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Fractal Measures on Infinite Dimensional Sets

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Fractal Measures on Infinite Dimensional Sets 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 <n: 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.
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