Fractal Measures on Infinite Dimensional Sets

Home , Packing dimension

Fractal Measures on Infinite

Dimensional Sets

DISSERTATION

Presented in Partial Ful llment of the Requirements for

the Degree Do ctor of Philosophy in the Graduate

Scho ol of The Ohio State University

Mark McClure, B.S.

*****

The Ohio State University

1994

Approved by Dissertation Committee:

Gerald A. Edgar

Joseph Rosenblatt

Adviser

Neil Falkner

Department of Mathematics

Vita

August 4, 1964 ...... Born in Columbus, Ohio.

1988 ...... B.S., Ohio State University, Columbus.

1989-1994 ...... Graduate Teaching Asso ciate,

Department of Mathematics,

The Ohio State University,

Columbus, Ohio.

Fields of Study

Ma jor Field: Mathematics

Studies in Fractal Geometry, Measure Theory ii

Table of Contents

VITA ...... ii

LIST OF SYMBOLS ...... v

CHAPTER PAGE

I Intro duction ...... 1

II Basic De nitions ...... 4

2.1 Measure Theoretic De nitions of Dimension ...... 4

2.1.1 Hausdor Dimension ...... 4

2.1.2 The Packing and Centered Covering Dimensions ...... 9

2.2 The Entropy Dimensions ...... 14

III Computational Metho ds ...... 21

3.1 Sequence Spaces ...... 21

3.2 Calculation of the EntropyDimensions...... 23

3.3 Computation of Hausdor Measures ...... 26

3.4 s-Nested Packings ...... 30

3.5 A Cartesian Pro duct Example ...... 34

IV Dimensions of Hyp erspaces ...... 38

4.1 Entropy Dimensions of the Space K (X ) ...... 38

4.2 Hausdor Dimension of K (X )...... 44

4.3 Entropy and Hausdor Dimensions of the Space C (R ) ...... 50 iii

V Function Spaces ...... 55

5.1 Entropy Dimensions of the Space of Smo oth Functions...... 55

5.2 Hausdor Dimension of the Space of Smo oth Functions ...... 62

BIBLIOGRAPHY ...... 65 iv

List of Symbols

Symbol Page

' '

;';H ; H ...... 5

; ; ...... 6

H ...... 8

' ' '

e e

C ; C ; C ...... 11

' ' '

e e

P ; P ; P ...... 11-12

N ; ; ...... 15

; ...... 16

; ; [ ] ...... 21-22

D (x; ( ) ) ...... 28

k k

' ' '

e e

Q ; Q ;Q ...... 34-35

K (X ); ...... 38

C (R ) ...... 50

C (E );fj ; kk ...... 55

F (c; q ; R );F(c; q ; E ) ...... 56 v

CHAPTER I

Intro duction

The oldest notion of dimension was clearly intuitive in nature. Mathematicians felt

that the line R was somehow di erent from the plane R long b efore they were proven

to b e non-homeomorphic. Intuition is less useful a guide for more complicated sets,

however. Cantor typ e sets illustrate this nicely. In some sense they are small b eing

ableto llupnopartofR. On the other hand, they seem big b eing uncountable and

measure theoretically rich. Thus the need arises for a rigorous de nition of dimension.

There are now manysuch de nitions. Fractal dimensions are those that concen-

trate on the metric structure of a set and lead to the p ossibilityofa sethaving a

non-integral dimension. Thus a fractal dimension allows for the p ossibility of dis-

tinction b etween sets with equal top ological dimension, which is necessarily integral.

Cantor's ternary set for example has top ological dimension 0, but fractal dimension

log 2

0:6309. (There are di erent de nitions of fractal dimension, but I knowof

log 3

none that assign some other dimension to Cantor's set.) Thus a fractal dimension

somehow captures the bigness of Cantor's set which the top ological dimension misses.

The oldest fractal dimension is Hausdor 's [Ha]. His de nition is based on a

measure theoretic construction of Caratheo dory [Ca ]. Caratheo dory was interested 1

in de ning a measure similar to Leb esgue's to measure the size of a smo oth m-

dimensional sub-manifold of R . In [Ha], Hausdor shows that this may b e extended

to a natural s-dimensional measure H for any real s>0. This in turn de nes

dimension bycho osing an appropriate parameter s to measure a given set E R .

More generally, the dimension of a set E may b e roughly asso ciated with a mono-

tone increasing function '(t) de ned for t 0 and with '(0) = 0. In this case write

dim(E ) '(t). Tosaythat E has nite dimension s,istosay that dim(E ) t .The

faster ' vanishes at the origin, the larger the corresp onding dimension. It is p ossible

if E is a subset of a large metric space, that dim(E ) ',where' disapp ears faster

at the origin than anypower of t.Thus we see that fractal dimensions maybeused

not only to distinguish b etween sets of equal integral dimension, but also p otentially

between sets of in nite dimension. It is on such sets that I will concentrate in this

dissertation.

The organization is as follows. In chapter two, I state de nitions making the

ab ove ideas precise. This chapter is based on much earlier work, but tailored to my

purp oses. There are many de nitions stated, but they are all interrelated and some

are essentially equivalent. I prove some comparison theorems which indicate that a

fairly complete dimensional picture emerges if we can understand two de nitions in

particular - the Hausdor dimension and the upp er entropy dimension.

In chapter three, I develop some imp ortant computational to ols and illustrate

their use on some spaces which are relatively easy to deal with called sequence spaces.

In section 3.4, sequence spaces are in turn used to de ne another imp ortant to ol for

calculating Hausdor dimension - the s-nested packing. Finally, in section 3.5 I discuss

an example involving Cartesian-pro ducts which illustrates why certain de nitions

haveevolved to their currentform.

Chapters four and ve form the heart of the original material dealing with hyp er-

spaces and function spaces resp ectively. These are two of the most natural examples

of in nite dimensional spaces. Hyp erspaces are metric spaces whose elements are

subsets of another metric space. Two early pap ers concerned with dimensions of hy-

p erspaces are [Boa] and [Bro]. In chapter four, I extend results proven there to other

notions of dimension and more general metric spaces.

The fth and nal chapter is based on [KolTi]which summarizes the earliest

dimensional estimates sp eci cally aimed at in nite dimensional spaces. The theorems

proven in [KolTi] deal with what are now called the entropy indices for various sets

of functions and are part of the resolution of some questions arising from Hilb ert's

thirteenth problem. I extend some of these theorems to other notions of dimension.

CHAPTER I I

Basic De nitions

In this dissertation I will b e de ning various notions of dimension in a general setting.

(X; )orsimplyX will denote an arbitrary separable metric space. I will frequently

sp ecify that X is complete. For E X ,diam(E ) will denote the diameter of E . That

is diam(E )=sup f(x; y )g.IfE; F X , then dist(E; F )= inf (x; y )

x2E; y2F

x;y 2X

denotes the distance from E to F .Given x 2 X and r>0, B (x) will denote the

closed ball of radius r ab out the p oint x.ThatisB (x)=fy 2 X : (x; y ) r g.

2.1 Measure Theoretic De nitions of Dimension

2.1.1 Hausdor Dimension

The rst de nition of a fractal dimension was Hausdor 's [Ha]. Although he was

primarily concerned with subsets of Euclidean space, his de nition generalizes readily

to -totally b ounded subsets of an arbitrary metric space. Here I followthevery

general approach of Rogers [Rog ].

For ">0an"-cover of E will b e a countable or nite collection of sets, E X ,

i 4

so that E [E and diam(E ) " for every i. Let denote the set of all non-

i i i

decreasing, continuous functions ' de ned on some interval [0;) so that '(0) = 0

and '(t) > 0 for t> 0. Such functions will b e called Hausdor functions. Given

' 2 , de ne a measure H on X as follows:

( )

H (E )= inf '(diam(E )) : fE g is an "-cover of E ;

i i i

' '

H (E )= limH (E ):

"&0

' '

Note that H (E ) increases as " decreases so that H (E )iswell de ned, though

p ossibly in nite. In [Rog ] it is proven that H is a metric outer measure on X .By

' ' ' '

a metric outer measure I mean that H satis es H (E [ F )= H (E )+ H (F ),

whenever dist(E; F ) > 0. This implies that all analytic (and in particular all Borel)

' ' '

subsets of X are H -measurable. Denote the restriction of H to the H -measurable

subsets of X also by H and call this the Hausdor '-measure on X .

The idea b ehind the Hausdor dimension is that the value of H (E )isgoverned

by the asymptotic prop erties of '(t)as t & 0inaway indicative of the dimension of

E .For example if is Leb esgue measure on R and (t)=t ,then

0 if >n

= c if = n (c constant) H

n n n

is non- - nite if

In this dissertation I am primarily concerned with in nite dimensional sets and so

need an appropriate de nition and metho d of comparing dimensions. Write:

dim(E ) ' if H (E )=0

dim(E ) ' if H (E )is - nite

dim(E ) ' if H (E )isnon- - nite

dim(E ) ' if H (E ) > 0

dim(E ) ' if H (E )ispositiveand - nite.

Iwould like to b e able to use these same symb ols to compare Hausdor functions. So

write:

(t)

=0 ' if lim

t&0

'(t)

(t)

' if lim sup < 1

t&0

'(t)

(t) (t)

' if 0 < lim inf lim sup < 1:

t&0

'(t) '(t)

The following lemma justi es the common use of these symb ols.

Lemma 2.1.1 If dim(E ) ' , then dim(E ) .

Pro of: Assume rst that H (E ) 0, cho ose ">0such

(t)

<=M , and cho ose an "-cover fE g of E such that that 0

i i

'(t)

'(diam(E ))

H (E ) (diam(E ))

(diam(E ))

= 0 and H Thus H (E )=0 as " maybechosen arbitrarily small.

' '

If H (E )is - nite, then E = [ E where H (E ) < 1 for every i.Sobythe

i i i

ab ove H (E ) H (E )=0:2

With minor mo di cations, the ab ove pro of shows the following:

If dim(E ) ' then dim(E )

There are reverse inequalities with similar pro ofs. For example:

Lemma 2.1.2 If dim(E ) ' , then dim(E ) .

(t)

>M. Then for Pro of: Let M>0 and cho ose "> 0such that 0

'(t)

any "-cover fU g of E ,

i i

X X

(diam(U ))

(diam(U )) = '(diam(U )) >MH (E ):

i i

'(diam(U ))

i i

Thus for any M>0, H (E ) M H (E )so H (E )= 1.

To see that H (E ) is non- - nite supp ose that E = [ E .ThenH (E ) > 0 for

i i i

some i and so H (E )=1 by the argumentabove.2

Again a similar pro of will show the following:

If dim(E ) ' then dim(E )

isavery rich set and the ab ove ordering is by no means total. It is consequently

not a tractable problem to understand howdim(E ) compares with every ' 2 .

What one sometimes do es, therefore, is de ne an appropriate one parameter family,

f' g , such that < implies ' ' . Then there is a critical value

2 [0; 1] such that

(

0 if >

H (E )=

1 if < :

Supp ose (t)= t . This family of functions is useful when studying subsets of

is usually abbreviated H .Ihave made the claim that in this Euclidean space and H

dissertation I am primarily interested in in nite dimensional metric spaces. I mean

by this more precisely that I will b e working with metric spaces (X; ) satisfying

dim(X ) for every >0. It so happ ens that a useful family of Hausdor

functions for many of the sets that I will b e studying is f g de ned by (t)=

1=t (1=t )

2 . Another useful family is f' g de ned by ' (t)=2 where s> 0is

xed.

In [Fal2]itisproven that Hausdor dimension is preserved by bi-Lipschitz trans-

formations. There, however, he is working with the family (t)= t . One needs

to b e more careful when working with more general functions. The following lemma

do es hold.

Lemma 2.1.3 Supp ose f : X ! X is bi-Lipschitz satisfying:

r (x; y ) (f (x);f(y )) r (x; y ): (2.1)

1 2

Then

dim(E ) ' ) dim(f (E )) '(t=r )

and

dim(E ) ' ) dim(f (E )) '(t=r ): 1

Pro of: If fE g is an "-cover of E ,thenff (E )g is an r "-cover of f (E ). If in addition

i i 2

'(diam(E )) < 1, then by the second part of inequality2.1,

! !

X X

r diam(E ) diam(f (E ))

2 i i

' < 1: '

r r

2 2

i i

If '(diam(E )) >c>0, then by the rst part of inequality 2.1,

! !

X X

diam(f (E )) r diam(E )

i 1 i

' ' >c:

r r

1 1

i i

The result follows.2

Consider, for example, the two parameter family of functions

(M=t)

' (t)= 2 :

For a xed >0 a bi-Lipschitz map with ratios r and r as ab ove can a ect the

1 2

critical value of M . But it can't b e raised by more than a factor 1=r and it cannot

be lowered by more than a factor of 1=r .For < ,however, ' ' for

1 1 2 M ; M ;

1 1 2 2

any M and M . So a bi-Lipschitz map won't haveany a ect on the critical value of

1 2

This bi-Lipschitz invariance of the Hausdor dimension is due to its dep endence

on the metric structure of the set in question. This is a general feature of fractal

dimensions and lemmas similar to 2.1.3 hold for all the notions of dimension discussed

here.

2.1.2 The Packing and Centered Covering Dimensions

The packing measure and dimension were intro duced byTaylor and Tricot [Tr, TayTr]

as notions (almost) dual to the Hausdor measure and dimension. Equalityofthe

Hausdor and packing dimensions of a set imply some regularity prop erties for the

set. The centered covering measure and dimension were intro duced by St. Raymond

and Tricot [RayTr] as notions precisely dual to the packing measure and dimen-

sion. In these pap ers it is assumed that Hausdor functions are blanketed. That is

lim sup '(2t)='(t) < 1. This is tantamount to restricting to the nite dimensional

t&0

case as the following lemma shows.

Lemma 2.1.4 If lim sup '(2t)='(t) < 1, then there is an >0suchthat' t .

t&0

Pro of: Cho ose t > 0suchthat0

0 0 0

n n n+1 1

'(2 t ) '(2 t ) '(2 t ) '(2 t ) '(t )

0 0 0 0 0

: M

'(t )

n n+1

Thus for 2 t

0 0

n+1

t t (2 t )

2 M 2 :

'(t) '(2 t ) '(t )

0 0

! 0whenever is large enough so that M 2 < 1:2 Nowas t & 0, n !1.Thus

'(t)

I'll now de ne the centered covering and packing measures and investigate the

relationships b etween these quantities and the Hausdor measure. Of course, I will

not assume that my Hausdor functions are blanketed. This will not a ect the

de nitions, but will slightly alter the comparison theorems.

The centeredcovering '-measure is very similar to the Hausdor measure. Rather

than covering the set E X with arbitrary sets, however, wecover E with closed

balls centered in E .We then measure the size of these balls using their radius rather

than their diameter. Thus de ne a centered "-cover of E to b e a nite or countable

collection of closed balls fB (x )g suchthatx 2 E and 2r " for every i.Then

r i i i i

let

( )

' 1

C (E )= inf '(2r ): fB (x )g is a centered "-cover of E ;

i r i

" i=1

' '

e e

(E ): C C (E )= lim

"&0

As with the Hausdor measure this is a well-de ned limit. However, a centered cover

of E maynotbea centered cover of some F E . Because of this it is p ossible to

' ' '

e e e

have F E with C (F ) > C (E ). So C is not an outer measure. Therefore de ne

' '

C (F ): C (E ) = sup

F E

' '

It is proved in [RayTr]thatC is a metric outer measure and I will also denote by C

' '

the restriction of C to the C -measurable sets.

For the packing '-measure, rather than covering E and taking an in mum, one

packs E with disjoint closed balls and takes a supremum. More precisely,for ">0an

"-packing of E will b e a nite or countable collection of disjoint closed balls fB (x )g

r i i

such that x 2 E and 2r " for every i. Then let

i i

) (

'(2r ):fB (x )g is an "-packing of E ; P (E ) = sup

i r i i

i "

' '

e e

P (E )= limP (E ):

"&0

This time the limit exists b ecause as " decreases P (E ) decreases as well. "

We shall see shortly that P is not sub- -additive and, so, not an outer measure.

So, for -totally b ounded E X de ne

( )

' '

P (E )=inf P (E ):E [E :

i i i

Note that although P is not an outer measure, it is nitely subadditive and mono-

n ' '

e e

tone. That is if E [ E ,thenP (E ) P (E )andifF E ,then

i i

i=1 i=1

' '

e e

P (F ) P (E ). Both these statements follow immediately from the fact that a

packing of a subset of E is also a packing of E . Also note that P is nite for a

broader class of sets than P . If E is -totally b ounded, but not totally b ounded

' '

e e

then P (E )=1 for any ' 2 . Next I will showthatP resp ects closure. That is

' '

e e

P (E )=P ( E ). Thus wemay assume that the E 's ab ove are closed sets.

' '

e e

Lemma 2.1.5 For all totally b ounded sets E , P (E )=P (E ).

Pro of: If fB (x )g is an "-packing of E ,thenwemaycho ose an "-packing fB (x )g

r i i i

P P

0 0 0

of E with (x ;x ) and r r > 0 as small as welike. Thus '(2r ) '(2r ) > 0

i i i

i i

i i i

' '

e e

E ):2 may b e made as small as welike. So P (E )=P (

The following example illustrates how the lackofsub- -additivityof P extends

into the in nite dimensional realm.

Example 2.1.1 Let ' 2 anda 2 ' (f1=ng). Note that a & 0. Let X =

n n

fx ;x ;x ;:::;x g be a countable set. De ne a metric on X by

0 1 2 1

a if n 6= m = 1

(x ;x )= a + a if 1 6= n 6= m 6= 1

n m n m

0 if n = m:

Of course, X is a countable union of singletons each satisfying P (fx g)=0.How-

ever, if a <" a , then considering the packing with n + 1 balls of radius "=2

n+1 n

centered at x ;x ;:::;x we get

0 1 n

n +1 "

=1: P (X ) (n +1)' 2

2 n +1

Thus P (X ) 1.

All the quantities intro duced so far re ect the asymptotic prop erties of '(t)as

t & 0inaway similar to lemma 2.1.1. There is also the bi-Lipschitz equivalence as

in lemma 2.1.3. The pro ofs are all similar. There are also relationships b etween these

quantities as the following theorem states.

Theorem 2.1.1 Supp ose that ; '; 2 satisfy

(2t) '(2t)

lim sup

'(t) (t)

t&0 t&0

Then for E X ,

' ' '

e e

C (E ) H (E ) C (E ) C (E ) M P (E ) M P (E ):

Pro of: I rst prove C (F ) H (F ) for every F E .LetF E ,let">0be

small enough so that 0

i i

of F .Wemay assume that E \ F 6= ; for all i. Cho ose for every i an x 2 E \ F .

i i i

Then fB (x )g forms a centered 2"-cover of F .Nowby the rst inequalityin

i i

diam(E )

2.2,

X X

(2diam(E )) A '(diam(E )):

i i

' ' '

e e

C (F ) H (F ). Now H is monotone so So C (F ) AH (F )and

2" "

1 1

' '

C (E )= sup C (F ) sup H (F )=H (E ):

A A

F E F E

' ' '

H (E ) C (E ) C (E ) is immediate. For the remaining inequalities, it suces

' '

e e e

to provethat C (E ) M P (E ). If this is the case then,by the monotonicityof P ,

for F E ,

e e e

C (F ) M P (F ) M P (E ):

Thus C (E ) M P (E ) up on taking supremums. Next, if E = [ E , then

i i

X X

' '

C (E ) C (E ) M P (E ):

i i

Thus C (E ) M P (E ) up on taking in mums.

e e e

So nally I show that C (E ) M P (E ). Wemay assume that P (E ) < 1 which

implies that E is totally b ounded. By the second inequalityof2.2,wemaycho ose

">0 small enough so that 0

"=2

i=1

an "-packing of E maximal in the sense that for any x 2 E; B (x) \ B (x ) 6= ;

"=2 "=2

for some i.ThenfB (x )g is a 2"-cover of E and

" i

i=1

n n

X X

'(2") M ("):

i=1 i=1

e e e e

So C (E ) M P (E )and C (E ) M P (E ) up on letting " & 0:2

2.2 The Entropy Dimensions

In this sections I de ne the entropy dimensions in terms of the earlier notion of an

entropy index. The entropy indices seem to have b een intro duced byanumber of

di erent authors. The earliest references I am aware of are [Bou] and [PonSc]. They

were generalized and studied in function space in [KolTi]. The mo di cation to obtain

the entropy dimensions go es back at least to [Weg].

The basic idea for the entropy indices is to intro duce for a totally b ounded subset

E X and ">0 a quantity N (E )which b ehaves asymptotically as " & 0inaway

indicative of the dimension of E . Here de ne

N (E )= maxnumb er of closed disjoint -balls centered in E;

where "=2 refers to the radius of the ball. Then, if I is an n-dimensional cub e,

n n

N (I ) (1=") ([KolTi] sec.4). I will b e interested in comparing N (E ) with more

" "

general Hausdor functions. So for ' 2 write:

(E ) ' if lim sup '(")N (E )=0

"&0

(E ) ' if lim sup '(")N (E )=1:

"&0

(E ) ' if lim sup '(")N (E ) < 1:

"&0

(E ) ' if lim sup '(")N (E ) > 0:

"&0

(E ) ' if 0 < lim sup '(")N (E ) < 1:

"&0

This de nes the upper entropy index, , through comparison of N with Hausdor

functions using the lim sup. The lower entropyindex, , is similar but we use the

lim inf . Thus:

(E ) ' if lim inf '(")N (E )= 0

"&0 "

(E ) ' if lim inf '(")N (E )= 1:

"&0 "

and similarly for , ,and .

As we shall see, is closely related to P and su ers from defects related to

P 's lack of sub- -additivity.We therefore mo dify the de nition to obtain the upper

entropy dimension as follows. By a decomposition of E , I mean a nite or countable

collection of sets fE g whose union is E .Write:

i i

b b

(E ) ' (resp. (E ) ') if there is a decomp osition E = [ E so that

i i

(E ) ' (resp. (E ) ') for every i

i i

b b

(E ) ' (resp. (E ) ')ifforevery decomp osition E = [ E there is an i

i i

so that (E ) ' (resp.(E ) ').

i i

Similar de nitions maybemadetode ne in terms of .

Some comments on these de nitions are in order:

1. Since a packing by "=2-balls of a subset of E is also one of E ,; ;; and

are all monotone. That is if F E , then (E ) ' implies (F ) ' and

(E ) ' implies (F ) '.

2. Although and are de ned only for totally b ounded sets, and are de ned

for -totally b ounded sets.

3. and resp ect closure. The pro of is similar to lemma 2.1.5 Wemay therefore

assume that the E 's in the de nition of and are closed.

The concept of -stability is a prop ertyof and de ned by the following lemma.

Lemma 2.2.1 Supp ose that E = [ E . Then (E ) ' for every n implies

n n n

b b b

(E ) '. (E ) ' implies there is an n suchthat (E ) '. Similar statements

hold with , ,or replacing , ,or resp ectively.

Pro of: For the rst part, if (E ) ' for every n, then each E may b e decomp osed

n n

E = [ E where (E ) ' for every i. Then E = [ [ E is a decomp osition

n i n;i n;i n i n;i

of E implying (E ) '.

For the second part, supp ose that each E is further decomp osed E = [ E .

n n k n;k

Then E = [ [ E .Since(E ) ', there are n and k such that (E ) '. E

n k n;k n;k n;k

is part of an arbitrary decomp osition of E .Thissays that this n satis es (E ) ':

n n

The other pro ofs are similar.2

Example 2.2.1 Let ' 2 andlet(X; ) b e the corresp onding metric space as in

example 2.1.1. Then the same argument used there shows that (X ) ' and even

that (X ) '.So and are not -stable as X is countable.

Next I investigate the relationship b etweenand P .Let' 2 . A blanketing

sequence for ' is a sequence a & 0suchthat '(a ) 2'(a ): Given ' 2 such

n n n+1

a sequence may always b e chosen recursively.

Theorem 2.2.1 Let E X b e totally b ounded. Then P (E ) < 1 implies (E )

'. Conversely, supp ose that (a ) is a blanketing sequence for ' and

n=1

(a )

< 1: (2.3)

'(a )

n=1

Then (E ) ' implies P (E )=0.

Pro of: For the rst statement note that for ">0wehave N (E )'(") P (E ) since

the supremum on the right is taken over more p ossible packings than the one on the

' '

e e

(E ) ! P (E ) < 1,then(E ) '. left. So if P

For the second statementlet >0andcho ose " ;M > 0suchthat0 <" "

0 0

implies N (E )'(")

" 0 n 0

(a )

< :

'(a ) 2M

n=n

Let " 2 (0;a ]andcho ose an "-packing B of E .For n n let B = fB (x) 2B:

n 0 n r

a < 2r a g. Note that #(B ) N (E ). So by these choices and inequality

n+1 n n a

n+1

2.3,

1 1

X X X X

(2r ) = (2r ) N (E ) (a )

a n

n+1

n=n n=n

0 0

B (x)2B B (x)2B

r r n

(a ) '(a )

n n

N (E )'(a ) =

a n+1

n+1

'(a ) '(a )

n n+1

n=n

(a )

< 2M = : 2M

'(a ) 2M

n=n

Thus P (E )=0 as >0was arbitrary. Since this is true for every ">0wehave

P (E )=0:2

This theorem says that the upp er entropy index and the packing premeasure lead

to almost the same notion of dimension. When working with a family of Hausdor

functions ( ) the packing premeasure and upp er entropy index will frequently

s s>0

s (n=s)

lead to the same critical value. For example if (")= " ,thena =2 de nes a

s n

blanketing sequence for .Ifs

X X

(a )

t n

n(1(t=s))

= 2 < 1:

(a )

s n

n n

And so the packing premeasure and upp er entropy index lead to the same critical

value for the family ( ) . More precisely

s s>0

e s

sup fs>0:P (E )= 1g = sup fs>0:(E ) g:

1=s (1=")

) is a blanketing As another example, supp ose (")= 2 . Then a =(

sequence for ( ) .If < , then

1=s s

((n= ) )

X X X

(a ) 2

n(1( = ))

= 2 < 1: =

1=s s

((n= ) )

(a )

n n n

So again the upp er entropy index and packing premeasure lead to the same critical

value for the family ( ) .

More imp ortant is the close relationship b etween the upp er entropy dimension

and the packing dimension stated in the following theorem.

Theorem 2.2.2 Let E X be -totally b ounded. If P (E )is - nite and ' ,

then (E ) .Conversely,if(a ) is a blanketing sequence for ' and

n n

(a )

< 1;

'(a )

n=1

then (E ) ' implies P (E )=0.

Pro of: For the rst part, assume rst that P (E )

' '

e e

decomp osition E = [ E where P (E )

i i i i

N (E )'(") P (E ), wehave

" i i

(E )

i " i

"&0

(")

lim sup N (E ) (") = lim sup N (E )'(")

" i "

'(")

"&0 "&0

(")

M lim sup < 1:

'(")

"&0

' '

So (E ) .IfP (E )is - nite, then E = [ E where P (E ) < 1 for every n.

n n n

b b

So (E ) for every n by the ab oveand(E ) by -stability (lemma 2.2.1).

For the second part, if (E ) ', then there is a decomp osition E = [ E such

i i

that (E ) ' for every i.SoP (E )=0 by theorem 2.2.1 and

i i

P (E ) P (E )= 0:2

i=1

CHAPTER I I I

Computational Metho ds

The de nitions of the preceding chapter are useful theoretically, but cumb ersome

computationally. Inthischapter, I'll develop some computational to ols and illustrate

them with some simple examples. As shown in theorems 2.1.1, 2.2.1, and 2.2.2, many

de nitions of dimension are essentially equivalent to others. From this p oint on, I will

concentrate on the Hausdor and upp er entropy dimensions. Conclusions ab out other

dimensions may b e drawn using the appropriate comparison theorem 2.1.1, 2.2.1, or

2.2.2.

3.1 Sequence Spaces

I b egin byintro ducing a family of compact, totally disconnected metric spaces which

+ +

are relatively easy to deal with and will aid in later analysis. For k 2 N let a 2 N ,

2 if A .Thus =( ) letA = f1;:::;a g b e a discrete set, and let =

j j k k

j =1 j =1

2 A for every j . I will de ne a metric d on inducing the pro duct top ology on

j j

. First I develop some useful notation. Given n 2 N an initial segment of length

with 2 A for every j =1;:::;n. There is by n is a nite sequence =( )

j j j

j =1 21

de nition one initial segment of length zero namely the empty segment denoted .

If is an initial segment, write j j to denote the length of . For n 2 N ,let

1 j

denote the set of all initial segments of length n.Let = [ b e the set of all

j =0

initial segments. If 2 write j for the initial segment( ;:::; ) 2 .We

n 1 n

may put a partial order on as follows: For ; 2 ,say =( ;:::; )and

1 j

=( ;:::; ), write < if j

1 k i i

n -

is said to b e a descendantof .If 2 ,thenlet denote the unique elementof

n1 - - n

suchthat < . is called the parent and the child.Alsoif 2 is an

initial segment, then let

[ ]= f 2 : = for i =1;:::;ng:

i i

To de ne a metric d on asso ciate to each 2 anumber r ( ) > 0 satisfying

the following:

1. r () > 0,

2. r ( ) ,

3. r ( j ) ! 0as n !1.

If ; 2 have as their longest common initial segment then de ne d(;)= r ( ).

De ne, also, d(;) = 0. Then d is a metric inducing the pro duct top ology on . For

details see [Ed1]. Note that if ; 2 and r ( j ) "

n n1

equivalentto = for i =1;:::;n.Thus B ( )= [ j ].

i i " n

There are two sp eci c typ es of sequence spaces which will b e of particular interest

k 1

here. For the rst, let s>0 and let A = f1;:::;2 g: Cho ose r 2 (0; 1) such that k

s n n

1=r =2.If 2 , de ne r ( )=r . The resulting sequence space will b e called

in nite s-space.We will see in the following sections that an appropriate Hausdor

s (1=")

function to describ e the dimension of this space is given by (")= 2 .

The other case of interest has b een used in the analysis of self-similar sets (see

[Ed1]). Let m 2 N b e xed and let A = f1;:::;mg for every k . Let (r ;:::;r )

k 1 m

satisfy 0

list.For 2 , let r ( )= r . In [Ed1 ] it is shown that the dimension of this

i=1

s s

sequence space is given by (")= " , where r =1.Infactitisshown that

i=1 i

s s

H ([ ]) = r ( ) (3.1)

for every 2 . This space is called self-similar sequencespace and I will b e using

it when studying hyp erspaces.

3.2 Calculation of the Entropy Dimensions

In this section I will calculate the entropy dimensions for in nite s-space. To estimate

the entropy indices of a totally b ounded set E it is necessary to obtain upp er and

lower b ounds for N (E ). As N is de ned in terms of a supremum, it is relatively easy

" "

to nd a lower b ound, but frequently dicult to nd an upp er b ound. The sp ecial

nature of sequence space alleviates this problem somewhat.

Theorem 3.2.1 Fix s>0, cho ose r 2 (0; 1) so that r = 2, and let ( ;)bethe

s s (1=")

corresp onding in nite s-space. Then ( ) (r"=2), where (")=2 .

k k 1 2 1

Pro of: Supp ose 2r "< 2r . I claim then that N ( ) = 2 . This is b ecause

k k 1 k

every ball of radius "=2 2 [r ;r ) is of the form [ ] for some 2 .Furthermore

if ; 2 then [ ] \ [ ]= ;.So

k k

Y Y

k 1

k 2

N ( ) = #( )= #(A )= 2

" k

i=1 i=1

k 1 k 2 k

2 +2 ++1 2 1

= 2 =2 :

k s k k s k k

s 2 1 (2=r ") 2 1 (2=2r ) 2 1 2

N ( ) (r"=2) = 2 2 : 2 2 =2 2 =

s s

and ( ) (r"=2). Thus lim sup N ( ) (r"=2)

"!0

k k 1 s

For the lower b ound, cho ose for every k 2 N an " 2 [2r ; 2r ) so that (r" =2)

k k

s k

(r ). Then

k k s

s 2 1 (1=r ) 1

N ( ) (r" =2) 2 2 =

" k

and

s s

: lim sup N ( ) (r"=2) lim sup N ( ) (r" =2)

" " k

"!0

k !1

s s

Thus ( ) (r"=2) and ( ) (r"=2)2:

s s s

A similar pro of shows that ( ) ("=2). Note that ("=2) (r"=2).

Next, we need to pass from the entropy index to the entropy dimension. Clearly

if (E ) ',then(E ) ', since (E ) is a decomp osition of E . A similar statement

holds with replaced by .Thekey to getting lower b ounds on is in the next

lemma.

Lemma 3.2.1 Supp ose that E is a closed subset of the complete metric space X

and (E \ U ) ' for every op en set U X . Then (E ) '. A similar statement

holds with replaced by .

Pro of: Let (E ) b e a decomp osition of E by closed sets. As E is a closed subset of

i i

a complete metric space, wemay apply the Baire category theorem to obtain an E

which is somewhere dense in E . This means there is an op en set U X such that

E \ U E .So(E ) ' by the monotonicity of and (E ) '. The same pro of

i i

works for :2

Theorem 3.2.2 If ( ;) is in nite s-space, then ( ) (r"=2).

Pro of: First I'll prove that ([ ]) (r"=2) for every 2 . This implies the

result when combined with lemma 3.2.1. Fix n 2 N and supp ose that 2 . If

k k 1

k>nand 2r "< 2r , then considerations similar to those when calculating

N ( ) show that

k k

Y Y

N ([ ]) = #(A )= 2

" i

i=n+1 i=n+1

k 1 k 2 n k n

2 +2 ++2 2 2

= 2 =2 :

s k k k 1 s

(r ). Then, Nowforevery k 2 N cho ose " 2 [2r ; 2r )suchthat (r" =2)

k k

k n

s 2 2 s k

N ([ ]) (r" =2) 2 (r )

" k

k n k k n k n

2 2 2 2 2 2 1 (2 +1)

= 2 2 =2 =2 :

s s (2 +1)

lim sup N ([ ]) (r"=2) lim sup N ([ ]) (r" =2) > 2 :

" " k

"!0

k !1

s s

Thus ([ ]) (r"=2) and ( ) (r"=2):2

A similar argumentshows that ( ) ("=2).

3.3 Computation of Hausdor Measures

s (1=t)

Now I turn to the computation of Hausdor measures. In this section (t)=2 :

As Hausdor measure is de ned in terms of in mums, upp er b ounds are relatively

straight forward.

Lemma 3.3.1 If ( ;) is in nite s-space, then H ( ) 1=2.

k s

Pro of: Let "> 0andcho ose k 2 N suchthat r <"where 1=r =2. Then

k 2 1 k

f[ ]: 2 g is an "-cover of with 2 sets of diameter r .So

s k k k s k k

2 1 s k 2 1 (1=r ) 2 1 2

H ( ) 2 (r )=2 2 =2 2 = :

s s

Thus H :2 ( )=lim H ( )

"!0

Lower b ounds may b e obtained by comparing H with other measures as in the

following lemma.

Lemma 3.3.2 Let ' 2 and supp ose there exists c; > 0 and a p ositive Borel

measure on E such that (U )

. Then (E ) cH (E ).

Pro of: Supp ose 0 <" and (U ) is an "-cover of E .Then

i i

X X

0 <(E ) (U ) c '(diam(U )):

i i

' '

So (E ) cH (E ) cH (E ) after taking in mums.2

Next, I apply this lemma to in nite s-space, ( ;). First some notation: If

k k +1

2 and 2 A ,let 2 denote the concatenation of with .Thus

k +1

k 2

e e

=( ;:::; ;). Nowfor 2 de ne ([ ]) = 2 . To extend to nite

1 k

unions of such sets, note that if 2 then

k k +1 k

2 2 2

e e

([ ]) = 2 2 =2 = ([ ]):

k +1

It follows by induction that if F2 is nite and [ ] \ [ ]= ;, for every ; 2F

with 6= , then

e e

([ [ ]) = ([ ])

de nes consistently on the algebra of sets A generated by f[ ]: 2 g.Wemay

use to de ne an outer measure by using what is sometimes called methodI(see

[Ed1])

( )

(E )=inf (A):E [ A and CA :

A2C

Finally, let denote the restriction of to the -measurable subsets of . Since is

nitely additive on disjoint subsets of A,itfollows that each A 2Ais -measurable

2 k

and = on A (see [Fol] prop. (1.13)). In particular ([ ]) = 2 if 2 . Later,

I will construct such measures without going into such detail. This measure may

b e used to obtain a lower b ound on H ( ).

Lemma 3.3.3 If ( ;) is in nite s-space, then H ( ) 1=2:

k +1 k k +1

Pro of: Let U satisfy r < diam(U ) r . Since diam(U ) >r , there are

k k

; 2 U so that 6= . So, in fact, diam(U )=r .If 2 U and (;) r ,

k +1 k +1

then = for i =1;:::;k.SoU [ j ]. Thus

i i k

k k s

2 (1=r ) s

(U ) ([ j ]) = 2 =2 = (diam(U ))

and H ( ) ( ) = 1=2, by lemma 3.3.2.2

Putting together lemmas 3.3.1 and 3.3.3 weget

( ) = 1=2. Theorem 3.3.1 If ( ;) is in nite s-space, then H

In fact, these arguments may b e strengthened to show that H = on .

Lemma 3.3.2 is go o d but not quite p owerful enough for some of my purp oses. The

next lemma is similar, but more widely applicable.

Lemma 3.3.4 For a separable metric space X with a p ositive Borel measure ,

x 2 X ,and >0let

(x) = supf(U ):x 2 U and diam(U ) g:

Let & 0. Supp ose that '; 2 satisfy

'( )

max

( )

k 2N

k +1

Let E X b e a Borel set which satis es

Pro of: Let k 2 N and cho ose 0 <"< .Let

0 k

E = fx 2 E : (x)

k k 0

Note that [ E = E . Supp ose that C is an "-cover of E and so of E .For k k

k k 0

k =1 0 0

write

C = fU 2C : < diam(U ) g:

k k +1 k

Imay assume that #(C ) < 1 for every k . Otherwise

(diam(U )) = 1

U 2C

and I'm done. For U 2C suchthatU \ E 6= ; Ihave (U )

M #(C )'( ) M (diam(U ))

Now (E ) ! (E )as k !1. Thus (diam(U )) >(E )=MA and

As an example, supp ose that (t)= 2 and = cu . Then for s

~~( ) (cu ) ( 1( ) )~~

~~for large k . This last term approaches zero as k !1. So given a Borel set E X ,to~~

~~(E )=1, it suces to nd a p ositive Borel measure on E , M>0, show that H~~

~~and an s >s suchthat~~

~~2 1~~

~~(x; (cu ))~~

~~s 0~~

~~for every x 2 E . In fact, this shows that dim(E ) , since if s~~

~~1 2~~

~~s s~~

~~(E )isnon- - nite. (E )= 1 implying that H H~~

~~3.4 s-Nested Packings~~

~~Part of the imp ortance of sequence space is that it may b e used to mo del many other~~

~~spaces. In [Ed1], for example, self-similar sequence spaces are used in the study of~~

~~E of a complete separable metric space X whichallows the construction of a subset~~

~~E E which is Lip eomorphic to a certain sequence space. This result will b e used~~

~~later to transfer results from sequence space to more general spaces.~~

~~Now let E b e as ab ove, x c; s > 0, " 2 (0; 1=4), and m> (1=") +1. Let~~

~~N n n~~

~~=f1;:::;mg with the metric d given by r ( )=c" for every 2 . An~~

~~(x )g satisfying: s-nestedpacking of E will b e a collection of closed balls fB~~

~~j j~~

~~1. x 2 E for every 2~~

~~n n~~

~~(x )=; for distinct ; 2 . (x ) \ B 2. B~~

~~c" c"~~

~~)forevery 2 . 3. B (x ) B (x~~

~~This de nition also dep ends on c; ",and m,however the imp ortant parameter is s~~

~~b ecause dimensional b ounds given later will b e in terms of s.IfE has suchan s-~~

~~0 0 1~~

~~(x ). De ne a map g : ! E by nested packing, then let E = \ [ B~~

~~Lemma 3.4.1 The map g ab oveisbi-Lipschitz.~~

~~Pro of: Let ! ;! 2 satisfy d(! ;! )=c" .Theng (! );g(! ) 2 B (x ), so~~

~~1 2 1 2 1 2 c"~~

~~! j~~

~~n+1~~

~~(g (! );g(! )) 2c" .For the lower b ound, note that g (! ) 2 B (x ) for~~

~~i =1; 2 and B (x ) \ B (x )=;.So~~

~~ (x ;g(! )) + (g (! );g(! )) + (g (! );x )~~

~~+ (g (! );g(! )) + : ~~

~~So (g (! );g(! )) = c" :2~~

~~1 2~~

~~2 2~~

~~Next, I would liketoshow that this condition is non-vacuous by constructing~~

~~an s-nested packing for a self similar set E X . Self similar sets are obtained as~~

~~follows: Let m 2 N and for i =1;:::;m let f : X ! X b e a similarity with ratio~~

~~1 1 i~~

~~r 2 (0; 1). This means that for every x; y 2 X wehave (f (x);f (y )) = r (x; y ).~~

~~i i i i~~

~~In this situation there exists a unique non-empty compact set E X such that~~

~~E = [ f (E ). A set obtained this wayissaidtobe self-similar. [Ed1] has more~~

~~i=1 details.~~

~~I will b e using two sequence spaces and to analyze the set E . The rst~~

~~1 2~~

~~one is the self-similar sequence space = f1;:::;m g with metric d given by~~

~~1 1 1~~

~~m m~~

~~1 1~~

~~the ratio list (r ) corresp onding to the contraction ratios of (f ) . Given =~~

~~i i~~

~~i=1 i=1~~

~~( ;:::; ) 2 , abbreviate f f (E )by (E ). I will construct the s-nested~~

~~1 n~~

~~packing in the metric space (E; ) rather than (X; ). The reason for this is b ecause~~

~~for x 2 E , 2 ,and ">0Ihave (B (x)) = B ( (x)) as long as only balls in~~

~~r ( )"~~

~~(E; ) are considered. This is due to the invariance of E under the transformations~~

~~and not generally true in the larger metric space (X; ). (f )~~

~~i=1~~

~~Now let s>0, let r =minfr g maxfdiam(E ); 1g. Fix 2 , and let c =~~

~~i=1~~

~~1 1~~

~~(0; minf ; diam(E )g) suchthatN (E ) > (c= ) +1. Sucha certainly exists if~~

~~4 4~~

~~(E ) t . Let " = =c and let m = N (E ). The other sequence space of interest~~

~~2 2~~

~~N n n~~

~~is = f1;:::;m g with metric d given by r ( )= c" for 2 .~~

~~2 2 2~~

~~I will construct an s-nested packing of E for the choices ab ove. Cho ose x 2 E~~

~~arbitrarily. This gives B (x ). The existence of fB (x )g is guaranteed bythe~~

~~c c"~~

~~fact that N (E ) > (c= ) + 1 since = c". The construction will pro ceed by induction~~

~~on the length of . Supp ose that B (x )have b een de ned for j j n.For 2 ,~~

~~j j~~

~~such that x 2 (E ), and cho ose 2~~

~~ 1~~

~~< diam( (E )): diam( (E )) ~~

~~8 c~~

~~Since r = minfr g wehave:~~

~~n n n~~

~~r r ~~

~~diam( (E )) = diam(E ) diam(E ):~~

~~n1 n1 n~~

~~8 c 8diam(E ) c c~~

~~n n~~

~~So N ( (E )) N (E )=m .Thus (E )maybepacked with m balls of~~

~~2 2 2~~

~~2 ( =c )~~

~~n+1 n n+1~~

~~radius =c = c" to continue the induction.~~

~~For a useful generalization, note that if f : X ! X is a bi-Lipschitz map satisfying~~

~~(f (x);f(y )) r(x; y ) for every x; y 2 X ,thenf (B (x)) B (f (x)). So if f : E !~~

~~" r"~~

~~F is a bi-Lipschitz bijection and E has an s-nested packing fB (x )g ,then~~

~~j j~~

~~. Putting all this f induces an s-nested packing of F namely fB (f (x ))g~~

~~j j~~

~~cr "~~

~~together we obtain:~~

~~Theorem 3.4.1 If E X has a subset which is Lip eomorphic to a self-similar set~~

~~F satisfying (F ) t ,thenE has an s-nested packing.~~

~~Finally, I will have o ccasion to extract a packing BfB (x )g from a~~

~~j j~~

~~given s-nested packing satisfying the condition describ ed in the following lemma.~~

~~(x )g Lemma 3.4.2 Given an s-nested packing there is a packing BfB~~

~~j j~~

~~such that for every n 2 N ,~~

~~n1 n~~

~~(x ) 2Bg)= (m 1) : #(f 2 : B~~

~~Pro of: Cho ose one 2 arbitrarily and put B (x ) 2B.Continuing recursively,~~

~~supp ose that B (x )have b een chosen for j jn suchthat~~

~~j j~~

~~k k 1~~

~~#(f 2 : B k (x ) 2Bg)=(m 1)~~

~~k n+1k~~

~~for each k =1;:::;n. Agiven B (x ) 2B where 2 contains m balls~~

~~n+1 k 1~~

~~n+1~~

~~B (x ) where 2 .For xed k , B contains (m 1) such balls B k (x ).~~

~~Thus for this xed k,~~

~~n+1 n+1k k 1~~

~~n+1~~

~~#(f 2 : B (x ) B k (x ) 2Bg)= m (m 1) :~~

~~c" c"~~

~~n+1~~

~~So then the number of 2 suchthatB (x )iscontained in no B k (x ) 2B~~

~~for any k =1;:::;n is~~

~~= m(m 1) > (m 1) :~~

~~Thus the induction maycontinue.2~~

~~3.5 A Cartesian Pro duct Example~~

~~The Hausdor measure and dimension were de ned in 1918. The related theory is~~

~~nowgreatlydevelop ed and the de nition has withsto o d the test of time. In contrast,~~

~~the packing measure and dimension havedevelop ed relatively recently. If one lo oks~~

~~at the recent references [Ed1 ], [TayTr], or [Tr], one nds a slightly di erent de nition~~

~~of packing measure based on the diameter of a ball rather than the radius of a ball, as~~

~~Ihave done. Sp eci cally,for ' 2 construct a diameter basedpacking measure Q~~

~~on a separable metric space (X; ) as follows: For E X an "-packing of E (for this~~

~~section only) is a nite or countable collection of closed balls fB (x )g with centers~~

~~r i i~~

~~in E such that diam(B (x )) " for every i.Let~~

~~r i~~

~~(E )=supf '(diam(B (x ))) : fB (x )g is an "-packing of E g; Q~~

~~(E ); Q Q (E )= lim~~

~~"!0~~

~~and~~

~~' 1 '~~

~~Q (E ):E [ E g: Q (E )=inff~~

~~i i~~

~~i=1~~

~~These de nitions are very similar to the de nitions for P given in section 2.1.2,~~

~~except that twice the radius of a ball 2r is replaced by the diameter of a ball~~

~~' '~~

~~diam(B (x )). Q is a reasonable measure which is clearly equal to P on Eu-~~

~~r i~~

~~clidean space R . In [Cut], however, it is argued that the radius based de nition~~

~~more closely preserves the desirable prop erties of packing measure and dimension on~~

~~Euclidean space. For example, she mentions that the value of the pre-measure Q is~~

~~sensitive to whether closed balls or op en balls are used, while P will not b e a ected~~

~~bythischoice. The freedom to pack with either closed balls or op en balls is frequently~~

~~convenient in pro ofs. In this section, I intend to give an example involving cartesian~~

~~pro ducts supp orting the view that the radius based de nition is preferable for general~~

~~metric spaces.~~

~~One very nice feature of the packing dimension in Euclidean space is its b ehavior~~

~~with resp ect to Cartesian pro ducts. Given metric spaces (X; )and(Y; ), a metric~~

~~x y~~

~~ may b e de ned on the set X Y = f(x; y ): x 2 X; y 2 Y g by~~

~~((x ;y ); (x ;y )) = maxf (x ;x ); (y ;y )g:~~

~~1 1 2 2 x 1 2 y 1 2~~

~~Let ' (t)=t .Part of [Tr] theorem 5 states that if Dim(E ) ' and Dim(F ) ' ,~~

~~s s s~~

~~1 2~~

~~then Dim(E F ) ' when E and F are subsets of Euclidean space. There is a~~

~~s +s~~

~~1 2~~

~~arbitrary metric spaces. By the comparison theorem 2.2.2, the radius based packing~~

~~dimension b ehaves in a similar manner. The following example shows that this is not~~

~~the case for the diameter based packing measure:~~

~~Example 3.5.1 Let b e the sequence space as describ ed in section 3.1 corresp ond-~~

~~ing to the sequence of natural numb ers (a ) with diameter function r : ! (0; 1]~~

~~n=1 1~~

~~for 2 and r () = 1. De ne similarly, but with the given by r ( )=~~

~~1 i=1~~

~~1 ' '~~

~~1 1~~

~~sequence (b ) . Let ' (t)= t.ThenQ ( ) 1andQ ( ) 1. But given any~~

~~n 1 1 2~~

~~n=1~~

~~' 2 the sequences (a ) and (b ) maybechosen to make Q ( )=1.~~

~~n n n n 1 2~~

~~Pro of: To showthatQ ( ) 1, I will use the notion of a re nement of a packing.~~

~~Given a packing B of a sequence space , a re nement B of B is a packing such~~

~~1 2 1~~

~~that for every [ ] 2B, there is an [ ] 2B suchthat[ ] [ ]. Note that any~~

~~2 1~~

~~packing of is a re nement of the trivial packing [].~~

~~Next note that if 2 and A denotes the set of all children of ,then~~

~~k k +1~~

~~X Y Y~~

~~1 1~~

~~= =diam([ ]): diam([ ]) = a~~

~~It follows that if B is a packing of and B is a re nementof B , then~~

~~1 1 2 1~~

~~X X~~

~~diam([ ]) diam([ ]):~~

~~[ ]2B [ ]2B~~

~~2 1~~

~~In particular, anypacking B of satis es diam([ ]) diam([]) = 1. So~~

~~[ ]2B~~

~~' '~~

~~Q ( ) Q ( ) 1. Similar considerations apply to .~~

~~1 1 2~~

~~1 1~~

~~Now let ' 2 . I will recursively de ne integers (a ) and (b ) so that~~

~~n n~~

~~n=1 n=1~~

~~Q ( )=1.Leta = 1, and cho ose b > maxfa ; 1='(a )g. Cho ose a 2 N~~

~~1 2 1 1 1 1 2~~

~~1 1 1 1 1 1~~

~~such that < < , then cho ose b 2=(a a b '( )) suchthat < .~~

~~2 1 2 1~~

~~a a b a a a b b a a~~

~~1 2 1 1 1 2 1 2 1 2~~

~~Supp ose that (a ;:::;a ) and (b ;:::;b )have b een chosen so that for every k =~~

~~) k: Cho ose a such that < ,thencho ose and ( a b )'(~~

~~Nowif = [ K where each K is closed, then by Baire category there~~

~~1 2 n n~~

~~n=1~~

~~k k~~

~~suchthat[ ] [ ] K . Thus to show that , 2 are k; n 2 N and 2~~

~~2 1~~

~~' ' k k~~

~~Q ( )=1, it suces to showthatQ ([ ] [ ]) = 1 for 2 , 2 .~~

~~1 2~~

~~k k 0 0~~

~~So supp ose that 2 , 2 .Letn>k and let and b e descendants of~~

~~and resp ectively of length n. Then diam([ ]) = and diam([ ]) = .~~

~~From the condition in equation 3.3, it follows that if (! ; ); (! ; ) 2 [ ] [ ], then~~

~~1 1 2 2~~

~~0 0~~

~~.Conversely,anypoint(! ; ) 2 n [ ] [ ] ((! ; ); (! ; )) ~~

~~3 3 1 2 1 1 2 2~~

~~i=1~~

~~0 0~~

~~satis es ((! ; ); (! ; )) > . So [ ] [ ] is a ball (closed and op en) of~~

~~diameter . The collection of all such[ ] [ ] forms a -packing of~~

~~, a b elements. So for " = [ ] [ ] with~~

~~Q ([ ] [ ]) ( a b )'( ) !1~~

~~as n !1.Thus Q ([ ] [ ]) = 1 and Q ( )=1:2~~

~~1 2~~

~~CHAPTER IV~~

~~Dimensions of Hyp erspaces~~

~~Given a metric space, (X; ), a hyp erspace asso ciated with X is a metric space whose~~

~~elements are subsets of X .Inthischapter I will discuss the relationship b etween the~~

~~dimension of a metric space and various hyp erspaces.~~

~~4.1 Entropy Dimensions of the Space K (X )~~

~~Given a metric space (X; ), let K (X ) denote the set of non-empty, compact subsets~~

~~of X . Endow K (X ) with a metric as follows: For A; B X let~~

~~dist(A; B ) = inf f(x; y ): x 2 A; y 2 B g:~~

~~Then for A; B 2K(X )let~~

~~(A; B )= maxfsupfdist(x; B )g; supfdist(y; A)gg:~~

~~x2A y 2B~~

~~The metric so de ned is generally called the Hausdor metric. Note that a tilde~~

~~will frequently indicate that I am working in the hyp erspace. The space (K (X ); )~~

~~inherits many nice features from (X; ). For example, K (X ) is complete whenever X 38~~

~~is complete and K (X ) is compact whenever X is compact. For a pro of of these facts,~~

~~including that is a metric, see [Ed1 ] section 2. 4.~~

~~The following lemmas describ e the relationship b etween (E )and(K (E )). Note~~

~~that all that follows holds for and with very similar pro ofs.~~

~~1='~~

~~Lemma 4.1.1 For totally b ounded E X ,(E ) ' implies (K (E )) 2 :~~

~~N (E )~~

~~Pro of: For ">0, let B = fB (x )g be an "-packing of E with N (E )balls~~

~~of radius "=2. If F fx g is non-empty,then F 2K(E )andwemay consider~~

~~i=1~~

~~B (F ) K(E ) the ball of radius "=2 centered on F. Note that~~

~~"=2~~

~~B (F )=fC 2K(E ): C [ B (x )and C \ B (x ) 6= ;8x 2 F g:~~

~~From this it is easy to see that given distinct, non-empty F ;F fx g we~~

~~have B (F ) \ B (F )=;.SoN (K (E )) 2 1and~~

~~1='(") N (E ) 1='(") "~~

~~'(") '(")~~

~~N (K (E ))2 (2 1)2 =2 2~~

~~lim sup N (K (E ))2 lim sup 2 2 = 1;~~

~~"!0 "!0~~

~~1='~~

~~since lim sup N (E )'(")=1: So (K (E )) 2 :2~~

~~"!0~~

~~For the reverse inequality it will b e useful to de ne the quantity M by~~

~~M (E )= minnumb er of sets of diameter " needed to cover E:~~

~~" "~~

~~Lemma 4.1.2 For totally b ounded E and "> 0, N (E ) M (E ) N (E ).~~

~~" "=2 "=4~~

~~Pro of: The rst inequality is b ecause anytwo closed disjoint balls of radius "=2must~~

~~havecenters separated by more than "=2. Thus an "=2-cover requires at least one set~~

~~for each elementofan "-packing.~~

~~The second inequalityfollows from the fact that a maximal packing by "=8-balls~~

~~induces an "=2-cover by doubling the radius of each of the balls.2~~

~~1='(t=4)~~

~~Lemma 4.1.3 For E totally b ounded, (E ) '(t) implies (K (E )) 2 .~~

~~N (E )~~

~~e e~~

~~Pro of: Let B = fB (x )g be an "-packing of E .ThenB = fB (F )g~~

~~forms a 2"-cover of K (E )with2 1elements. So~~

~~N (E )~~

~~N (K (E )) M (K (E )) 2 1~~

~~4" 2"~~

~~for every ">0. So~~

~~1='(") N (E ) 1='(")~~

~~N (K (E ))2 (2 1)2~~

~~as " ! 0. Thus (K (E )) 2 :2~~

~~1='~~

~~It is natural to ask if (E ) ' implies (K (E )) 2 . The answer is no,~~

~~basically b ecause the relation is to o sensitive. For example, if I R is a closed~~

~~interval of length `, then the optimal "-packing of I consists of [`="] balls with centers~~

~~separated by ". So N (I )= [`="]andlim "N (I )= `. So (I ) " for any~~

~~" "!0 "~~

~~bounded interval I . But, as we see in the pro ofs of lemmas 4.1.1 and 4.1.3,~~

~~N (E ) N (E )~~

~~"=4~~

~~2 1 N (K (E )) 2 1: "~~

(

~~[`="] 1="~~

~~ (2 1)2 !1 if `>1~~

~~The next step is to pass to the entropy dimensions.~~

~~1='~~

~~b b~~

~~Theorem 4.1.1 If E X is -compact and (E ) ',then(K (E )) 2 .~~

~~Pro of: Supp ose that (E ) ' and E is compact. The extension of the theorem~~

~~to -compact sets is straightforward as any such set contains a compact subset E~~

~~satisfying (E ) ' by -stability. I will need to highlight a subset of E with a~~

~~certain regularity prop erty. Let~~

~~E = fx 2 E : (E \ B (x)) ' 8 r>0g:~~

~~' r~~

~~E nE is op en in E practically by de nition. So E is closed in E and, therefore,~~

~~' '~~

~~ -compact. Let x 2 E and let r>0. I claim that (E \ B (x)) '. (This~~

~~' ' r~~

~~is the needed regularity prop ertyof E .) (E \ B (x)) ' by de nition. For~~

~~' r~~

~~every y 2 (E nE ) \ B (x), cho ose an r suchthat (E \ B (y )) 6 '. Then~~

~~' r y r~~

~~fB (y )g is an op en cover of the -compact set (E nE ) \ B (x). So~~

~~r ' r~~

~~y 2(E nE )\B (x)~~

~~' r~~

~~there is a countable sub cover fB (y )g . Since~~

~~r k~~

~~k =1~~

~~E \ B (x)=([ B (y ) \ E ) [ (E \ B (x))~~

~~r r k ' r~~

~~k =1~~

~~b b b~~

~~and (B (y ) \ E ) 6 ' 8 k ,itmust b e that (E \ B (x)) ' by -stabilityof .~~

~~I'll nowshowthat(K (E )) 2 from which it easily follows that (K (E ))~~

~~1=' 1~~

~~f f~~

~~2 . Supp ose that K (E )= [ K .Itmay b e assumed that each K is closed. By~~

~~' n n~~

~~the Baire category theorem, one of the K 's is somewhere dense. This means that~~

~~f e f~~

~~there is a set D 2 K and an r>0suchthatB (D ) K .Letx 2 D and de ne~~

~~n r n~~

~~e f~~

~~A = fC 2 K : C =(D nB ) [ F; where F B (x) \ E is compactg:~~

~~x;r n r r=2 '~~

~~e f~~

~~A K is naturally isometric to K (E \ B (x)) which satis es (K (E \ B (x)))~~

~~2 , since (E \ B (x)) '. Therefore, (K ) 2 and (K (E ))~~

~~The next example shows that the converse inequality do es not hold.~~

~~Example 4.1.1 Recall the situation from example 2.1.1: Let a 2 ' (f1=ng) where~~

~~' 2 . Let X = fx ;x ;x ;:::;x g be a countable set. De ne a metric on X by~~

~~0 1 2 1~~

~~a if n 6= m = 1~~

~~a + a if 1 6= n 6= m 6= 1~~

~~(x ;x )=~~

~~n m~~

~~0 if n = m:~~

~~Then (X ) for every 2 as X is the countable union of singletons. But~~

~~1='~~

~~(K (X )) 2 .~~

~~f e~~

~~Pro of: Write K (X )=K [ I where~~

~~f e~~

~~K = fC 2K(X ):x 2 C g and I = fC 2K(X ):x 62 C g:~~

~~1 1~~

~~e f~~

~~I consists of the countably many isolated p oints of K (X ). So K is closed in K (X ).~~

~~f f f~~

~~Any decomp osition of K (X ) induces one of K , so supp ose that K = [ K , where~~

~~n=1~~

~~f f~~

~~the K may b e assumed closed. By the Baire Category Theorem, one of the K is~~

~~n n~~

~~f e f~~

~~somewhere dense. So there exists n 2 N , C 2 K ,and r>0such that B (C ) K .~~

~~n r n~~

~~Let~~

~~X = fx 2 X : (x; x ) r g; C = fx 2 C : (x; x ) >rg;~~

~~r 1 r 1~~

~~and let~~

~~A = fD 2K(E ):D = C [ F where F 2K(X )g:~~

~~r r~~

~~e e f e~~

~~Then A B (C ) K and A is naturally isometric to K (X ). Now(X ) ' by~~

~~r n r r~~

~~1=' 1='~~

~~example 2.2.1, so (K (X )) 2 by lemma 4.1.1. Thus (K (X )) 2 :2~~

~~In spite of example 4.1.1 wedohave the following:~~

~~Lemma 4.1.4 Supp ose that E is totally b ounded and (E ) '(t). Then (E ) ~~

~~1='(t=4)~~

~~2 .~~

~~Pro of: This immediate by lemma4.1.3.2~~

~~Thus sets which are \regular enough" satisfy the exp ected typ e of upp er b ound.~~

~~Another interesting example is K ([0; 1]). For c>0, let ' (t)= 2 . In [Go o1 ],~~

~~it is shown that dim(K ([0; 1])) ' for every c>0. A careful reading of the pro of~~

~~there shows, in fact, (K ([0; 1])) ' for every c>0. I will investigate a converse~~

~~to this statement using theorem 4.1.1 and the following lemma. Logarithms are to~~

~~the base 2.~~

~~and supp ose that ' ' for every c>0. Let Lemma 4.1.5 Let ' (t)= 2~~

~~c c~~

~~(t)= 1= log ('(t)). Then t.~~

~~' (t)~~

~~ 1. Then Pro of: Let c>0andcho ose t > 0suchthat0~~

~~c c~~

~~'(t)~~

~~= t log '(t) t log ' (t)=log(2 ) = c:~~

~~(t)~~

~~So ! 0as t ! 0, since c>0 is arbitrary.2~~

~~(t)~~

~~Corollary 4.1.1 (K ([0; 1])) ', where ' is as ab ove.~~

~~b b~~

~~Pro of: Since (t) t,Ihave ([0; 1]) and so (K ([0; 1])) 2 = ',by~~

~~theorem 4.1.1.2.~~

~~4.2 Hausdor Dimension of K (X )~~

~~(1=")~~

~~In this section I investigate conditions on E to ensure that dim(K (E )) 2 .~~

~~Early results in this direction are in [Boa, Go o1 , Go o2 ] which deal with K ([0; 1]).~~

~~Calculations of Hausdor dimensions are generally much more dicult than those~~

~~for entropy dimensions. Assumptions made on E will b e much more stringent. I~~

~~b egin with self-similar sequence space and will then transfer the results to other~~

~~sets mo deled by . In this section, = (1;:::;m) is a xed self-similar sequence~~

~~space with contraction ratio list (r ;:::;r )sothat r =1.~~

~~1 m~~

~~M (1=")~~

~~. Then there exists an M large Theorem 4.2.1 For M>0 let ' (")= 2~~

~~enough so that H (K ( )) < 1.~~

~~Pro of: Cho ose 0~~

~~k -~~

~~L = f 2 : r ( ) u~~

~~Each 2 L satis es~~

~~k +1 k~~

~~and~~

~~(k +1) s k~~

~~n < H ([ ]) n (4.1)~~

~~by equation 3.1. Supp ose that #(L )=L. Then since each 2 L satis es n <~~

~~1 1~~

~~s 1~~

~~H ([ ]) n , for each k 2 N the numb er of descendants of in L cannot exceed~~

~~n .So~~

~~k 2 k~~

~~Ln #(L ) Ln :~~

~~Let A L b e nonempty. Asso ciate with A a set A K( ) de ned by:~~

~~A = fC 2K( ) : f 2 L :[ ] \ C 6= ;g = Ag:~~

~~k k~~

~~e e~~

~~Such a set A is called a k-set and satis es diam(A) u .Since#(L ) Ln , there~~

~~are no more than 2 1such k -sets. This leads to the following estimate:~~

~~ ~~

~~Ln k~~

~~H (K ( )) 2 1 ' (u )~~

~~k M~~

~~k k s k k~~

~~Ln M (1=u ) Ln Mn~~

~~ 2 2 =2 2 1~~

~~as long as M L.Thus for M L,Ihave H (K ( )) 1:2~~

~~s (1=")~~

~~Now for the lower b ound, let (")=2 for s>0.~~

~~Theorem 4.2.2 H (K ( )) > 0, whenever s~~

~~Pro of: L ;L;u;n, and k -sets are as in the preceding pro of. Given A L , de ne~~

~~k k~~

~~k 2 0 s 0~~

~~ (A)= #(A)=Ln . Let n =1=u~~

~~ ~~

~~1 n~~

~~< . I will construct a measure concen- cho ose j 2 N large enough so that~~

~~pn n~~

~~kj k kj +1~~

~~trated on those kj -sets A with (A) p =n so that satis es D (x; (u )) 1 ~~

~~for every x 2K( ) implying the result by lemma 3.3.4. Recall that the de nition of~~

~~kj +1~~

~~D (x; (u )) is given in lemma 3.3.4.~~

~~The measure will b e constructed recursively. The emptyword is the only~~

~~string of length 0 leading to the one 0-set =K ( ). De ne (K ( )) = 1. Fix~~

~~e e~~

~~k 2 N and supp ose that has b een de ned for all kj -sets A suchthat(A) > 0~~

~~kj k~~

~~e e~~

~~only if (A) p =n .IfA is suchakj -set, then distribute (A) among all those~~

~~(k +1)j~~

~~e e e~~

~~(k +1)j -sets B A suchthat (B ) .Sucha set B will b e called an eligible~~

~~k +1~~

~~e e~~

~~descendentof A. I need a lower b ound on the numb er of eligible descendants of A.I~~

~~kj kj~~

~~p p~~

~~kj 2~~

~~have#(A) Ln , since (A) .If 2 A L , then~~

~~by equation 4.1 while if 2 L , then similarly~~

~~(k +1)j~~

~~(k +1)j 1 s (k +1)j~~

~~n < H ([ ]) n :~~

~~Thus if L is the set of descendants of in L , then~~

~~To form an eligible descendent B A pro ceed as follows: Take p Ln of~~

~~the 's 2 A and cho ose all p ossible descendants to form part of the set B .This~~

~~kj 2 j 1 (k +1)j 3~~

~~Ln n = Ln ; #(B ) ~~

~~k k~~

~~n n~~

~~(k +1)j~~

~~as required. Iamnow free to cho ose descendants of the so that (B ) ~~

~~Ln Ln (1 p )( Ln ) 1~~

~~k k k~~

~~n n n~~

~~j 1~~

~~'s 2 A as I like. Since each 2 A has at least n descendants 2 L and I~~

~~(k +1)j~~

~~maycho ose any p ossible non-empty subset of these as p ossible descendants, I get at~~

~~eligible descendants B A. Thismeansthatanysuch B satis es~~

~~Applying this recursively I get that a kj -set satis es~~

~~j 1 j 2 j j 1 j j 1 k 1~~

~~n (1p )Ln (1+p n ++(p n ) )k~~

~~where L is a large enough constant.~~

~~kj +1 0 s~~

~~The diameter of a kj -set is u .Letn =1=u~~

~~Since p n > (n ) by assumption.2~~

~~The next order of business is to extend these theorems to more general sets E .~~

~~For the upp er b ound, let us supp ose that E F where F is the self-similar set given~~

~~by the maps (f ;:::;f ) with ratio list (r ;:::;r ). Let ( ;) b e the corresp onding~~

~~1 m 1 m~~

~~surjective Lipschitz map h : ! F . Since a Lipschitz map is continuous and the~~

~~continuous image of a compact set is compact, h extends naturally to a Lipschitz~~

~~map h : K ( ) !K(F ). Thus the upp er b ound for K ( ) should hold for K (E ).~~

~~By comp osing the map h with another if necessary,itisalsoclearthat F need not~~

~~b e strictly self-similar, but only the Lipschitz image of a self similar set. This is~~

~~summarized as the following theorem.~~

~~Theorem 4.2.3 Let E F , where F is the Lipschitz image of a self-similar set with~~

~~= 1. Then there is an M>0 large enough r ratio list (r ;:::;r )such that~~

~~1 m~~

~~i=1~~

~~(K (E )) < 1. so that H~~

~~For the lower b ound, supp ose that E has an s -nested packing. Then wemay~~

~~extract a subset E E which is bi-Lipschitz equivalent to a self-similar sequence~~

~~space ( ;) of nite Hausdor dimension s by lemma 3.4.1. Again, the bi-Lipschitz~~

~~0 0~~

~~map g : ! E extends to a bi-Lipschitz map g : K ( ) !K(E ). Thus I havethe~~

~~following theorem.~~

~~Theorem 4.2.4 Supp ose that E has an s -nested packing. Then for s< s ,Ihave~~

~~0 0~~

~~H (K (E )) > 0.~~

~~Next I turn to more concrete examples. Supp ose that X =(x ;x ;:::;x )isa~~

~~0 1 1~~

~~countable metric space with metric satisfying (x ;x )=a & 0and(x ;x ) ~~

~~n 1 n n m~~

~~a for m< n. Clearly H (X ) = 0 for every ' 2 . But the following is also true: n~~

~~Theorem 4.2.5 Let (X; )beasabove and supp ose ' 2 satis es '(a )=2 .~~

~~Then dim(K (X )) '.~~

~~Pro of: A set T 2K(X ) is isolated if and only if x 62 T .Let~~

~~K (X )=fT 2K(X ):x 2 T g:~~

~~0 ' 0~~

~~Then K (X ) nK (X )iscountable so that H (K (X ) nK (X )) = 0.~~

~~Turn nowto K (X ). For xed n 2 N ,each set A fx ;:::;x g determines a~~

~~0 n1~~

~~set~~

~~A = fT 2K(X ):A = T \fx ;:::;x gg:~~

~~n 0 n1~~

~~Note that if S; T 2 A ,thenanypoint x 2 S with k n satis es (x ;x ) a .So~~

~~n k k 1 n~~

~~dist(x ;T) a , since x 2 T .SinceS and T agree on A,Ihavethatdist(x ;T) ~~

~~k n 1 k~~

~~a for every x 2 S and vice versa. So diam(A ) a . In fact, A [fx g and~~

~~n k n n 1~~

~~e e e~~

~~A [fx ;x g2 A ,sothatdiam(A )=a .Now there are 2 such A's contained in~~

~~H (K (X )) 2 '(a )=2 2 =1:~~

~~' 0~~

~~So H (K (X )) 1.~~

~~For the lower b ound, I will construct a measure on K (X ) recursively. Let~~

~~(K (X )) = 1. Fix m 2 N and supp ose that has b een constructed so that A ~~

~~fx ;:::;x g implies (A )= 2 for every n m. Note that if A fx ;:::;x g,~~

~~0 n1 n 0 m1~~

~~then~~

~~e e e~~

~~A = fT 2 A : x 2 T g[fS 2 A : x 62 S g:~~

~~m m m m m~~

~~Divide (A )evenly b etween these two sets. In this way is constructed so that~~

~~(A )=2 for any n 2 N and A fx ;:::;x g.~~

~~n 0 n1~~

~~e e e~~

~~Now supp ose that B K(X ) satis es a < diam(B ) a .LetT 2 B and let~~

~~n+1 n~~

~~e e~~

~~A = T \fx ;:::;x g. Then B A ,so~~

~~0 n1 n~~

~~n (n+1)~~

~~e e~~

~~(B ) 2 =2 2 =2'(a ) < 2'(diam(B )):~~

~~n+1~~

~~' 0~~

~~Thus H (K (X )) 1=2, by 3.3.2.2~~

~~4.3 Entropy and Hausdor Dimensions of the Space C (R )~~

~~In this section I investigate the entropy and Hausdor dimensions of the set of compact~~

~~d d~~

~~convex subsets of R , denoted C (R ), endowed with the Hausdor metric .Thisis~~

~~a closed subspace of K (R ) and so is a complete separable metric space. A good~~

~~reference for general facts ab out C (R )is[Sch].~~

~~The case d = 1 is easy.Any closed convex subset of R is a closed interval which~~

~~is uniquely determined by its endp oints. C (R) is therefore two dimensional. Results~~

~~for d 2 rest on the indep endentwork in [Bro]and[Dud]which essentially compute~~

~~the entropy indices. In particular, theorem 5 in [Bro] states the following:~~

~~Theorem 4.3.1 Let T = B (0) b e the op en ball of radius 1 ab out the origin in R .~~

~~d 1~~

~~Then for 0 <" 1=(10 (d 1)),~~

~~,then In my notation, this implies that if ' (t)= 2~~

~~' (C (T )) (C (T )) ' :~~

~~a d d b~~

~~d d~~

~~My rst contribution is to extend this to the entropy dimensions.~~

~~a(1=t)~~

~~and supp ose that ' ' for every Theorem 4.3.2 Let ' (t)= 2~~

~~a a~~

~~a> 0. Then for every r>0,~~

~~' (C (B (0))) (C (R )) :~~

~~Pro of: Let T denote the op en unit ball in R .Iff : T ! B (0) is a similaritywith~~

~~d d r~~

~~e e~~

~~ratio r>0, then f : C (T ) !C(B (0)) given by f (E )=f (E ) is also a similaritywith~~

~~d r~~

~~ratio r . It then follows from theorem 4.3.1 that~~

~~d1 d1 d1 d1~~

~~2 2 2 2~~

~~r r a (1=") b (1=")~~

~~d d~~

~~2 N (C (B (0))) 2 :~~

~~" r~~

~~' (C (B (0))) (C (B (0))) ' ; (4.2)~~

~~For the lower b ound I will apply a Baire category argument to the set C (B (0))~~

~~which is not complete. C (B (0)) is easily seen to b e op en in C (R ) and is there-~~

~~fore top ologically complete by theorem 12. 1 in [Oxt]. This means that it maybe~~

~~remetrized by means of a complete metric and so the Baire category theorem still~~

~~holds.~~

~~e e~~

~~Fix r>0 and supp ose C (B (0)) = [ C , where each C is closed. By Baire~~

~~r n n n~~

~~e e~~

~~category, there are n 2 N , "> 0, and E 2C(B (0)) suchthatB (E ) C . I claim~~

~~r " n~~

~~that B (E ) fE g + C (B (0)) where + stands for set addition (E + A = fx + y : x 2~~

~~" "~~

~~E; y 2 Ag). This is b ecause set addition is an isometry in C (R )(see[Sch] page 59).~~

~~So if A B (0), then (E; E + A) <".We also see that fE g + C (B (0)) is isometric~~

~~" "~~

~~to C (B (0)) which satis es ' ' (C (B (0))) for some a>0by equation 4.2.~~

~~" a "~~

~~Thus ' (C )and ' (C (B (0))).~~

~~n r~~

~~For the upp er b ound, write C (R )=[ C (B (0)). Then by equation 4.2, for~~

~~n2N~~

~~every n 2 N , there is a b > 0 so that (C (B (0))) ' .So(C (R )) :2~~

~~n n b~~

~~Inow turn to the computation of the Hausdor dimension. The upp er b ound given~~

~~ab ove for the entropy dimensions holds for the Hausdor dimension as well bythe~~

~~comparison theorems 2.1.1 and 2.2.2. The lower b ound for the Hausdor dimension~~

~~is somewhat weaker and the pro of is considerably more dicult. A go o d reference for~~

~~the Riesz representation theorem and Alaoglu's theorem as used in the next pro of is~~

~~[Con].~~

~~s (1=t) d~~

~~Theorem 4.3.3 Let (t)= 2 . Then dim(C (R )) and supp ose that s<~~

~~Pro of: Let T = B (0) R b e the closed unit ball. I will need to obtain a measure~~

~~d1 d~~

~~ on C (T )astheweak- limit of some discrete measures. Let S = @T b e the unit~~

~~sphere. Takea d 1 dimensional cub e of side length < 1 and pro ject it orthogonally~~

~~d1 d1~~

~~onto S . Denote this region of S so obtained by C .ThenC is the bi-Lipschitz~~

~~image of a self-similar d 1 dimensional set and so has a 2s-nested packing since~~

~~j j~~

~~2 c"~~

~~such that for all n 2 N , Then I may extract a packing BfB (x )g~~

~~j j~~

~~n n1~~

~~#(f 2 : B (x ) 2Bg)= (m 1)~~

~~by lemma 3.4.2. For n 2 N ,let~~

~~B = fx 2 C : B (x ) 2B and j j ng:~~

~~j j~~

~~Let x 2 S be a point opp osite C (i.e. x 2C ). Each subset F B determines~~

~~0 0 n~~

~~e e~~

~~aconvex subset F T given by F = conv(F [fx g), where conv denotes the closed~~

~~convex hull. Let G denote the set of all suchconvex sets generated by B .Then~~

~~n n~~

~~(m 1) 1~~

~~2 n1~~

~~#(B )=1+(m 1) + (m 1) + +(m 1) =~~

~~m 2~~

~~(m1) 1~~

~~and #(G )= 2 . The elements of G are all distinct. In fact, there is a~~

~~n n~~

~~convenientlower b ound on their separation. If x 2 E n F , where E; F B and~~

~~j j = k , then anypointof F is separated from x byatleast c" . The argument~~

~~used in [Bro] in the pro of of 4.3.1 then shows that there is an a> 0 indep endentof~~

~~e e~~

~~k suchthat (E; F ) a" .Let b e normalized counting measure on G . Each ~~

~~n n n~~

~~is a continuous linear functional on C (C (T )) of norm 1 by the Riesz representation~~

~~theorem. The unit ball in C (C (T )) is weak- compact by Alaoglu's theorem, so~~

~~has a weak- cluster p oint, say . ( )~~

~~n=1~~

~~The measure ab ove will b e used in a density argument, so I need to b e able~~

~~to compare the -measure of a set with its diameter. So x n 2 N and supp ose~~

~~A C(T ) is op en and satis es~~

~~2(n+1) 2n~~

~~a" diam(A)~~

~~e e~~

~~To estimate (A), it suces to estimate (A) for large j as is a weak- cluster p oint~~

~~e e~~

~~of ( ) . This in turn dep ends on #(A \ G ). The restriction on diam(A) guarantees~~

~~j j j~~

~~that A cannot contain sets E; F 2 G which disagree at some p oint x 2B . On~~

~~j n~~

~~the other hand, A is p otentially large enough to accommo date all p ossible choices for~~

~~x 2B nB .Note~~

~~j n~~

~~j 1 j 2 n~~

~~#(B nB ) = (m 1) +(m 1) + +(m 1)~~

~~(m1) (m1) 1(m1)~~

~~m2 m2~~

~~so #(A \ G ) 2 . An element E 2 G has (fE g)=2 .So~~

~~j j j~~

~~j n j n~~

~~(m1) (m1) 1(m1) 1(m1)~~

~~m2 m2 m2~~

~~ (A) 2 2 =2 ;~~

~~1(m1)~~

~~. for every j>n.So(A) 2~~

~~2(n+1) s 0 2s n~~

~~s (1=a" ) a (1=" ) 0~~

~~Now (diam(A)) 2 =2 , where a > 0 is a constant. So~~

~~as n !1, since 1="~~

~~the result by lemma 3.3.4.2~~

~~CHAPTER V~~

~~Function Spaces~~

~~In [KolTi]theentropy indices are investigated for various compact sets of functions.~~

~~Of central imp ortance there are sets of functions of some prescrib ed degree of smo oth-~~

~~d s~~

~~ness. Roughly sp eaking, if E R has dim(E ) t and F is the set of functions~~

~~s=q~~

~~(1=t)~~

~~de ned on E and smo oth of order q>0, then dim(F ) 2 .Thus as E in-~~

~~creases in dimension, so do es F . While an increase in the order of smo othness of the~~

~~functions in F , decreases the dimension of F .Inthischapter I extend these theorems~~

~~to some other notions of dimension.~~

~~5.1 Entropy Dimensions of the Space of Smo oth Functions~~

~~In order to make the ab ove ideas precise, I will rst need to carefully de ne the~~

~~set of functions under consideration. Let c>0, d 2 N , E R b e compact, and~~

~~let q = p + where p 2 N and 2 (0; 1]. Denote the set of real valued b ounded~~

~~d d~~

~~continuous functions de ned on R by C (R ). Similarly, C (E ) denotes the set of real~~

~~valued continuous (and necessarily b ounded) functions on E . If f 2 C (R ), then~~

~~f j will denote the restriction of f to E . The uniform norm is denoted by kk~~

~~E 55~~

~~and will b e applied to functions in b oth C (R )andC (E ). Iwant to de ne a set~~

~~F (c; q ; E ) C (E )which are to b e thought of as smo oth of order q and b ound by c.~~

~~d d~~

~~First I need to de ne a similar set F (c; q ; R ) C (R ).~~

~~To de ne F (c; q ; R )a convenient notation for the partial derivatives of a function~~

~~d d~~

~~f 2 C (R ) will b e needed. Amulti-index is a vector in N . This is not to b e~~

~~confused with the notion of an initial segment. If =( ;:::; )isa multi-index,~~

~~1 d~~

~~then write j j = + + .Iff 2 C (R )isj j-times di erentiable, then D f (x)~~

~~1 d~~

~~j j~~

~~@ f (x)~~

~~denotes the partial derivative where x =(x ;:::;x ).~~

~~1 n~~

~~@ x @ x~~

~~d d~~

~~F (c; q ; R ) denotes the set of all functions f 2 C (R )suchthat:~~

~~1. D f exists and satis es kD f kc for all multi-indices with j j p.~~

~~ d~~

~~2. jD f (x) D f (y )j cjx y j for all x; y 2 R and multi-indices with j j = p.~~

~~Note that condition 1 ab ove includes the case j j = 0 so that f itself satis es kf kc.~~

~~The set F (c; q ; E ) is de ned by restricting functions from F (c; q ; R )to E . So write~~

~~g 2 F (c; q ; E )if g = f j , where f 2 F (c; q ; R ). The following theorem is a slight~~

~~mo di cation of theorem 15 in [KolTi].~~

~~Theorem 5.1.1 Fix c; q ; s ;s > 0 and supp ose that E R is a compact set~~

~~t (E ) (E ) t :~~

~~Then there are a; b > 0 suchthat~~

~~s =q s =q~~

~~1 2~~

~~a(1=t) b(1=t)~~

~~2 (F (c; q ; E )) (F (c; q ; E )) 2 :~~

~~I should p oint out that the de nition of F (c; q ; E ) in [KolTi] is somewhat more ab-~~

~~stract. However, a close reading of the pro of there yields this theorem for the de nition~~

~~of F (c; q ; E )given here as well.~~

~~The goal in this section is to extend theorem 5.1.1 to the entropy dimensions. As~~

~~ in general, the upp er b ound is not a diculty.To extend the lower b ound~~

~~to will require a Baire category argument. This in turn requires that F (c; q ; E )~~

~~is uniformly closed in C (E ). I have not b een able to nd this in the literature and~~

~~so proveitnow. This will require several lemmas. The rst goal will b e to show~~

~~d d~~

~~that F (c; q ; R ) is closed in C (R ) with resp ect to uniform convergence on compact~~

~~subsets of R . Thismodeofconvergence will henceforth b e abbreviated convergence~~

~~UCS. The notion of equicontinuity will play a crucial role. A family F of real valued~~

~~functions de ned on a metric space (X; )issaidtobeequicontinuous if for every~~

~~">0 there is a >0suchthatjf (x) f (y )j <" whenever f 2 F and (x; y ) <.~~

~~Note that a subset of an equicontinuous family of functions is again equicontinuous.~~

~~Lemma 5.1.1 Let c>0, q 2 (0; 1], and f 2 F (c; q ; R )forevery n 2 N such that~~

~~f ! f as n !1 pointwise. Then f 2 F (c; q ; R ).~~

~~Pro of: Clearly kf kc.To establish the Holder condition, let x; y 2 R , ">0, and~~

~~cho ose n 2 N such that n>n implies that jf (x) f (x)j <"=2andjf (y ) f (y )j <~~

~~0 0 n n~~

~~"=2. Then for n>n ,~~

~~jf (x) f (y )j jf (x) f (x)j + jf (x) f (y )j + jf (y ) f (y )j~~

~~n n n n~~

~~" "~~

~~q q~~

~~+ cjx y j + = cjx y j + ": ~~

~~2 2~~

~~Thus jf (x) f (y )jcjx y j ,as ">0 is arbitrary.2~~

~~Lemma 5.1.2 Let c>0 and let q = p + where p 2 N and 2 (0; 1]. If is a~~

~~multi-index with j jp, then the family fD f : f 2 F (c; q ; R )g is equicontinuous.~~

~~Pro of: If j j = p,thenImay apply the Holder condition. So let ">0 and cho ose~~

~~1=~~

~~0 < <("=c) .Thenjx y j < implies~~

~~ ~~

~~jD f (x) D f (y )jcjx y j~~

~~If j j~~

~~j j =1,wehavethat D f satis es the Lipschitz condition~~

~~jD f (x) D f (y )j kgrad (D f )kjx y j dcjx y j:~~

~~Thus the same argument applies.2~~

~~The following lemma is commonly known as the Arzela-Ascoli Theorem and ap-~~

~~pears in [Fol] as theorem 4.44 (b).~~

~~Lemma 5.1.3 If (f ) is an equicontinuous p ointwise b ounded sequence of real val-~~

~~n n~~

~~d d~~

~~ued functions de ned of R , then there is a subsequence (f ) and f 2 C (R )such~~

~~n j~~

~~! f UCS. that f~~

~~The nal lemma app ears as theorem 4.56 in [Str].~~

~~Lemma 5.1.4 Let I R b e a compact interval and let (f ) b e a sequence of real~~

~~n n~~

~~valued di erentiable functions on I . Supp ose that f ! g uniformly on I and that n~~

~~for some c 2 I , f (c)converges. Then there is a di erentiable function f de ned on~~

~~I such that f ! f uniformly on I and f = g .~~

~~Corollary 5.1.1 Supp ose that f ! f pointwise on R and f ! g UCS. Then f is~~

~~di erentiable and f = g .~~

~~d d~~

~~Theorem 5.1.2 F (c; q ; R ) is closed in C (R ) with resp ect to the top ology of uniform~~

~~convergence on compact subsets of R .~~

~~Pro of: The case 0 1. For~~

~~every n 2 N , let f 2 F (c; q ; R ) and supp ose that f ! f UCS. I want to show that~~

~~n n~~

~~f 2 F (c; q ; R ).~~

~~I will recursively de ne a sequence n %1 such that D f converges UCS as~~

~~j n~~

~~j !1 for every multi-index with j jp.LetM be the number of multi-indices~~

~~of length p and let ( ) bealistofthosemulti-indices with j j =0. Let~~

~~i 1~~

~~i=1~~

~~j (1) %1b e just the sequence of natural numb ers. Then D f = f ! f~~

~~n n~~

~~j (1) j (1)~~

~~UCS as j (1) !1. Supp ose that for k 1~~

~~b een de ned so that D f converges UCS to say g as j (k 1) !1 for every~~

~~j (k 1)~~

~~i =1;:::;k 1. By lemma 5.1.2 the set fD f g is equicontinuous. It~~

~~j (k 1)~~

~~is also uniformly b ounded by c and so by lemma 5.1.3 there is a subsequence j (k )~~

~~of j (k 1) so that (D f ) converges UCS as j (k ) !1to say g . This~~

~~j (k )~~

~~pro cess terminates when j (M ) is reached to generate the nal sequence (n ) such~~

~~p j j~~

~~! g UCS as j !1 for all multi-indices with j jp. that D f~~

~~n j~~

~~Next supp ose that is a multi-index with j j =1. Then viewing (f ) as a~~

~~n j~~

~~sequence of functions of the single variable with resp ect to which D di erentiates,~~

~~we see that corollary 5.1.1 applies to the sequences f ! f UCS and D f ! g~~

~~n n~~

~~j j~~

~~UCS. Thus f is di erentiable and D f = g . Clearly kD f kc since kD f kc~~

~~for every j . This same argumentmay b e recursively applied to obtain D f = g and~~

~~kD f kc for all multi-indices with j jp.~~

~~Finally, for j j = p, D f satis es the necessary Holder condition bylemma 5.1.1.2~~

~~Corollary 5.1.2 F (c; q ; E ) is compact (and therefore closed) with resp ect to the~~

~~uniform norm.~~

~~Pro of: Let g 2 F (c; q ; E ) for every n 2 N .Sog = f j where f 2 F (c; q ; R ).~~

~~n n n E n~~

~~The sequence (f ) is uniformly b ounded by c. By lemma 5.1.2, (f ) forms an~~

~~n n n n~~

~~equicontinuous family.Thus by lemma 5.1.3, there is a function f 2 C (R ) and a~~

~~subsequence (f ) of (f ) suchthat f ! f UCS as j !1. By theorem 5.1.2,~~

~~n j n n n~~

~~j j~~

~~f 2 F (c; q ; R ). Thus the function g = f j satis es g 2 F (c; q ; E )and g ! g~~

~~E n~~

~~uniformly on E:2~~

~~One more lemma is necessary for the main theorem of the section.~~

~~d d~~

~~Lemma 5.1.5 Let c ;c ;q; > 0, f 2 F (c ;q;R ), and g 2 F (c ;q;R ). Then~~

~~1 2 1 2~~

~~d d~~

~~f + g 2 F (c + c ;q;R ) and f 2 F ( c ;q;R ).~~

~~1 2 1~~

~~Pro of: If is a multi-index with j j p,thenD (f + g )=D f + D g and~~

~~kD (f + g )kkD f k + kD g k c + c :~~

~~1 2~~

~~If j j = p and x; y 2 R , then~~

~~jD (f + g )(x) D (f + g )(y )j jD f (x) D f (y )j + jD g (x) D g (y )j~~

~~ ~~

~~ c jx y j + c jx y j =(c + c )jx y j :~~

~~1 2 1 2~~

~~Similarly, kD ( f )k c and~~

~~kD ( f (x)) D ( f (y ))j c jx y j :2~~

~~Corollary 5.1.3 Let c ;c ;q; > 0, E R b e compact, f 2 F (c ;q;E), and~~

~~1 2 1~~

~~g 2 F (c ;q;E). Then f + g 2 F (c + c ;q;E)and f 2 F ( c ;q;E).~~

~~2 1 2 1~~

~~Finally, the following theorem givesalower b ound found for (F (c; q ; E )).~~

~~s=q~~

~~a(1=t)~~

~~for every Theorem 5.1.3 Let s; q ; c > 0 b e xed and supp ose that ' 2~~

~~d s~~

~~a> 0. If the compact set E R satis es (E ) t then (F (c; q ; E )) '.~~

~~Pro of: Supp ose that F (c; q ; E )= [ F where each F is closed. By Baire category~~

~~n n n~~

~~there is an n 2 N an f 2 F , and an r>0suchthatB (f ) F .Imay assume, in~~

~~n r n~~

~~0 0~~

~~fact, that f 2 F (c ;q;E) where 0~~

~~<1 is close enough to 1 so that f 2 B (f ). Then f 2 F ( c; q; E) and there is~~

~~an r > 0suchthat B ( f ) B (f ).~~

~~r r~~

~~Consider now g 2 F (min(r;c c );q;E). By corollary 5.1.3, g + f 2 F (c; q ; E ).~~

~~Thus~~

~~g 2 B (f ) ff g = fh f : h 2 B (f )g~~

~~r r~~

~~since wemay take h = g + f . B (f ) ff g is clearly isometric to B (f ) and we see~~

~~r r~~

~~that~~

~~B (f ) ff gF (min(r;c c );q;E):~~

~~s=q~~

~~a(1=t)~~

~~Thus by theorem 5.1.1, there is an a> 0 suchthat (F ) (B (f )) 2 '~~

~~n r~~

~~and (F (c; q ; E )) ':2~~

~~It is interesting to note that the b ound on (F (c; q ; E )) is expressed in terms of~~

~~b b~~

~~ (E ) rather than (E ). thus a set small with resp ect to (but large with resp ect~~

~~to ) can lead to a set of functions large with resp ect to . For example, the set~~

~~1 1~~

~~;:::; ;:::gR satis es (E ) (E ) ' for every ' 2 sinceE E = f0; 1;~~

~~2 n~~

~~1=2~~

~~is countable. E also satis es (E ) (E ) t ([Fal2] example 3.5). Thus for~~

~~s=q~~

~~(1=t)~~

~~0 0wehave (F (c; q ; E )) 2 .~~

~~5.2 Hausdor Dimension of the Space of Smo oth Functions~~

~~In this section, I investigate the Hausdor dimension of F (c; q ; E ). The upp er b ound~~

~~given in theorem 5.1.1 holds for the Hausdor dimension as well, so I concentrate here~~

~~on the lower b ound. As with the space K (E ), I need to make stronger assumptions~~

~~on the set E .~~

~~Theorem 5.2.1 Fix s; q ; c > 0 and supp ose the compact set E has an s-nested~~

~~s=q~~

~~(1=t)~~

~~packing. Then dim(F (c; q ; E )) ' (t)=2 .~~

~~a (1 + x ) (1 x ) if all jx j < 1~~

~~where x =(x ;:::;x ) 2 R and a>0 is a constant. Note that if a is chosen~~

~~0 d~~

~~small enough, then g 2 F (c; q ; E ). Note also that for b 2 (0; 1) wehave b g (x=b) 2~~

~~F (c; q ; E ).~~

~~. Extract from this a packing Now E has an s-nested packing say fB (x )g~~

~~#(f 2 : B (x ) 2Bg)=(m 1) ;~~

~~c "~~

~~as guaranteed by lemma 3.4.2. Recall that m; c ;",and s are all xed and satisfy~~

~~s n~~

~~m>(1=") + 1. De ne B = fB (x ) 2B: 2 g: I'll now de ne some discrete~~

~~n c "~~

~~sets and argue in a wayvery similar to theorem 4.3.3. Let G b e the set of all those~~

~~functions f 2 F (c; q ; E ) of the form:~~

~~d(xx )~~

~~c "~~

~~( ) g ( ) for x 2 B (x ) 2B~~

~~) g ( ( (x ) 2B ) for x 2 B~~

~~n n c "~~

~~c "~~

~~> 0~~

~~0 for all other x:~~

~~k 1 n~~

~~Now#(B )=(m 1) and for each B 2[ B there are twochoices to form an~~

~~k k~~

~~k =1~~

~~f 2 G induced bythe ab ove. So~~

~~m1 (m1)~~

~~#(G ) = 2 2 2~~

~~(m1) 1~~

~~1+(m1)++(m1)~~

~~= 2 : =2~~

~~As in the pro of of theorem 4.3.3, let b e normalized counting measure on G and~~

~~n n~~

~~let be a weak- cluster p ointof f g. Let A F (c; q ; E ) b e op en and satisfy~~

~~c c~~

~~0 0~~

~~q (n+1)q q nq~~

~~p p~~

~~) " diam(A) < 2a( ) " : 2a(~~

~~d d~~

~~Iwant an upp er b ound on (A). For this it suces to havea boundon (A) for~~

~~large j , which I get by estimating #(A \ G ). Note that if two functions f ;f 2 G~~

~~j 1 2 j~~

~~q nq~~

~~) " .SoA is big disagree on some B 2B , then they will disagree by at least 2a(~~

~~enough to allow functions to do what they likeonany B 2B if k = n +1;:::;j,but~~

~~functions in A \ G are restricted to just one choice on any B 2B for k =1;:::;n.~~

~~n k~~

~~So for j>n,~~

~~j 1 j 2 n~~

~~(m1) (m1) (m1)~~

~~#(A \ G ) 2 2 2~~

~~n j n1 j n2~~

~~(m1) ((m1) +(m1) ++1~~

~~Any f 2 G satis es (ff g)=2 ,so~~

~~j j~~

~~j n j n~~

~~1(m1) 1(m1) (m1) (m1)~~

~~) " ,so Now diam(A) 2a(~~

~~' (diam(A)) 2 =2 ;~~

~~s=q~~

~~where c is constant. Thus~~

~~as n !1, since m 1 > 1=" . This implies the result by lemma 3.3.4.2~~

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