractal Notions 0. Introduction

87

1. Views and Similarities 2. Relative Strength of View Structures and Similarity View Structures

89 95

3. View Self-Similarity 4. Self-Similarity of Some w-Extensions of Invariant Sets 5. The View Topology 6. A Definition of 'Visually Fractal' 7. Notes, Questions, and Suggestions for Further Work

98 102 107 116 125

Appendices 1. Topological Monoid Actions 2. 3. 4. 5. 6. 7. 8. 9.

Finite and Infinite Sequences in a Hausdorff Space Continuity of fix : Contrac X ---,X Reductions of a Metric Space Nonoverlapping Sets, and Tilings The Body Topology The S-Compact Topology The Hyperspace of Convex Bodies Similitudes

132 134 137 139 144 149 152 155 158

References

160

Notation Index

162

Term Index

165

Introduction As briefly outlined in the Foreword, the main theme of this monograph is the study of fractals and fractal notions, namely comprising Chapters 3 and 4, aided by the prior development of certain hyperspace theory in Chapters 1 and 2. To understand the motivation for the latter, consider now the first part of the fractal study. The notion of self-similarity has always been one of the central themes in the subject of fractals, and in Chapter 3 we consider the particular type of self-similarity captured by J. E. Hutchinson's 'invariant sets'. These are nonempty compact subsets K of a complete metric space X, which are composed of smaller images of themselves in the sense that

K =f__[~JF fK

for some finite set F of contractions of X.

Strictly

speaking it is not so much K that is studied, rather the properties of K dictated by F . A general treatment is given, blending in the already existing basic results with contributions of the author, one aspect of which is the generalization from finite to certain compact sets F of contractions (or even more generally, 'reductions'). Further aspects include consideration of the regularity or residuality of invariant sets, and a study of those which are tiled by their images under the maps involved. The natural setting for invariant sets is the s p a c e ~ X of nonempty compact subsets of X equipped with the Hausdorff metric; indeed, the existence of invariant sets follows from an elegant apphcation in JgX of Banach's contraction mapping theorem. We therefore precede the study of invariant sets with a chapter introducing the Hausdorff metric and using nonstandard analysis to quickly prove everything needed later on. In particular we give short nonstandard proofs of the facts that if X is complete so is ~ X , and that if X is boundedly compact so is ~ ' X (the latter result being the Blaschke Selection Theorem in the case X = ~n ). In turn however, since the

topology of the Hausdorff metric is an example of the 'Vietoris topology' defined in terms of the topology of X, the study of the topological aspect o f ~ X is relegated to a preceding chapter on the Vietoris topology, which is defined on the nonempty compact subsets (or more generally, on the closed subsets) of any Hausdorff space. Extensive use is again made of nonstandard analysis, demonstrating its potential in the study of hyperspaces and intended to be as much a part of the work as the results themselves, many of which are in any case well-known, such as the results that if X is compact or locally compact, so respectively is ~ X . Hyperspace theory is also drawn upon in the fourth and final chapter which, comprising the second part of the fractal work, introduces the formal concept of a 'view' as part of a general framework concerned with studying the structure and perception of sets within a given space X. Roughly speaking, a 'view' of a subset A of X consists of a region of X together with the part of A lying within that region. In the

case of X = ~2 this closely models what one might expect to see upon looking down and surveying A. Using views we define a new notion of self-similarity, being a formal interpretation of the idea that wherever one looks at the set in question one can see the same sort of structure.

This property is shown to hold for certain 'w-extensions' of

invariant sets, obtained roughly speaking by 'growing out' an invariant set K indefinitely. With the aid of a natural topology on a set of views we go on to define a second fractal notion, that of a subset A of ~n being 'visually fractal' at a point z e A. This is an interpretation of the idea that as we 'zoom in' on z we see detail beyond detail;

more precisely it expresses that no visual convergence takes place, and in

particular we show that many residual invariant sets are visually fractal at all points. A number of other view-related notions are more briefly touched upon and may begin to make clear the wider potential apphcation of views, facilitating what might be thought of in general as 'visual analysis'.

Chapters 3 and 4 each conclude with a

number of notes, questions, and suggestions for further work. Several appendices to the monograph are provided and cover an assortment of background topics including further hyperspace work, of which we take advantage in an appendix on tilings when we give a nonstandard proof of a fairly general tiling existence theorem. Since nonstandard analysis is used throughout in studying any topological or metric space, we provide for convenience a 'Preliminaries' section which gives a round-up of the nonstandard formulations used, including proofs of some new ones. Also included are one or two reminders of standard definitions and theory.

Numbering Convention : Each chapter consists of sections numbered 0, 1, 2 etc., Section 0 being a short introduction and summary of the work covered. In Section n the results are numbered n .1, n .2, n .3 etc. with the chapter number suppressed, and reference to say '2.4' will naturally mean result 2.4 of the present chapter. Occasional reference to results from other chapters will explicitly mention the chapter involved. After the four chapters there follow nine appendices, A1 to Ag, the results in appendix n being numbered An .1, An .2 and so on.

Preliminaries

Set,ion 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction Monads Standard and Substandard Parts of Sets Monads of Subsets Subspaces Compactness and Local Compactness Convergence of Sequences Properties of Functions Open Identification Maps Ordering of Topologies Products Induced Topologies Generating a Topology Bounded Compactness of Metric Spaces The Compact-Open Topology

15. 16. 17. 18. 19.

Topologies of Uniform Convergence Over Subsets Uniform Continuity Lipschitz Maps and Contractions Kegular Sets, and Bodies Residual Closed Sets

Page 4 5 5 6 6 6

7 7 8

9 9 9 9 I0 10 10 11 11 12 12

O. Introduction In this monograph nonstandard analysis will be routinely used in the study of any topological or metric space, and the reader who wishes to follow all the proofs will therefore need to have a degree of familiarity with basic nonstandard methods. We shall be using the 'superstructure' approach, the objects O under investigation always being assumed to lie in the 'standard universe', thus having a corresponding image O * in the 'nonstandard universe' (O * just being O if O is an 'atom'). The most extensive introductory accounts of this approach lie in Chapter 2 of Hurd and Loeb's book [HL], Lindstr0m's 'An Invitation to Nonstandard Analysis' [Li,3], the section on Logic and Superstructures in Keisler's book [Ke], and Davis' book IDa]. A briefer account can be found in Cutland's survey paper [Cu] however which gives a clear descripton of the axiomatic approach which is perhaps the simplest path into the nonstandard world. Another relatively brief account lies in [AFHL]. In using nonstandard methods we'll assume we always have sufficient 'saturation' for the job in hand; technically, we can just assume 'polysaturation' (see [Li,3]). In particular this imphes the useful fact that every set A in the standard universe has a hyperfinite intermediary (Terminology: an intermediary of A is an internal set I such that A a C I c A * , w h e r e A a d e n o t e s {a* [ a e A } . Such a set is thus intermediate between the pointwise image A a of A and the .-transform A * of A ; and being internal we have a handle on it. As yet the literature seems to have gone without a term for the concept and by now its introduction is well overdue). A further consequence is that for any lower-directed set (X,_ 0, h(A,B) < 6. Hence h(A,B) .~ 0 as required, And as a consequence, noting that A "z = (A *) ~ for A e .Y6"X,...

o

33 1.._55 Proposition The topology o n , ~ X induced by h is the Vietoris topology.

Proof: For A E J6X and B E J~X *,

B~A*

¢~ B and A * are infinitesimally indistinguishable B and A are infinitesimally indistinguishable ~=~ B E #A in the Vietoris topology (as in general b ~ a ¢~ b E #a ).

O

The results of Chapter 1 (Sections 2 onwards) thus apply in the present situation and will be assumed in the following work. A few of the necessarily topologicalresults of that chapter can be improved to metric results. In particular, in connection with 4.1 of Chapter 1, for any subset A of X, ~ A moreover forms a under the Hausdorff metric induced from the metric on A.

metric subspace

of 2gX And regarding 5.2 of

Chapter 1 it's quite easy to show (using the usual nonstandard formulation) that the union map U : ~ ( X ~ ~ X is uniformly continuous, g i v i n g ~ J g X the ttausdorff metric induced by the ttausdorff metric h on ¢VX. We end this section with an inequality on the Hausdorff distance between unions, to be used later ;

Proof: Each a E U Ai is in some within distance

Aj hence within distance h(Aj ,B. ) of V h(Ai ,Bi ) of this b E U Bi" Likewise vice versa,

some b E Bj hence

o

34 2_. Boundedness and Bounded Compactness of,YgX Recalling that we identify the embedded image of X inddX with X itself, the following note gives the recipe for the Hausdorff distance of an element ofddX from a point z of X. Recall that Ix ]g denotes the closed ~-ball on z in X. 2.1 Note F o r A E d g X a n d z E X ,

h(A,z)< ~ ¢:~ A c [ z ] 6 ,

h(A,z) = V{d(a,x)

so

I E A}.

Proof: The first part is simple, and from this it follows that h(A,x) is the least 6 such that A C [z]6 , i.e. such that VaE A d(a,z) < 6. o As a trivial corollary, in ddX the dosed b-ball on {x} consists of the nonempty compact subsets of [z ]~. Now, recalling that for a metric space Ythe set of 'bounded' points of Y* is bd Y* = {a E Y* I a is finitely distant from some (hence all) y E Y}, which is the union monad of the ideal Bd Y of bounded subsets of Y, we have the following criterion for the bounded subsets of JdX, equivalently (bearing in mind Section 6 of Chapter 1) for the bounded elements ofddX *; 2._.22 Proposition BdddX = (Bd X) dd , i.e. for ~/g C~"X, ,/g is bounded ~ U Jg" is bounded in X, i.e. f o r A E d d X * , AE bdddX* ¢~ AC b d X * Pro of: We prove the last line, which as explained in Section 6 of Chapter 1, amounts to the first. Taking any x E X, we have

A e bddgX * ¢ , h( A,z) is finite ¢~ V{d(a,x) [ a e A} is finite ¢:~ Va E A d(a,x) is finite ¢~ A c bd X *, using for the third '¢~' that the least upper bound of any internal set of finite hyperreals is finite, o And since [J ~ d x = x, we immediately have t h a t . . . 2.___33Coronary

ddXis bounded ¢=~ X is bounded.

o

From knowledge of the Vietoris topology we already know that JdX is compact iff X is. The nonstandard formulation of when a metric space Y is boundedly compact (namely that every bounded element of Y* be nearstandard) allows the following simple proof of the fact that ddX is boundedly compact iff X is ;

35 2..._44Proposition

d d X is boundedly compact ¢0 X is boundedly compact.

Proof:

* : As X is dosed in JdX. ¢ : Then Y B E b d J d X *, B C bd X * = n, X * so (by 3.1 of Chapter 1) B E naddX *. o

An alternative standard proof uses (along with 2.2 above, and 7.1 of Chapter 1) the easily proved result that for ideals J" and J on X , J j ~ _CJj~ (:¢ J_C J ,

as

follows ;

BdJ~X C subCp,TdX {~ ( B d X ) ~ C (subCp X ) ~

~ X i s boundedly compact ~

~:~ Bd X C subCp X 0 , where B E ~ X with h(B *,A) _< e/2 there's /~ E B * with d(/~,a) _< e/2, and in turn /~ is near some b E B , giving d(b,a) < e. So a is pre-nearstandard,

o

• .. then we have the following criterion for completeness of2~'X ;

3.__.22 Proposition

JdXis complete ¢~ X is complete• Proof: : As X is closed in ,TdX. ¢:AEpns~X* , AC_pnsX*=nsX* ~ AEnsJdX*. Using this we find that the completion of J~X is ~ ( X c ) ; 3._33 Proposition

(~X)c = ~(X c )

Proof: Xis dense in X c so by 4.2 of Chapter 1 ~ X is dense in ~ ( X c ), which is complete as X c is, hence J f ( X c ) is a completion of ~ X . o And using also the aforementioned result that a E Y* is pre-nearstandard in Y* iff a is nearstandard in ( y C ) . we can provide the converse of 3.1 to round off the picture ; 3.4 Proposition For A E dgX*,

Proof: A E pnsJgX *

A E pnsJ~X * ¢~ A c_pns X *.

A (=) AC n s ( X C ) * ¢:~ A C pns X *.

)*

*

as

)

37

4. Limits of Sequences in J~X For a sequence A = (An) of nonempty subsets of X we'll say z is a limit point of (An) with respect to d if (d,(z, An)) ---, 0. We'll call the set of limit points of (An) the limit set of (An) with respect to d , denoted by lira d (An) ; the subscript d is there so as not to confuse with the possible limit h m (An) of (An) in,Y~X when the sets A are in J d X . n

4.1 .proposition

For a sequence A = (An) of nonempty subsets of X,

(1) z e lim d (An) ~

A n intersects #z for all infinite n.

(2) lira d (An) = n {st A n [ n is infinite}. (3) lira d (An) is closed.

Proof: (1) z E lim d (An)

(d(~,An)> -~ 0 ¢¢ ¥ infinite n d(z, An) z 0 ¢~ V infinite n z E A n ~ ¢=~ V infinite n A nintersects # z .

(2) This is just a concise restatement of (1) since A n intersects #x iff z E st A n . (3) By (2), since each st A n is closed. The reason we're interested in lira d (An) is that . . . 4.__.22 Proposition

Every convergent sequence in J ~ X converges to its limit set.

Proof: Suppose (An) ~ B . Then for all infinite n , A n E #B so B = st A n ; thus

B = [7 {st A n [ n is infinite} = lim d (An). Bear in mind though that for a sequence (An) i n ~ X

lim d (An) need not belong

and even if it does, (An) need not converge to it with respect to h .

to~X

As an

example of the latter, if X is $ and A n is {0} if n is even and {0,1} if n is odd, then lira d (An) = {0}, but of course (An) is not convergent. More can be said in the case of increasing or decreasing sequences however. In g e n e r a l . . . 4.___33Proposition For a sequence (An) of nonempty subsets of X, (1) [1 A n c_lim d (An) with equality if (An) is decreasing. n

(2) lim d (An) C ~n An with equality if (An) is increasing.

Pro of: (1) 'C' is trivial.

Now further suppose (An) is decreasing.

Then for z E lim d (An) ,

z E s t A m for infinite m', and hence for all finite n we have (as A me_An *)

z E s t ( A n * ) = "~n .

38 (2) 'C' is trivial.

And if (An) is increasing we easily see that [J A n C lim d (An) , n

hence (as the latter is dosed)

[J A n c lim d (An).

o

n

And in p a r t i c u l a r . . . 4.4 Proposition

For a sequence (An) in~dX,

(I) If (An) is decreasing, (An) -~ ~ A n . (2) If (An) is increasing and bounded above in,~gX, (An) -~ ~n A n . Pro of: (1) Let B = r~A nn

Taking any infinite m

'

we must show that A

m

E ~d3

"

Since

A m c ns X * (as A m C A 0 * ) it remains to show (recalling 2.1 of Chapter 2) that st A m = B . F o r a l l f i n i t e n , A Conversely, B ' C A

incA n *so

s t A m -C s t ( A n * ) = A

n"' so st A m o-b

m so B = st B * C st A m .

(2) Let B = [J A n and let C E ddX with all A c C. Taking any infinite m n

n-

show A m E # B . A moB*

so

"

we must '

A mc_ ns X * as A m c C*, so it remains to show that st A m = B .

st A m c_s t B *

= -B = B .

Conversely, for all finite n , A n * c-A r u s o

A n = st (A n *) c st A m ; so [J A n c_ st A m , so as the latter's closed, B c_ st A m .

o

n

Also note t h a t . . . 4.__5 Proposition

InddX, if ( A n ) ~

B then ( [ J A n ) U B = n

[JAnEddX. n

Proof: Let C = [J A n . As (An} --, B then {A n ] n E w} U {B} is a compact subset ofddX n

hence the union By 4.3(2), B C C

C U B is a compact subset of X. We now show that it equals C . giving

CUBC_C.

For the reverse, if

zEC-C

then where

c E C * with c E # z , since x ~ C it follows that c belongs to some A n with n infinite, so z E st A = B . o n

Another result of interest i s . . . 4.__66 Proposition

For convergent sequences and inddX,

( Vn A n intersects B n ) ~ lira (An) intersects lira (Bn) . Proof: Let _ 2, if k is injective (in particular if F reduces

some A E ~ X such that ( f A [ f e F ) is disjoint with all f e F are injective on A) then Kis totally disconnected, o The above corollary is essentially also proved in Theorem 4.4 of [Ha]. Note incidentally that even when every element of F is a direct similitude of X = ~ n K can be disconnected without being totally disconnected. As an example in ~ we could have K = [0,1] U [2,3] , formed as the union of four nonoverlapping copies of itself, each a quarter the size of K. Figure 12 of Section 1 shows another example, this time in ~ . The main result on when K is connected is due to Hata in the above-mentioned paper [Ha], and we now give a proof of this result along with some corollaries, first of all preparing with a lemma which generalizes a step used in Hata's proof. For sets A and B we may write A~, B for 'A intersects B ' . We'll say a set~/$ of sets is interlinked if for all A , B E ~ 4 there is a linking chain from A to B i n ~ , namely a finite sequence C I . . . C n of elements o f ~ such that A = C 1 . . . . . 5.4 Lemma For A E ~ X , if F expands A and { f A I f E F }

Cn = B .

is interlinked then

{fA I f E F n } is interlinked for all n . Proof:

First note that (1) For all fE F n, ( f g A I g E F } is interlinked. For, whenever g 1A . . . . . (2) For all f , g e f n w i t h f A

gm A then .fg 1A . . . . .

^gA,{fhA

I hEF}'linkswith'(ghA

in the sense that some f h A intersects some g h A. IAc

[j f h A hEF

and g A C

fgm A .

[j g h A (by 2.6(2)). hEF

I hEF}

This is just because

65 We are now ready to prove the lemma by induction on n . The hypothesis is true for n = 1 by assumption (and also trivially true for n = 0). Now let n > 1 and assume true for n.

Take any (n+l)th-level images f i g 1A and f~ g 2A of A ( f i , f ~ E F n

and g t ,g 2 E F ) . By the hypothesis for n there is a linking chain h 1.4 . . . . . from f l A

to f~A in {hA [ hER n} (with h t = f t a n d

each set {hi g A [ g E f }

hm=f2);

hmA

now by (1)

is interlinked, and by (2) each {hi g A I g E E }

is

'linked with' {hi+ 1 g A i g E F } , hence as required there is a linking chain of (n+l)th-level images of A from h l g 1A = f i g 1A to hm g 2A = f 2 g 2A , proving the hypothesis for n+l. o Now to Hata's result, with the slight addition of (4) by the author (as well as the case for infinite F ). The proof is closely that of Hata's, who uses the special case of the above lemma in which A = K ; 5.___55Proposition [Ha] For finite F the following are equivalent, whilst for infinite F we have the implications (3) ¢# (4) ~ (1).

(1) K is connected. (2) K is connected and locally connected. (3) {fK I fE F } is interlinked. (4) { f g ] IE F n } is interlinked for all n. Proof: Since F expands K the last lemma gives (3) ¢~ (4). Now if (4) holds then bearing in mind that for each n K is the union of its nth-level images and that these have diameters < r n diam K, it follows that K is 'chain-connected' (i.e. Vz,y E K Ve > 0 there's an 'e-chain from x to y in K ' , i.e. a finite sequence in K starting at x and ending at y such that successive points are distance at most e apart), and since K is compact this is equivalent to K being connected. This establishes (4) ~ (1). For the remaining implications assume F is finite ; (1) ~ (3) : Let fE F and G = {g E F [ there's a linking chain from f K to g K in { h K I h E F } } . If G # F t h e n f o r a l l g E G a n d h ~ G g K i s d i s j o i n t f r o m h K , thus ~JG, K is disjoint from ~JGhK,andsincethesearenonemptyclosedsetsKisnot g h connected, contradiction ; so G = F as required. (1) $ (2) : As Kis connected so is each nth-level image of K (being a continuous image of K ) , and since Kis their union we have that Ye > 0 Kis the union of finitely many connected sets of diameter _< e ; hence by a general result K is locally connected, o As also noted by Hata, (2) in other words says that K is a locally connected continuum (a 'continuum' being a nonempty connected compact space), which by the Hahn-Mazurkiewicz theorem is equivalent to K being a continuous image of [0,1].

66 Also note that the implication '(1) ~ (2)' cannot hold for arbitrary F, simply by 1.2 and the fact that there can exist connected A E~ X without A being locally connected. Hata's result easily yields the following sufficient condition for K to be connected ; 5._66 Corollary If some A E ~ X is expanded by F with {fA ] fe F) interlinked, then K is connected.

Proo] : We know A c K, so {fK IfE F} is interlinked too.

o

In particular, as (fix f l fE F ) is expanded by F, and is compact being the image of Funder the continuous map fix : Semigroup F ~ K in 4.5, t h e n . . . 5._2 Corollary Letting A = (fix f I fE F }, if ( f A I fE F } is interlinked then K is connected, o For example the above applies to show that the Sierpinski Gasket is connected. Alternatively 5.6 could have been used, by taking A to be the boundary of the triangle involved. Here's a further sufficient condition for K to be connected. Since the set of connected elements o f ~ X is (by 4.3 of Chapter 1) closed i n , X , and K = lira (F hA) for all A E ~ X , t h e n . . . 5.8 Note If there's A E ~ X such that all iterates F nA of A under U F are connected, then K is connected, o 5._99 Corollary

If there's connected A E ~ X such that ( f A I f E F n } is interlinked

for all n , then K is connected.

Proof: As {fA ] fE F n ) is interlinked then since each ] A is connected (being a continuous image of A) their union F nA is connected (by a general result),

o

Note that in the case where F expands connected A E ~ X we can replace the condition in the above by the condition that {fA I f E F } be interlinked, thus giving an alternative path to 5.6. On the other hand, if F is finite and reduces connected A E ~ X , then K is connected iff {fA I f E F n } is interlinked for all n , since if K is connected then each { I K I f E F n } is interlinked hence so is each {fA [ fE F n } as in general f K c fA.

67 6. Regulaxity and Residuality of K In this section we show that subject to the 'homeomorphism condition' below, K is either regular or residual (see 'Preliminaries' for a reminder on these terms). Throughout let hF denote the set of elements of F which are homeomorphisms of X. The homeomorphism condition on F, expressing that the elements of F which are not homeomorphisms are nevertheless tied down to some extent by those which

are, is that

F - hF c Semigroup hF (implicitly demanding that hF be nonempty), equivalently that Semigroup hF is dense in Semigroup F (noting that if the first condition holds, then F C SemigrouphF, so as SemigrouphF is a semigroup, Semigroup F C Semigroup hF , i.e. Semigroup hF is dense in Semigroup F ). In the case where every element of F is a homeomorphism, the condition is of course trivially satisfied and we can expect the general theory concerned with K to be simpler than it otherwise might.

The main reason we widen the scope of the work below from this

simpler situation to the situation in which merely the homeomorphism condition is required, is for the sake of an application at the end of Section 4 of Chapter 4 when we show that certain invariant sets may be 'view self-similar'. 6.1 Note

If F satisfies the homeomorphism condition,

(1) {fix f l fE Semigroup hF } is dense in K. (2) For any z E K , any neighbourhood N of x, and any bounded B c X , there's f E Semigroup hF with f B c N.

Proof: (1) Semigroup hF is dense in Semigroup F, and the map fix : Semigroup F ---, K is continuous by 4.5, so {fix f l f e Semigroup hE } is dense in {fix f l f e Semigroup F } ; and as the latter is dense in K (4.3) so is the former. (2) By (1) it suffices to prove this for the case where x= fix g

for some

g E Semigroup hF. Being bounded, B is a subset of some ball on x, and if we apply the contraction g repeatedly then the ball will be mapped inside N ; thus we can take f to be some power of g.

o

The homeomorphism condition is of little interest in the case of finite F since hF has the same attractor as F , thus allowing us to reduce things to the case where all the contractions involved are homeomorphisms ; 6.22 Corollary If F is finite and satisfies the homeomorphism condition, K F = KhF.

Proof: Being nonempty and finite, hF is admissible. And by the last note, K F is the closure of {fix f I fE Semigroup hF } ; but so is KhF of course,

o

6B 6.__33 Lemr-a

If F satisfies the homeomorphism condition, the following are equivalent

for any A E J~X reduced by F ; (1) K O A is dense in K . (2) Kintersects A . (3) For some fE Semigroup hF, fix f E A . (4) For some f e Semigroup hF, f A C_A .

Pro of:

(1), (2): Trivial. (2) $ (1) : Let z e K, and N be a neighbourhood of x.

As A is bounded there's

SESemigrouphF with f A C N , so (as K N A # O ) O~f(KNA_~cfACN; and f ( K f l ,4) = f g f l f A C g o A , so g f l Aintersects g . (2) ¢=~ (3) : Since {fix f[ fE Semigroup hF } is dense in K, it intersects A i f f K does. (2) $ (4) : Let z E K O A .

Then as A_ is a neighbourhood of z and A is bounded, there's

f e Semigroup hF with f A C_A . (4) ~ (2) : Then since f g c f A c A , gintersects A__.

o

We now have the main result ; 6.__A4Proposition

If F satisfies the homeomorphism condition, K is regular or residual.

Proof: If K is not residual, i.e. K ~ 0, then K intersects K so by 6.3 K A K is dense in K, i.e. K i s dense in K, i.e. K i s regular,

o

The following result gives a sufficient condition for K to be residual in the case of X = g{n For A C ~n, dim A denotes the topological dimension of A (for the definition of which, see for example [HW]). 6._.55 Proposition

Let X = ~n and F be finite with F = hF.

Then if for all distinct S,g E F

dim (fKN g K ) < n-2, equivalently if some

A E ~ X i s reduced by F s u c h that for all distinct f,g E F dim ( f A N g A) < n-2, K is residual.

Proof: The equivalence of the two conditions is easily seen, using that if the second holds then so will the first as K c A. Assume now the condition on K. By induction on n (using

F = hF ) it follows that for all n >_ 1, for all distinct f,g E F n dim ( f K N g K) < n-2. Suppose now for a contradiction that K # ~3 ; let B be a closed ball with B c K. Let 0 < e < diam B , and take n such that r n diam K 2 K will be disconnected of course (totally disconnected if F = hF ; recall 5.3). Towards establishing a more generally applicable sufficient condition for K to be tiled with respect to F we introduce the following definition. The compact set condition on F is that there exists a regular nonempty compact set A (or 'body' for short; see 'Preliminaries') reduced by F with (fA I f e F) nonoverlapping. K is a subset of any such set of course, hence ( f K ]fE F ~ will be nonoverlapping in X too, but this need not be saying much ; for if every f K i s residual, trivially (fK] fE F ) is nonoverlapping in X; and it certainly does not follow that ( f K I f E F ) is nonoverlapping in K. Related to the compact set condition is a condition appearing in the literature in various forms, namely the open set condition that there exists a body-interior D closed under all f e F with (fD IfE F ) disjoint. Indeed, bearing in mind that the bodies correspond bijectiveIy with the body-interiors via the mutually inverse maps A ~ ,4 and D ~-* D, we have the following ; 7.5 Note If F = hF then (1) If the compact set condition holds with A, the open set condition holds with A . (2) If the open set condition holds with D, the compact set condition holds with D . (3) The compact set condition and the open set condition are equivalent.

Proof: (1) As A is closed under all fE Fso is A . And as (fA [ fE F ) is nonoverlapping then (fA [ f e F ) is disjoint, i.e. ( f A [ f E F ) is disjoint. (2) As D is closed under all f e F so is D. And as (fD [ fE F ) is disjoint with each f D open, (f---D [ fE F ) is nonoverlapping, i.e. ( f D [ fE F ) is nonoverlapping. (3) By (1) and. (2). o In the case F = hF the open set condition is also equivalent to merely the existence of a nonempty subcompact open set V closed under all fE F with (f V IfE F ) disjoint, since F will then satisfy the compact set condition with V (and hence the open set condition with the regularization (V) of V). It is usually this

72 statement of the open set condition that has appeared in the literature (e.g. see [Fa] and [Li,1] or [Li,2] ), though with 'subcompact' equivalently replaced by 'bounded' since the interest has only been in ~n In [ttu] however, the requirement of boundedness is omitted altogether. The literature seems to have dealt solely in the open set condition, but for our purposes the compact set condition seems more natural. A few notes on the condition follow. 7.6 Note If F = hF and F satisfies the compact set condition, F must be finite.

Proof: Suppose F satisfies the compact set condition with A. Then { f A I fE F } is a cover of PA by nonempty compact sets with each f A regular in FA (being regular in X ) and with { f A I f E F } nonoverlapping in FA (being nonoverlapping in X with each f A regular in X (see A5.1)) ; thus it's a tiling of FA. And with reasoning similar to that in 7.1, it follows that F must be finite, o

7.7 Note If F = hF and F satisfies the compact set condition with A, then for all n ( f A I fE F n ) is nonoverlapping with each f A regular. Proof: As A is regular so is each f A = (of)A since of is a homeomorphism. And as ( f A I fE P ) is nonoverlapping in X, it's equivalently nonoverlapping in A as each f A is regular; so by 7.2(2) ( f A t fE F n ) is nonoverlapping in A, equivalently in X.

o

7.8 Note If F = hF then U F preserves regularity, i.e. if A e ~ X is regular so is FA.

Proof: If A E ~ X is regular, so is each first-level image f A of A, hence so is the union FA by the simple result that any union of regular closed sets which is closed is regular, o 7.9 Note If F = hF and F satisfies the compact set condition with A, it also satisfies it with FA.

Proof: By the last note FA is regular, and as A is reduced by F so is FA.

Lastly, as

(f A l f E F } is nonoverlapping so is ( f FA [ f E F} since each f FA c_f A.

o

7.10 Lemma If F = hF and F satisfies the compact set condition with A, then

VfE Monoidf , KN ~ A = f K N f__A_=f ( K N A ) . Proof: Let f = o f where fE F n. Since (g A I g E F n ) is nonoverlapping (by 7.7) then for g E F n _ { f } , g A is disjoint from LA_, so g K is disjoint from ]_.Atoo. So as K is the union of its nth-level images, K N/.4 = f K N f_A. Lastly, as o f is a homeomorphism,

f K n f A = f K n fA_= f ( K n A A_). We now have the first main result of this section;

o

73 If F = hF and F satisfies the compact set condition with some A

7.11 Proposition

such that K intersects A , then K is tiled with respect to F.

Proof: Taking fE F, let V = K n f A , equivalently V = f K fl f_A = f ( K n A). Then V is an open-in-K subset of f K , and is dense in f K as

K fl A is dense in K (by 6.3(1)).

So f K i s regular in K. And for g e F - { f } , f K i s disjoint from £--K-K( - - K denoting 'interior in K ' ), otherwise the set V (being dense in f K ) implying that f A intersects g A (as

V c fA

would intersect g K g ,

and 9 KK C_g K c g A ), contradicting

that f A and g A are nonoverlapping,

o

Whether or not the condition that K intersects A can be omitted in the above is at present unknown to me.

Certainly K need not intersect the interior of every A

satisfying the compact set condition; i.e. K could be a subset of the boundary OA of A. As an example of this let X = ~2, F = { f l , f 2 } where fl

and f2

contract

by factor

1/2

about

A

zt~z 2

respectively, so K = [z~ ,z 2 ], and let A be a rectangle with base K. Although K c OA, K is nevertheless tiled with respect to F. Indeed, there is a set B satisfying the

z~

compact set condition with K intersecting _BB; for

W ~

z~

~

example the square with diagonal K. Also note that in this example there's a one-dimensional affine subspace

z1 ~

z2

of ~ closed under the elements of F (within which K therefore lies);

this could be relevant.

However, a

more fundamental open question is the following.

If F is finite and F = hF, and

( f K I f E F ) is nonoverlapping in K, does it follow automatically that the sets f K are regular in K, i.e. that K i s tiled with respect to F ? For I F I = 2 at least, the answer is yes. See note 10.11 for some remarks on the matter. Our second main result is concerned with regular K being tiled with respect to F ; 7.12 Proposition If F = h F a n d Kis regular, then (1) K i s tiled with respect to F ¢:~ ( f K I fE F ) is nonoverlapping. (2) F can satisfy the compact set condition only with K ; and it does so iff Kis tiled with respect to F.

Proof: (1) The first-level images of K a t e regular in X, hence (as K i s regular) also in K. So K is tiled with respect to F iff ( f K I f e F ) is nonoverlapping in K. But since the sets

f K are regular,

(f K I f e F)

is

nonoverlapping

in

K

iff

(f K I f e F )

is

nonoverlapping in X. (2) Suppose F satisfies the compact set condition with A. We must show A = K, i.e. A c K. As K ~ O, there's f e Semigroup F w i t h f A _c K (Kis a neighbourhood of some

74 xE K , and we can apply 6.1(2) to B = A ) , so f A = K N f A = K n f A = f ( K f l A ) (using 7.10 for the last equality) giving A = K f] A , i.e. A c K , so A c K as required. Lastly, by (1) and the regularity in K of the first-level images of K, F satisfies the compact set condition with K iff K is tiled with respect to F. o As a consequence we have the following quite widely applicable sufficient condition for residuality of K ; 7.13 Corollary If F = hF and F satisfies the compact set condition with some A such that FA ~ A (i.e. A ~ K ), then K is residual. Proof: By 6.4 K is either regular or residual. But by 7.12 it cannot be regular as F satisfies the compact set condition with some A ~ K. o For example the above shows that the well-known Sierpinski Carpet is residual (something for which 6.5 was too weak), by taking A to be the square involved. It also shows that the Sierpinski Gasket is residual (though this also followed from the more specialized result 6.6) by taking A to be the triangle involved. Using 7.12 and a few earlier results we observe also the following equivalences ; 7.14 Proposition If F = hF, the following are equivalent ; (1) K is regular and is tiled with respect to F.

(2) (3) (4) (5)

K is regular and F satisfies the compact set condition. F satisfies the compact set condition with a unique A. F satisfies the compact set condition with a least A. F satisfies the compact set condition with K.

Proof:

(1), (2) (2), (3) (3), (4)

: By 7.12(2) Fsatisfies the compact set condition with K. : By 7.12(2) again. : Trivial.

( 4 ) , (5) : If F satisfies the compact set condition with least A, then since it also satisfies it with FA (by 7.9) we have A c_ FA, hence FA = A so A = K. (5) ~ (1) : By 7.11, taking A = K. o Finally we point out the connection of the above subject matter with 'reptiles', which have been known in the literature of tilings for some time, the term being short for 'replicating tile'. A reptile (see [Mar] for example) is a polygon in ~2 which is tiled by smaller mutually congruent copies of itself; in our terms it's a polygon of the form KF for some finite set F of similitudes of ~2 having the same scale factor, such that KF is tiled with respect to F. By an existence theorem for tilings (a general form of which is proved in the latter half of Appendix 5 using nonstandard analysis and various hyperspace work), any reptile can be used to tile ~2 The condition of being a polygon

75 is superfluous in this respect, as is the dimension of the space; we can generalize for example as follows. By an isometric copy of K = K E we'll mean an image of K under an isometry of the space, whilst by a direct isometric copy we'll mean an image under a

direct isometry. 7.15 Proposition

For any finite nonempty set F of [direct] similitudes of s n with

common scale factor r < 1 such that K is regular and tiled with respect to F, ~n can be tiled by [direct] isometric copies of K.

Proof: (Note that in the following, 'copy' will mean 'isometric copy' or 'direct isometric copy' according to which version of the result we're proving.) By an existence theorem (e.g. A5.13 with ~ = { K } ) it suffices to show that every bounded subset of ~n can be covered by mutually nonoverlapping copies of K . Taking any x E K let g be the dilation about z by factor 1/r. Then for all n , letting /

be the set of nth-level images of K , g n /

is a set of mutually nonoverlapping

copies of K with union g n K . Since every bounded set is covered by a set of the form g n K , the proof is complete,

o

In connection with this result, see also note 7.8 of Chapter 4, concerned with obtaining explicit examples of tilings of X by isometric copies of K.

76 8~ Continuity of the Attractor Map Recall that for each admissible set F of contractions of X we have the 'attractor'

KF of F. We thus have a map K : Admis X ~ ~ X taking each F to its attractor KF . We shall call this the attractor map. Admis X is a subset of u~'C (X,X), and giving the latter the Vietoris topology induced from the compact-uniform topology on

C (X,X), Admis X becomes a topological space, namely a subspace of JgC (X,X), and we are now free to consider the possible continuity of the attractor map. The main result of this section is t h a t . . . 8-_! Proposition

If X i s locally compact, the attractor map is continuous.

Proof: The attractor map is the composition of

[J : Admis X ~

ContracJgX followed by

fix : Contrac~X ---,JgX. And these are both continuous, the first by 6.6 of Chapter 2 (local compactness not being needed here), and the second by A3.2 applied t o ~ X (noting that ~ ' X is locally compact as X is).

o

For non-locally-compact X the attractor map need not be continuous, due to the fact that

fix: C o n t r a c X ~ X

need not be. For i f g E # f

with f i x g ¢ # f i x f ,

then

taking F = {f} and G = {g } we have G E # F i n Admis X, but since K G = {fix g } and K F = {fix f } then KG¢ # K F. However, by placing an upper bound u on the Lipschitz ratios allowed, we can obtain the following restricted result. For u < 1 let

A d m i s u X = { F e A d m i s X l VfE F rf < u, i.e.

\ / r e_< u } , feF J space Y let Gon~racuY= {f e Contrac Y I rf 2 }, F i s compact by (3). However, for all n > 2, en+ 1 < cn

so (17%)~n+l = In ~n+l ~ F {%+1 } so h(F {~n+l }' F {0}) = h(F{cn+ 1 }, {0)) > (1-~n)~n+ 1 = (1-~ n) h({%+ 1 },{0}) ~ving fUR>1-%. Thus rUE>1, so [J F is not a contraction. 10.3 For F EJgContrac X, we know that if F is admissible then [J F is a contraction; the converse however does not hold. For example, take X = [0,1] and let

F = {fn I n >_ 2 } U {c x I x E X } where fn is as described in the last note and cxis the constant map with value x. Then U F is the constant map with value X, but f ~ ; ~ s f = 1 so F is not admissible, ts it true though that for F C~ R e d u c X such that a reduction, F is reduction-admissible? If so, we could shorten the proof of 9.3 since, using A4.5 applied to J~X, 9.3(3) implies [J F is a reduction. 10.4 It's not generally true that F is determined by its union function U F ; haveF~ Gyet UF=

we can

UG- For example l e t X = ~ , F = { f ~ , f 2 } a n d G = { g l , g 2 }

where f l x = x/2 , f2 x = - ~ / 2 , g I agrees with f2 on (-oo,0] and with f l on [0,oo), and g 2 agrees with f l on (-c~,0] and with f2 on [0,oo). To see that U F = U G use the easy general result that [J F = [J G ¢~ Vx E X Fx = Gx (Fx and Gx denoting

F{x) and G{x) ).

However, the maps gi are not bijective. Is there an example in

82 which all the elements of F U G are bijective? Note also that in general LJ F = LJ G {fix.t'l fE F } = ,{fix g I gE G } . 10,5 For any similitude g of X = ~ n

g(KF)= K(go Fo g -1)

where g o F o g ' l

denotes {g o f o g -1 [ fE F }.

Section 2 : nth-level Imag.~ 10.6 Suppose all elements of F axe bijective.

Recall that for fE F n we defined

o f = f0 o . • . o f n - 1 ' the terms of f being applied in reverse order of their appeaxancein f . It may be helpful to realise that o f can also be expressed as a h composition of suitable conjugates of f 0 , - • , f n - 1 in this order. Namely, letting g denote the c o n j u g a t e A o g o h ' l o f g b y

h, and using that

Aog=gAoh,

we have

for example that The pattern is obvious. 10.7 The following result essentially appears in [Ha] on page injective and I KI >- 2 then K is perfect. A more basic proof is for all ~ z belongs to some nth-level image f K of K, and ~th-level images decrease to 0, and every nth-level image

389; if all fE F are to use that for x E K, the diameters of the of K has the same

cardinality as K due to injectivity of the elements of F. Section 5 : C0nnectedness of K 10.8 Note that we can have A E ~ X reduced by F with ( f A [ f E F ) interlinked yet (fA IfE F 2 ) no~ interlinked. An example is shown below in which X = U{~ and

F = {.ft ,.f~}, where fi contracts about zi by factor 1/2 then rotates about Pi by ~r/2. In this example K F is totally disconnected since F reduces FA and 2.

83 Section

6 :

Regularity and Residuality of K

10.10 Try and find practical sufficient conditions for K to be regular. Section 7 : Tiling of K 10.11 If F is finite and every element of F is a homeomorphism, and 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

.~ 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 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 ~ 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 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 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 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 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) 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 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) 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

144

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

145

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).

o

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

149

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#}.

150

(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

,

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