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
By
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
d
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
"
b
b
; ...... 16
-
; ; [ ] ...... 21-22
'
D (x; ( ) ) ...... 28
k k
' ' '
e e
Q ; Q ;Q ...... 34-35
"
e
K (X ); ...... 38
d
C (R ) ...... 50
C (E );fj ; kk ...... 55
E
d
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
2
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
2
in de ning a measure similar to Leb esgue's to measure the size of a smo oth m-
n
dimensional sub-manifold of R . In [Ha], Hausdor shows that this may b e extended
s
to a natural s-dimensional measure H for any real s>0. This in turn de nes
n
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
s
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
3
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
r
closed ball of radius r ab out the p oint x.ThatisB (x)=fy 2 X : (x; y ) r g.
r
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
5
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:
( )
X
'
H (E )= inf '(diam(E )) : fE g is an "-cover of E ;
i 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
n
E .For example if is Leb esgue measure on R and (t)=t ,then
n
8
>
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 6 ' 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&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 ) (t) <=M , and cho ose an "-cover fE g of E such that that 0 i i '(t) P '(diam(E )) i i X H (E ) (diam(E )) i " i X (diam(E )) i i " (diam(E )) M< p> i i = 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 P ab ove H (E ) H (E )=0:2 i i With minor mo di cations, the ab ove pro of shows the following: 7 If dim(E ) ' then dim(E ) If dim(E ) ' then dim(E ) 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 )) i ' (diam(U )) = '(diam(U )) >MH (E ): i i " '(diam(U )) i 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 ) If dim(E ) ' then dim(E ) 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, 8 f' g , such that < implies ' ' . Then there is a critical value >0 2 [0; 1] such that 0 ( 0 if > 0 ' H (E )= 1 if < : 0 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)= >0 s