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This is to certify that the

dissertation entitled

Hausdorff Dimension of Invariant Sets for
Expanding and Hyperbolic Systems

presented by

Yingjie Zhang

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Applied Mathematics

 

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Date 8/20/96

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Michigan State
University

 

 

 

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HAUSDORFF DIMENSION OF INVARIANT SETS FOR
EXPANDING AND HYPERBOLIC SYSTEMS

By

Yingjie Zhang

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1996

ABSTRACT

HAUSDORFF DIMENSION OF INVARIANT SETS FOR EXPANDING AND
HYPERBOLIC SYSTEMS
By

Yingjie Zhang

Consider a compact invariant set of a differentiable dynamical system. The Haus-
dorff dimension of such a set is closely related to the dynamical behavior of the map
defining the system. This thesis studies systems with strict expanding property, which
include the finite-dimensional and infinite-dimensional expanding maps and the hy-
perbolic sets. We focus on non-conformal systems. Various upper and lower bounds
for the Hausdorff dimension are provided. The major upper bound is constructed
using the expanding ratios and the topological pressure, which generalizes a formula
of R. Bowen and D. Ruelle on conformal maps. For some well-known invariant sets
the bound has the property that it exactly equals the dimension in typical cases.
Other bounds are also obtained by means of topological entropy, measure-theoretical
entropy and Lyapunov exponents. These bounds have improved or supplemented

current results.

To my dear parents and my lovely wife.

iii

ACKNOWLEDGMENTS

I thank Prof. T.-Y. Li for leading me into this fascinating research field. He has
played a major role in guiding and supporting my career at Michigan State. His
knowledge and mathematical insight have been invaluable. I was constantly inspired

by his enthusiasm and professional approach to research.

I thank Prof. Sheldon Newhouse for expert guidance and encouragement. The

many discussions I had with him have brought my research work to a higher level.

My thanks also go out to Professors Bill Sledd, Cliff Weil and Zhengfang Zhou for
carefully reviewing the first draft of my thesis and pointing out many errors. Without

their help this final version would be impossible.

iv

Contents

1 Introduction
1.1 Conformal Expanding Systems ......................
1.1.1 Classical Examples ........................
1.1.2 A Formula of Bowen and Ruelle .................
1.2 Some Nonconformal Cases ........................

1.3 Main Question and Organization of the Thesis .............

2 Expanding Systems
2.1 Definitions and Notation .........................
2.2 Upper Bound by Topological Pressure ..................
2.3 Proof of Theorem 2.1 ...........................
2.4 Upper Bound by Lyapunov Exponents and Entropy ..........

2.5 Lower Bounds ...............................

3 Hausdorff Dimension of Hyperbolic Sets on Unstable Manifolds
3.1 Hyperbolic Sets ..............................

3.2 Upper Bound by Topological Pressure ..................

13

13

17

21

27

32

39

39

40

3.3 Upper Bound by Lyapunov Exponents ................. 47

3.4 Lower Bounds ............................... 48
4 Expanding Systems in Hilbert Spaces 55
4.1 Expanding Ratios and Lyapunov Exponents .............. 55
4.2 Upper Bounds ............................... 60
4.3 Lower Bounds ............................... 64
4.4 Example: A Hilbert Space Self-affine Set ................ 65
BIBLIOGRAPHY 69

vi

Chapter 1

Introduction

This thesis studies the relationship between the Hausdorff dimension of a compact
invariant set and the metric property of the system. For many systems such a relation
is useful to estimate the Hausdorff dimension. Let us first consider a discrete—time
dynamical system defined by an expanding differentiable map. A compact invariant
set for such a system is usually an irregular set with an intricate structure, known
as a fractal set. Consequently, it has a fractional dimension. There is a variety of
definitions of dimensions for irregular sets. Among them the Hausdorff dimension is
the most widely used. It is known that the Hausdorff dimension is closely related
to the expanding ratios of the map. This thesis focuses on nonconformal expanding
maps, meaning that the expanding ratios in different directions are different. This
work generalizes some recent results in two special situations —— a study on nonlinear
conformal expanding maps by Bowen and Ruelle, and several interesting formulas
about non-conformal piecewise linear expanding systems by K. Falconer and others.

Let us begin with a review of these past results.

2
1.1 Conformal Expanding Systems

1 . l . 1 Classical Examples

We first describe two well-known fractal sets which are among the earliest examples
of irregular sets that have a fractional Hausdorff dimension. Since these sets are made
up of small pieces each geometrically similar to the whole, they are called self-similar
sets. The usual construction of a self-similar set is to iterate a family of contracting

maps. Here, instead, we view it as an invariant set of an appropriate expanding map.

Example 1.1 Let f be the following piecewise linear map vspace0.2in

 

 

0 1/3 2/3 1
Figure 1

f is defined on the set C1 = [0, %] U[§-, 1]. It is expanding since the slope is 3, which is

greater than 1. Define a sequence of subsets C’n by taking the inverse of f successively

c. = Io,§1UI§-,u=rl({o,u)
c. = f“Cn_1=f'"([0,ll)

— — C1
— — — —— C2
-- -- -- -- C3
Figure2
Thelimitset
“In.“ DC

is the middle-third Cantor set. Clearly, C Is invariant; that is, fC = C.

The Hausdorff dimension of C is well-known

dimH C = log2

 

log3'

Further, we can consider a piecewise linear map as follows

 

 

Figure 3

where the domain of f consists of p disjoint intervals each of length 1/ X (2 S p _<_ x),
and on each interval f has slope x. Hence, f‘"([0, 1]) is the union of p“ disjoint

intervals with identical length l/X". The limit set

“333111011011 "(1011)

has a similar dimension formula

 

lo
dimHA= 0gp. (1.1)
logx

Example 1.2 Consider a more general map f that may have different slopes on

different subintervals.

 

 

Figure 4

f is defined on p disjoint closed intervals. The slope of f on the j-th interval is Xj,

I le > 1. So the length of the j-th interval is l/Ile. Similarly define the limit set

A=gggof (I011)=flf ([)01]

The Hausdorff dimension of A is given by the value of d satisfying

1 1
—+---+——=1 1.2
ded prld . ( )

This famous formula is due to Moran [22]. When all Xj,3 are equal and positive, (1.2)

reduces to (1.1).

1.1.2 A Formula of Bowen and Ruelle

Formulas (1.1) and (1.2) clearly show that the slope of f plays a major role in deter-

mining the Hausdorff dimension. The number of subintervals, 19, being equal, invariant

sets A with larger slope X (or IXJ-I’s) have smaller Hausdorff dimension. When dealing
with a nonlinear expanding map, e. g., the map in Figure 5, the one thing that stands
out is that the slope of the map varies from point to point. The Hausdorff dimension
of a given invariant set should involve, in some sense, an average effect of the slope
function over the whole set. A fundamental contribution was made by Bowen [4] and
Ruelle [30], where they found the the right tool—the topological pressure—to cope

with the nonconstant slope function.

 

 

Figure 5

The concept of topological pressure was introduced earlier by Ruelle [27] to study
the statistical mechanics of lattice systems. Walters developed the general theory
soon afterwards [35]. We review the definition of topological pressure as well as some
of its properties in Chapter 2. The work of Bowen and Ruelle originally dealt with
multi-dimensional conformal maps, which included one-dimensional real maps as a
special case. A conformal expanding map has, at each point, a constant expanding
ratio in all directions. So it is sometimes said to be essentially one-dimensional. Their

celebrated formula, as stated in [30], is as follows.

Let f be a C 1+“ conformal expanding map and A be a compact invariant set with

the following properties.

(i) A is isolated, i.e., there is a neighborhood U of A satisfying
fl f‘”U = A.
11:0

(ii) f is topologically mixing on A.

Then the Hausdorff dimension of A is given by the number d satisfying the equation

P(flAa_d10gllD:rfll) = 0 (13)

where P( f | A, ¢) denotes the topological pressure of a function o5 with respect to the
system (A, f).

The smoothness assumption was recently relaxed to C1 in [13]. An alternative
proof is also given in this thesis (Chapter 2). Note that, in (1.3), the quantity
log ||Dxf|| is the exponential expanding rate of f at :r, and P(f|A,—dlog||Dxf||)
calculates, in some sense, an average effect of the function —dlog ”Dr f I] on the set

A. (1.3) reduces to (1.2) in the piecewise linear case.

1.2 Some Nonconformal Cases

Most maps are not conformal. The dimension problems for nonconformal systems
are much more complicated. In fact, there does not exist such precise and general
relations as those in Section 1 for conformal systems. Therefore, one starts off with
some special situations. In recent years, a class of simple yet interesting fractal sets,
known as the self-afine sets, has drawn considerable attention. A self—affine set can
be viewed as an invariant set for a piecewise linear expanding map. In this section,

we review two well-studied examples that largely motivated our work.

The construction of a self-affine set is similar to that in Section 1.1.1. Denote

the unit cube [O,1]m in Rm by R. Let T1, ..., T; be 1 linear operators on Rm that

are expanding, i.e., [ITJ-xll > ”x“ for 0 75 .1: 6 Rm. Then for b1, ...,b1 E R“, Fja: =
Tja: + bj defines l expanding affine maps. Assume b1, ...,b1 are so chosen that the
parallelepipeds F l‘lR, , FI‘IR are all contained in R and are pairwise disjoint. Set
R1 = UL, Fj"1R. Define a map f on R1 by leJ-IR = Fj. Then f is expanding and
R1 = f‘1(R). The limit set

00

A = 33,130 f'"(R) = fl f‘"(R)

is called a self-affine set. A typical self-affine set is shown below.

 

 

 

 

"o 0.2 0.4 0.3 0.0 1

 

 

 

 

 

 

 

 

 

Figure 6
Example 1.3 Consider a special case where m = 2, l = 2 and T1 = T2 2
X1 0

, 1 < < .
( 0 X2 ) X1 _ X2

R

1/X2

“X1

 

 

 

Figure 7

(i) If X1 2 2, then dim” A = £51 except when one of FflR and F’s—1R is directly

108X1 ’

above the other, in which case dimH A = 1951 < 351.
losxz — IOSXI

(ii) 1 < x1 < 2 < x2. It is easy to show that dimHA g 1+_L_"’32 X1

108 X2

. A more complete

relation is contained in the following conjecture [1, 25]: For Lebesgue almost

all X1 6 (1,2), dimHA = 1+ £83m, except when one of Fl-IR and Fz'lR is

108 X2

directly above the other. A recent work by Solomyak [32] seems to contain a

proof of the conjecture.

Example 1.4 Return to a general self-affine set. The expanding ratios of F j (F j$ =

Tia: + bj) are given by the singular values of T], or equivalently, by the lengths of

the principle axes of the ellipsoid T ,B, where B is a unit ball in R“. To determine

dimHA we need not only the expanding ratios of Fj, but also those of all iterates

F j" o . - . o Fjl. The discussion below is due to Falconer [9].

Let T be an expanding linear operator with singular values 1 < X1 S S Xm-

For 0 S d S m define

d—
¢>d(T) = X1"°X[d]XId]i-dl

where [d] is the largest integer less than or equal to d. Then ¢d(T) is continuous and

strictly increasing in d. For each n = 1, 2, set

I
2f. = Z l¢d(Tjn o - - - 0 ml“.

jl ..... j..=1
The sequence Eff, has the submultiplicative property
Bin. S $3.31
which implies that the limit

2:" = lim (29%

00
71—)00

exists and equals infn(§3i)%. Ego is continuous and strictly decreasing in d, and

2g S 1. So there is a unique 0 S d S m satisfying
Egzr 04)

Denote this (1 by d(T1, ..., T1). Falconer showed [9]

(i) dimHA _<_ d(T1,...,T,).

(ii) If [ITj-lll < %, j = 1,...,l, then for Lebesgue almost all 51,...,bl, dimHA =
d(T1,...,T1).

In the two special cases discussed in Example 1.3, d(T1, ..., TI) is equal to fiffi or

1 + 533%?“ respectively. Attempts have been made to remove the %-condition in (ii).
But none has succeeded. It is also easy to see that for conformal Tj’s, d = d(T1, ..., Tl)

satisfies an equation similar to (1.2)

1 1

__ + . . . + _ =1
IITIII“ lszlld

and by the result of Bowen and Ruelle, dimH A = d(T1, ..., Tl).

1.3 Main Question and Organization of the Thesis

This thesis studies nonlinear and non-conformal maps. We will find relations between
the Hausdorff dimension and the expanding property of the map. It is easy to believe
that the non-conformal case is much more subtle than the conformal case. First,
from the two examples in Section 1.2 we see that the expanding ratios of the map
can not completely determine the Hausdorff dimension. In Example 1.3, keeping the
size of the rectangles F {IR and Fz’lR fixed and moving one directly above the other,
the resulting invariant set A has the same expanding ratios but smaller Hausdorff

dimension. Further, the new A is topologically conjugate to the old one. (They both

10

are topologically conjugate to a one-sided full shift on two symbols.) Therefore, we
have here two invariant sets that are equivalent topologically, have the same expanding
ratios, but differ in dimension. This example shows there does not exist a dimension

formula, like that of Bowen and Ruelle, which covers all non-conformal cases.

On the other hand, the two examples also show a bright side of the topic. In both
cases, there is a dynamical quantity defined by the expanding ratios that is an upper
bound for the Hausdorff dimension. Moreover, it is much more than just an ordinary
upper bound. It actually equals the Hausdorff dimension frequently. This surprising
fact has attracted a great deal of attention. In Example 1.3 where 1 < X1 < 2 < X2,
a lot of work had been done before Solomyak to determine the size of the set of X1 for
which the Hausdorff dimension equals 1 + %’2‘—‘. (Earlier work was on an essentially
equivalent topic, called infinite Bernoulli convolutions. See [1] for an excellent survey.)

In Example 1.4, the g-condition is technical and seems unnatural. Whether it can be

improved or even removed is currently under investigation.

Now it should be clear how to proceed to study the Hausdorff dimension of invari-
ant sets for nonlinear and non-conformal expanding systems. We expect to construct
a dynamical quantity using the expanding ratios and topological invariants, which

satisfies the following conditions.

(i) The dynamical quantity is an upper bound for the Hausdorff dimension.
(ii) It is an extention of all those quantities surveyed in last two sections.

(iii) It should equal the Hausdorff dimension for a system that is typical in some

sense.

The goal is partially achieved in this thesis. We use topological pressure to nonlin-

earize Falconer’s procedure in Section 1.2 to produce an upper bound for the Hausdorff

11

dimension. The upper bound coincides with the solution of (1.2) and (1.3) for con-
formal systems; and in all the piecewise linear non-conformal situations in Section
1.2 it equals the upper bounds there. It remains to be verified that this upper bound
satisfies (iii) as well. Finding a suitable measurement to describe the typicality seems

to be a challenging task.
The thesis is organized as follows.

Chapter 2 studies the Hausdorff dimension of invariant sets for finite-dimensional
non-conformal expanding maps. First, the definitions of dimension, entropy and
topological pressure are reviewed. Then we show the construction of the upper bound
mentioned above, whose proof takes up a large part of the chapter. Following the
tradition of smooth ergodic theory, we then derive another upper bound in terms of
Lyapunov exponents and entropy. The well-known Lyapunov dimension is adapted
to expanding systems, the relation between which and the Hausdorff dimension stays
the same as in diffeomorphic systems. Finally, some lower bounds for the Hausdorff
dimension are obtained via studying the dimension of ergodic measures. In particular,

our method provides an alternative proof for the formula of Bowen and Ruelle.

A similar idea of using topological pressure to obtain an upper bound appeared

in a recent paper of Falconer [11]. But our technique and result are different.

In Chapter 3, we consider the dimension of the intersection of a hyperbolic set
with a piece of unstable manifold. Since the map is expanding on the unstable
manifolds, the set of intersection has exactly the same properties as an invariant set
of an expanding map. It is thus not hard to adapt the techniques in Chapter 2 to

construct upper bounds here. Lower bounds are discussed using a different approach.

Chapter 4 deals with an expanding system in a Hilbert space. For a compact

invariant set in an infinite-dimensional space, the first concern is whether it has finite

12

Hausdorff dimension. The question is answered by showing that the dimension is
always bounded above by a finite dynamical quantity, which is constructed using the
same topological pressure approach. The sigular values for a noncompact infinite-
dimensional operator need to be carefully defined. Then the proofs in Chapter 2

largely carry through.

Part of Chapter 2 and Chapter 3 will appear in Ergodic Theory 59' Dynamical
Systems [38].

Chapter 2

Expanding Systems

In this chapter we study compact invariant sets for finite dimensional expanding
maps. The main result is an upper bound for the Hausdorff dimension which is de-
fined by the expanding ratios and the topological pressure. The bound is a natural
generalization of those that appeared in Section 1.2. Our method combines the topo-
logical pressure technique of Bowen-Ruelle and the singular value function used by
Falconer. Several other bounds are also discussed, which involve topological entropy,

measure-theoretical entropy and Lyapunov exponents.

2.1 Definitions and Notation

This section first reviews some background materials in dimension theory and ergodic
theory. The definitions are given in a general setting. Then we discuss the logarithmic

singular value function for an expanding linear operator — a key concept in this thesis.

Let X be a compact metric space. Use B(a:,r) to denote the open ball centered

at a: E X with radius r. Fix d > 0. For Y C X and r > 0 define

’HflY) = inf{2:r3-i : U B($j,rj) I) Y, :13,- E X and rj S r}.

J J

13

14
Obviously, if r < r' then HflY) 2 ’Hf,(Y). The limit

Hdm = lim 7am

r-+O

is called the d—dimensional Hausdorfir (ball) measure. The Hausdorfir dimension of Y
is given by
dim” Y = inf{d : ’Hd(Y) = 0}.

Hausdorff dimension generalizes the usual dimension concept in Euclidean spaces. Its

properties are nicely surveyed in [10].

Let f : X —> X be a continuous map. For :1: E X and r > 0, denote
Bn(a:,r) = {y E X : fig 6 B(fia:,r), for all i = 0,-~,n — 1}.

This set is called an (n,r)-ball (with respect to f) at a). If 45 is a real continuous
function on X and n = 1,2, ..., let
n—l .
(and) = Z¢(f’x)
i=0
and for r > 0, set
Q(¢,n,r) = inf{}:exp{(5n¢)(xj)} = U 84w) 2 X}.
j j

Then the topological pressure of d) is given by

P(f, <15) = mum sup 110g Q(¢,n,r)-

n—ioo n
The maps considered in this thesis, either expanding or hyperbolic, possess the so—
called expansive property [36, Definition 5.1]. Consequently there is some r0 > 0 such

that for any 7' < r0 and any continuous function qt

P(f,¢) = Iimsup 1 logQI¢,n.r) (2.1)

n—)oo n

[36, Chapter 9]. For an expansive map, P( f, qt) is always finite.

15

In particular, if 45 E 0, the topological pressure, P(f,0), is called the topological
entropy of f, denoted by htop( f) Entropy is one of the most important concepts in

ergodic theory.

We also study the measure—theoretic version of the above notation. A Borel prob-

ability measure ,u on X is f—invariant, if for any Borel set Y C X

It(f"lY) = IJY-

An invariant measure a is ergodic, if it can not be written in the form $111 + %/12,

where [11 and m are two different invariant measures.

The Hausdorff dimension of an invariant measure a, dimH p, is defined as

(11me = inf{dimHY : Y C X,pY =1}.

The entropy of an invariant measure was first introduced by Kolmogorov and
Sinai. Now there are several equivalent definitions. The following one from [5] is

most convenient for us. Define the local entropy of p at a: E X as

hp(:1:,f) = limliminf—llongn(:r,r). (2.2)

r—)0 n-)oo n
It is an invariant function: hu(f:c,f) = h,,(:c,f). The entropy of p (with respect to
f) is given by

W) = [X h.(x.f)du(a:).

If p is an ergodic measure, then for ,u-a.e. :1:
hu(f) = hu($,f). (2-3)

Topological pressure and measure-theoretic entropy are connected by the following

Variational Principle [36]

P(f,<I>)= sup {hu(f)+ /X MI} (2.4)

ue€(X,f)

16

where 8 (X, f) is the set of ergodic probability measures. In particular

ht0P(f) :- sup hu(f)°
#€£(X,f)

Moreover, if f is expansive, then for any real continuous function 45 the sup in

(2.4) is assumed by some p E 5(X, f), which is called an equilibrium state of gt [36]

Now we define the logarithmic singular value function gd(L) for an expanding
linear map L. Let V and V’ be two real inner product spaces with the same dimension
m. A linear map L : V —> V’ is said to be expanding if ||Lv|| > ”v” for all u E V and
v # 0. A singular value of L is an eigenvalue of the positive definite matrix (L‘L)1/2.

We write the logarithms of the singular values of L in decreasing order
A1 2A22WZAm,
where Am > 0 since L is expanding. Then for any d E [0,m] set

gd(L)= Z Ai+(d—ldl)’\m—Id]

i=m—[d]+l

where [d] is the largest integer S d. Obviously, gd(L) is continuous and strictly
increasing in d. g°(L) = 0 and gm(L) = 2:1)“- 2 log [Jac(L)], where Jac(L) is the
J acobian of L.
gd has the following important superadditive property. If L : V ——> V’ and L’ :
V' —-) V" are two linear expanding maps, where dimV = dim V’ = dim V" = m,
then for any d E [0, m]
9d(L'L) Z gd(L) + gd(L')- (2.5)

See [17, 9] for proof.

17

2.2 Upper Bound by Topological Pressure

We study the main upper bound in this section. Let M be a C °° Riemannian manifold,
dimM = m. Let U be an open subset of M and f : U ——> M be a C1 map. Suppose
A C U is a compact invariant set on which f is expanding, that is, f A = A and there

is k > 1 such that for all x E A and v E TxM,
llefvll 2 kllvll

where I] - I] is the norm induced by an adapted Riemannian metric. Here, we need not

assume either A is isolated or f | A has any transitivity property.

If a: E A, then Dxf : TxM -—) foM is a linear expanding map. Denote the

logarithms of the singular values of Dr f by

and for d E [0,m], write

gd(a=)Egd(Dxf)= Z A:'(:I«')+(d-[61])/\m—I<II($)-

i=m—[d]+l

Since f is C1, the functions a: -—> A,(:r) and a: —> gd(x) are all continuous.

For each d E [0, m], consider P1 (d) E P( f | A, —gd). It is obvious from the definition
of topological pressure that P1(d) is continuous in d and is strictly decreasing. P1(O) =
htop(f|A) Z 0. The sign of P1(m) = P(f|A, — log |Jac(D$f)|) is in general unknown.

However, when f is C 2 and A is an isolated invariant set, P1(m) is nonpositive [3, 28].

Our upper bound will be defined by the logarithmic singular value functions for
all Dxf", n > 0. fA = A implies f"A = A. f“ is also expanding on A. Let the

logarithms of the singular values of Dxf" be

A1(:c,n) > Z Am(:r,n)

18

and set

gum): 2 AIM) —[d])m-Id1(:v,n)-

i=m—[d]+1

In particular, A;(:c,1) = A;(:r) and gd(:c, 1) = gd(x). Since the function :2: —> gd(:1:,n)

is continuous, we can consider the topological pressure

Pn(d) = P(f|A, —;fi-gII-.n)).

It turns out that the sequence of functions Pn(d), n = 1,2, converges to a nice

function.

Lemma 2.1 For every d E [0,m], the following limit exists

lim Pn(d) = inf Pn(d).

n—too n>0

Set
P*(d) : lim Pn(d).

n—too

Then P“ is continuous and strictly decreasing on [0,m]. P*(0) = htop(f|A) 2 0, and

P*(m) = P(f|A,-108|JaC(Dxf)|)-
Proof. The proof follows standard properties of topological pressure. Since
Dxfn“ = D f... f’Dx f"

from (2.5) we have

gd(a:,n + l) 2 gd(.r,n) + gd(f":c, l). (2.6)

By [36, Theorem 9.7],

Pn+z(d) = P(f|A,— fir——,ngx,n+t))

 

S P-——(flA, (— $9396.71» +

n+l

19

< n
_ n+1

 

PIfIA, gm, n)) + $7M)“ —%ngf"x, 1))
n l
= n—+an(d) + WHW)

This, together with [36, Theorem 4.9], proves the existance of the limit

P*(d) = lim Pn(d) = inf Pn(d).

n—ioo n>0

For each n, Pn(0) = htop(f|A) and

Pn(m) = PIfIA.—%IogIJacID.f“)I)
n—l
= P(f|A,—;1;ZloglJaC(Df-‘xf)|)
i=0

= P(f[A,—10g|JaC(Dxf)l)a

where the last equality follows from the definition of topological pressure. So P"'(0) =
htop(f|A) and P"‘(m) = P(f|A,—log |Jac(Dxf)|). Set K 2 supra [lef||. Then for
anyOSd1 <d2Sm,

gd’(x,f") - gd‘(:c,f")

<
nlogk_ dg—dl

 

SnlogK.

By [36, Theorem 9.7],

Pn(d2) — Pn(dl)

— <
logK _ d2 —d1

 

S — 10g k,
which implies that P“ is continuous and strictly decreasing in d. D
Our main upper bound for the Hausdorff dimension is given by the zeros of func-
tion 1"“.
Theorem 2.1 Let
D(f, A) = max{d E [0,m] : P“(d) 2 0}.

Then

20

The proof will be given in the next section. Theorem 2.1 implies that if P( f I A, — log [.1 ac(Dx f )|

0, then D( f, A) is the unique number d satisfying

otherwise, D(f, A) = m.
A C1 map f is said to be conformal if ||Dva|| = ||Dxf||Hv|| for any a: and v. If f
is conformal, then so is f", and all singular values of Dz f ” equals ||Dxf"||. Therefore,
flan) = dlog llef”||

n—l
dz: 10g llDf'xfll'

i=0

By the definition of topological pressure we have
P’(d) = P1(d), d E [0,m].
Hence D( f, A) is the solution for d in the following equation

PHIL-£1103 llefll) = 0-

This is exactly equation (1.3). So the result of Bowen-Ruelle says that, under mild
additional assumptions (condition (1) and (ii) in Section 1.1.2), ’D(f, A) = dim” A for

conformal maps.

Even in nonconformal situations, D(f,A) often has an unusual property which
makes it a particularly interesting upper bound. For the piecewise linear systems
we studied in Section 1.2, D(f,A) can be more explicitely calculated. In Example
1.3, ’D(f,A) equals $1531- or 111% according as X1 2 2 or I < X1 < 2 < X2- In
Example 1.4, D(f,A) = d(T1,...,T1). Therefore, in these two classes of systems,
D( f, A) = dim}; A holds almost surely in the sense of Lebesgue measure. Therefore,
D(f,A) is an dynamical quantity satisfying the desired properties (i) and (ii) in

Section 1.3.

21

A recent paper by Falconer [11] proves a related result. If we denote the box
dimension of A by dimB A, then it is well-known that dimHA S dimB A, and strict

inequality generally holds. [11] shows that if f satisfies a bounded distortion condition

||(Dxf)'1||2||Dxf|| < 1,

then dimBA S D( f, A). However, the definition of ’D(f,A) in [11] is more compli-
cated, and the method there does not apply to systems which do not satisfy the

condition. On the other hand, our technique does not work for box dimension.

2.3 Proof of Theorem 2.1

Theorem 2.1 is proved by several lemmas. First we show how the Hausdorff measure

of a small set changes under an expanding map.

Lemma 2.2 Let U C Rm be an open set and f : U —> Rm be a Cl map. Assume at a
point x E U, Dxf : Rm —-> Rm is an expanding linear map. Then for every d E [0,m],

there exists r0 > 0, such that B(x,ro) C U and for any set A C B(x,r0) we have

"Hid/1) S CHfUA) (2-7)

for all r < r0, where
b = 2mepr—Am-me)}, (2.8)
C = 2dm‘i/2 exp{—gd(x)}. (2.9)

Proof. We follow an idea in [8]. Let c be a small positive number with (1 + e)e‘ < 2.
Since Dy f : Rm -—> Rm is continuous in y, we can choose r0 satisfying the following

conditions.

(i) f maps B(x,2ro) diffeomorphically onto fB(x,2ro). ||fz — fyH > “z — y||, for

all y,z E B(x,2r0) with y 7b 2.

22
(ii) |/\.-(y) — /\,-(x)| < e, for all y E B(x,r0) and i = 1, - - - ,m.

(iii) Ilz — y — (om-1w — fy)“ < eIIz — yII, for all y e BIm) and z e Ewe).

where condition (iii) is because (DE/f)"l is also continuous in y. Assume A C B(x, r0).
Fix any r < r0. Let a = Hfl f A) (a is obviously finite). Then for any 17 > 0, there are
finitely many balls B(zj,rj) , zj 6 fB(x,r0) , rJ- S r, such that U,‘ B(Zj,7'j) D fA
and Z]. r? < a+77. Let B;- = {y E B(x,ro) : fy E B(Zj,rj)} . Then U]. B;- D A. By
(i) and (iii) B;- C yj+(1+ e)(DyJ.f)'lB(O,rj), where yj E B(x,ro) and f(yj) = Zj
. The definition of the singular values implies that ( Dy] f )‘IB(0, r,) is an ellipsoid

which has principle axes of lengths

2T1 expf-Am(yj)} Z Z 27‘jexpl-A1(yj)}o

Therefore, B; is contained in a rectangular box with sides

2(1 + €)7‘jeXP{—/\m(yj)}.~ - ' °,2(1 + C)I‘J'€><P{—/\1(yj)}

which can be covered by

eXpi- Z Ai(yj)}expildl/\m—Id1(w)}

i=m—[d]+l
balls with radius
x/Wl + 6)nexp{--/\m-Id1(311)}-

By (ii) exp{—/\m_[d](yj)} S e‘exp{—Am_[d](x)}. Summing up over allj and using

the assumption (1 + e)e‘ < 2 we get

Hid/1) S ZeXPi- Z /\i(yj)}eXP{ldl)‘m-[d](yj)}

i=m—[d]+l

.md/2(1 + c)dr;-i exp{—d)\m—[d](yj)}

23

= (1+6)dmd/2erexp{-gd(yj)}

J

s (1 + ereprdomm Zr? epr-gdm}

J

|/\

rants—gun»:
j

S C(a + 17).
Since 17 > 0 is arbitrary, this proves the lemma. D

By a routine procedure we can extend the inequality to any compact set.

Corollary 2.1 Let M be a Riemannian manifold, dimM = m, and let f : M —> M
be a C1 map. Assume X C M is compact and f is expanding on X. {X need not
be f invariant.) Fix d E [0,m]. Then for any be > 2\/r_n and Co > 2dmd/2, there is
r0 > 0, such that for all x E X and all A C B(x,r0) (2.7) holds for every r < r0,

where b and C (depending on x) are given by

b = boexp{—/\m_[d](x)}, (2.10)

C = Coexp{—gd(x)}. (2.11)

Proof. Since XUfX is compact, there is r1 > 0, such that for any x E XUfX, expx
maps E40, r1), the rl-ball at 0 in TxM, diffeomorphically onto the ball B (x, r1) C M.
Choose r2 < r1 so that for every x E X, fB(x,r2) C B(fx, r1). Then f; E exp}; ofo
expz is defined on E$(0,r2). Obviously, Do}: can be identified with Dz. f. Again by
compactness, both expx and exp}; uniformly C1 approach isometries on §x(0,r)
and B(fx,r), as r —> 0. So there is r0 < r2, such that at every x E X, f; satisfies
the conditions (i), (ii) and (iii) in the proof of Lemma 2.2. Applying Lemma 2.2 to

Tim for every A C Ex(O,ro) we have

utIfi) 3 mm» (2.12)

24

~

for all r < r0, where b and C are given by (2.8) and (2.9). Set A = expr(A). Then
£(A) = exp}; 0 f (A) Reducing r0 one more time if necessary, we may make all expx
and exp]; so close to isometries that (2.12) can be replaced by (2.7) with slightly

larger b and C, which are preassigned by (2.10) and (2.11). This completes the proof.

C)

The next lemma provides a rough upper bound for dimH A using the expanding

ratios and topological pressure.

Lemma 2.3 Set

DI = max{d 6 [0,177.] : P1(d) Z 0}.

Then
dim” A S 01.

Proof. For simplicity we will assume that f is defined and is C1 on the whole of
M. Assume D1 < m, otherwise the assertion is obvious. Then P1(Dl) = 0. Take
any d > D1. Since P1(-) is strictly decreasing, e E —%P1(d) = —%P(f|A,—gd) is a
positive number. Choose be > 2\/r—n and Co > 2dmd/2. Since Ileva 2 k||v|| for all
x E A and v E TxM, llefan _>_ k"||v||. This implies that A,(x,f”) Z nlogk for all

i=1,---,mandn>0.

Pick an integer N > 0 satisfying the following inequalities
S E bolt-N < 1, (2-13)

Co exp{—Ne} < 1. (2.14)

Applying Corollary 2.1 to the map fN, we can find r0 > 0 such that for all x E A
and A C B(x,ro)
HtIA) s Coepr—ngx, N)}H:‘IINA) (2.15)

25

for all r < r0. Also assume r0 is so small that the topological pressure can be

calculated by (2.1) for any 1' < r0. Fix such a r. By the definition of e,

limsupg logQ(— —gd,n,r)= —2e.

n-too

Hence there exists no such that n > no implies
Q(—gd,nN,r) < exp{—nNe}.
So there are points {xj} C A, U]. BnN(xJ-,r) D A, such that

nN—l

Zexp{— Z gd( ij) )} < exp{—nNe}.

i=0

Since B,N(x,r) C B(x,ro) and fNB;N(x,r) C B(,-_1)N(fo,r) for all i = 1, - - - ,n and

all x E A, applying (2.15) n — 1 times to BnN(xJ-,r) yields

n—2
Hgnr(BnN(-Tja7‘)) S 03—16%“—ZQdUWCCj,N)}H%r(BN(f("’l)N$j,T))
i=0
n-l .
s 03-‘Clczepr—ngIrNxmn (2.16)
i=0

where C1 = supxeA ng(B(x,r)) and 02 = supxeA exp{gd(x, N)}. By (2.6),

11—1 . n— —l N—l nN-l
ngU'ijJV) > 2:93 f'N+’:I:J-) =2: 9(
{:0 i=0 i=0

Summing up (2.16) over j we have
Hind/U S ZflbnrwnmxnrD
j

nN—l

S 010208—1 Z€Xp{- Zd g(

i=0

< C1C2C(’,‘_1 exp{—nNe}.

26

By (2.13) and (2.14), letting n -—) 00 we get ’Hd(A) = 0. Therefore d 2 dim” A. Since

d can be made arbitrarily close to D1, this proves dimH A S D1. D
Now we are ready to prove Theorem 2.1.

Proof of Theorem 2.1. Without loss of generality, we assume P( f | A, —— log [J ac(D$ f )l) S

0. For n > 1, consider the equation
13nd); P(f"|m —ng-,n)) = 0.
Since 811(0) = htop(fnlA) = nhtop(flA) Z 0 and
13,,(m) = P(f"|)\, -10g IJaC(Dxf")|) = nP(flA, -10g IJaC(Dxf)|) S 0

the equation has a unique solution, which we denote by D... Applying Lemma 2.3 to
the expanding map f" yields dimHA S D”. So dimHA S infn>o Dn. The proof of

Theorem 2.1 will be completed if the following inequality is verified
333 D7, S D(f,A). (2.17)
For small 6 > 0, let d = inf Dn — e. As in the proof of Lemma 2.1, for any d1 < d2,
13,.(d2)— 13,.(d1) g —nlogk - (d2 — d1).
Substituting d1 = d and d2 = D”, yields

Pn(d) 2 nlogk~(Dn —d) 2 nlogk-e.

Fix any integer N > 0. We show PN(d) > 0.

Set C = max{gd(x, N) : x E A}. Let an integer l be so big that lNlogk - e > C.

For eachj = 0,1, - - - , N — 1, using (2.6) we have

l-l
Z gd(fiN+jx, N)

i=0

27

l—2

s (gums) + ng(f‘”+jx, N) + ngf<’-”N+Ix, N — 1)) + 0

i=0
3 gd(x, IN) + C.

Summing over j, we get
1 lN-l .
N Z gd(f‘x,N) S gd(x,lN) + C.

i=0
Now by [36, Theorem 9.8 (i) and Theorem 9.7]
1
PN(d) = P(flA, —N9d($, Ni)

1 1lN-l
_ _ IN __ d i
— mm It. Ngngw»

1

Z WP(fINlA,—gd($,lN) — C)
= 21iv'(EN(d) " C)
I

IV

lN(lNlogk-e- C)
which is > 0 by the definition of l.

Since PN(d) > 0 for all N, P*(d) Z 0. So d S D(f,A). As 6 = infDn — d can be

arbitrarily small, proving (2.17), which completes the proof of Theorem 2.1. D

With a little more effort it can actually be shown D( f, A) = infn D”.

2.4 Upper Bound by Lyapunov Exponents and En-
tropy

Using Lyapunov exponents and topological entropy, we give another upper bound for

dimH A in this section. First let us recall some basic facts about Lyapunov exponents.

A point x E A is called a regular point, if for every v E TxM

1
lim —log Ilefan E x(x,v)
n—too n

28

exists. For a regular point x, x(x, u) can take on at most m different values. In fact,

there is a family of subspaces
fiMzEmDEmDmDEM

such that x(x,v) is constant for v E E(l)\E(i+1) and x(x,v(i)) > x(x,v(‘+1)), v“) E
E(‘)\E(‘+1), i = 1,...,r — 1. The numbers x(x,v(‘)) are called Lyapunov exponents
of f at x, and dim E“) — dim ES“) is the multiplicity of x(x,v(‘)). Let us write the

Lyapunov exponents, counting multiplicity, by /\'{(x) 2 A;(x) 2 2 A1,,(x).

If p is an invariant measure on A, the Multiplicative Ergodic Theorem [24] says

that ,u-almost every point x is regular and

lim —1-)I,-(x,n) = X’(x), ,u -a.e. and in L101). (2.18)

n-+oo n '
In particular, when p is ergodic, the set of Lyapunov exponents are the same for

p-almost all x. Therefore we may call them the Lyapunov exponents of p and write
A101) 2 Z Am(l‘)-

Let 8(A, f) be the set of f—invariant ergodic measures with support in A. If
,u E 8(A, f), by expansion the /\,-(p)’s are all positive. Write

gd(u)= Z /\i(II)+(d-[dl)/\m-Id1(#)-

i=m-[d]+l

By (2.18)

lim —1—gd(x,n) = gd(p), ,u -a.e. and in L1(,u).

n—+oo n

For each u E €(A, f), we define a number dA(p) by its Lyapunov exponents and

topological entropy htop( f I A)

dA(II) = max{d 6 [0m] = 9%) S h.op(f|)\)}.

The main theorem in this section is

29

Theorem 2.2

dimHAS sup dA(p).
“66(AJ)

The definition of dA(p) is motivated by the Lyapunov dimension [16, 17, 1]. If

2:1 AIM) > htop(f[A), we can rewrite

htoplflA) — Am(l1)—"'— )‘m-H-IU")
Ant-10‘)

where l = max{i : Am(,u) + + Am_,+1(p) S htop(f|A)}. Let us compare (2.19)

 

did/I) = 1+ (2.19)

with the definition of Lyapunov dimension. Suppose V is an ergodic measure of any
differentiable map f with Lyapunov exponents A1 2 2 Au > 0 2 Au“ 2 2 Am.

Then the Lyapunov dimension is

A1+~~+Au+Au+1+~~+Az
|/\I+1|

 

Lyap dim(1/) = l +

where l = max{i : A1 + + )1,- Z 0}. When f is a diffeomorphism and l/ is abso—
lutely continuous with respect to Lebesgue measure (or only absolutely continuous

on unstable manifolds), by Pesin’s formula [24] we have

hu(f) +z\u+1 +-~+/\z

Lyap dim(t/) = l + [AI+1l

 

where h,,(f) is the metric entropy and l = max{i : —Au+1 — — A.- S hu(f)}. This
expression is similar to (2.19). In the definition of dA(/1), however, we use htop( f | A)
instead of hp( f) in order to prove Theorem 2.2. Notice that Lyap dim(,u) does not
apply to expanding systems, since all exponents of p are positive. So dA(p) may
be considered as a reasonable substitute. Thus, Theorem 2.2 is a supplement to the
following inequality of Ledrappier [17]

dimHA S sup Lyap dim(1/), (2.20)

uE£(A,f)

where A is a compact invariant set of any differentiable map f.

30

A construction similar to dA(p) also appears in [12], where, instead of gd(p), a

function consisting of uniform Lyapunov exponents is used.

The rest of this section is devoted to the proof of Theorem 2.2. A lemma similar
to Proposition 2 in [17] is needed. Mimicing the definition of dA(p), for any x E A

and integer n > 0 we set
dAIm) = max{d 6 [0m] =ngx,n) s mum}.

Notice that htop(f"|A) = nhtop(f|A).

Lemma 2.4

infsupdA(x,n)= sup dA(p).
">0 xEA p€€(A,f)

Proof. Let d = infn>o supxeA dA(x,n). We first show d S supp€£(AJ)dA(/t). For
simplicity let us denote h = htop( f | A). Since every dA(x, n) is uniformly continuous
on A, there is a sequence {xn} C A such that dA(xn,n) = supxeA dA(x,n) Z d for all
n. This implies gd(xn,n) S nh. Now fix an integer N > 0. If n > N and 0 S j < N,

we have by (2.6)

ln/Nl-HJ')

Z ngf‘vinw)

i=0

[n/Nl-kU) . _ g
s ngx..j)+ Z gd(f‘N+’$mN)+9d(f“’)rcmn-l(j))
i=0

|/\

gd(xmn)

S nh,

where k(j) = 1, ifO S j S n —N[n/N]; k(j) = 2, ifn — N[n/N] < j < N, and

1(3) = N([n/N] — k(j) + 1) +j. Summing overj yields

n—N
Z gd(fix,,, N) S nNh.
.=0

31

1 n—

Letting Vn = ;; {:1 6px", where 6, is the Dirac measure at x, we rewrite the in-
equality as

] ngm, N)du.Ix) s Nh + 50 (2.21)
A n

where C = supxeA gd(x,N). Let V be a vague limit point of {Vn}. Then V is an

invariant probability measure on A (See Theorem 6.9 in [36].) and from (2.21)

jgd(x,N)dV(x) S Nh
A
1 d
[ANg (x,N)dV(x) S h. (2.22)

Let V = A pde(x) be an ergodic decomposition of V, where at 6 £(A,f), V-a.e.

x E A. Then limNnoo figd(x, N) = gd(,ur), V—a.e. and in Ll(l/). Thus by (2.22)

i. gd(#2)dv(w) s h.
Therefore gd(px) S h for some pa, 6 8(A,f), i.e., d S dA(px).

Now we prove the converse. Obviously, d 2 supxeAinfn dA(x,n). For any [1 E
€(A,f), there is x E A such that lim.,,_m igd'(x,n) = gd'(p), all d’ E [0,m]. So
limn“mo dA(x,n) = dA(p). By (2.6), dA(x,n + l) S sup{dA(x,n),dA(f"x,l)}, for
all n,l > 0. Applying this inequality repeatedly we obtain limnnoo dA(x,n) S

supn inf; dA(f"x,l) S d, which proves dA(p) S d. Cl

Proof of Theorem 2.2. By Lemma 2.4 and the fact dimH A S infn Dn (See the

proof of Theorem 2.1.) we only need to show that for each n

Dn S sup dA(xa Tl).
xEA

Let d = supxeA dA(x,n). Thengd(x,n) Z nh for allx E A. So Pn(d) = P(fnlA, —gd(-,n))
P(fnlA,—nh) = P(f"|A,0) — nh = nh — nh = 0. By the definition of Du, we get

d 2 D", as desired. Cl

32

As mentioned at the end of Section 2.3, D(f,A) = infn 0". Therefore we have

DU, A) S supp€5(A,f) dA(H)-
2.5 Lower Bounds

In the construction of the two lower bounds, the least expanding ratios are used. As
expected, by considering the largest expanding ratios we can obtain lower bounds for
dim” A. A common approach to the study of lower bounds is to calculate a lower

bound for the dimension of ergodic measures. We begin with the following lemma.

Lemma 2.5 (a) Up 6 5(A,f), then the following function ofx

_ . . loguBkw)
ASCII”) — 11?;ng

is constant almost everywhere. Denote the constant by _d_p(p) and call it the lower

pointwise dimension of ,u.

(b)
_,, ‘ L. 10s IIszlldMIv)

 

Proof. Pick a small 6 > 0. We claim that there is r0 > 0 such that for any x E A

andr<ro
l—e

In fact, by the continuity of D, f and the compactness of A, it is possible to choose

r0 so small that if x E A and y E B(x,ro),

llfy - fa: — D2f(y - x)“ S filly - :6“-

Then it is easy to check that if lly—xll < “fir, r < r0, then ||fy— fo| < r, proving

(2.23).

33

 

(a) For x E A set c(x )=: IIDZIII‘ By (2. 23) and the invariance of u, ,uB(y, r-) —
Itf“B(y,r) Z Exef-ny#3($26(x)r)- SO for any x,
uB(-’B,C(x)r) S IIB(f:v,r)-
This gives
dp(x,#) 2 dp(fw.#)-
But It is invariant. So fdp( (x ,xp)(d,u zfdp (fx, p)d,u(x). Thus
4.45%) = dp(f:v,#)
p-a.e.. By ergodicity _d_p(x, p) is constant, ,u-a.e..
(b) Applying (2.23) repeatedly, yields

"(_1—6)nr)CB(ka,r), k=0,1,...,n—1.

ka(x,
3:01”thfo

 

Namely,
(1 - e)
Hizol ”Df'xf“

 

B(x, r) C Bn(x, r). (2.24)

Therefore

loguB(a: ,—(l;‘L,— )

mt. IID,.- III

10g 7:0:er

Hszo ”Df'x f”

 

l
-- ..logr -10g(1 - 6) +- "EL—0 10s llefll
By the Birkhoff Ergodic Theorem (see [36, p. 34]), we have for p-a.e. x

 

Z —% 10g ”Bub/Ba 7‘)

lim— gleam“: [logllDMflldItH

i=0

So for p-a.e. x,

108#B($,—£11;CL—|7‘)

 

lim inf 11:; |lesz
l—e "
"if” log 11:23 “0,5111"

 

> 11m 1nf—-log,uB (x, r) 1

. 2.25
a...» —log(1-€)+fllogl|Dxf||d#($) ( )

34

Following [37] we know that the left-hand side is equal to liminfpno log—WM =

108 )0

dp(x,p). Taking r —> 0 in (2.25) and using (2.2), (2.3) we get

1
' —log(1 — e) + L, log Ileflldu(w)°

Since 6 can be arbitrarily small, this proves (b). D

.dp(x’ I”) Z hu(f)

 

Following the lemma, two lower bounds will be constructed for dimH A. The first
one is given by topological pressure. To simplify later statements in this section we
assume P(fnlA, —nlog ||D,,f”||) S 0. This condition holds automatically if f is C2

and A is an isolated invariant set.

Theorem 2.3 Denote by Dn the unique number d satisfying
754d) 5 P(fnlm —d108 llefnll) = 0-

Then

dimH A Z sup Dn = lim supDn.

71—)00

Here the existance and uniqueness of a zero (1 of the function Pn(d) can be estab—
lished the same way as in last two sections for Pn(d) and Pn(d). The latter equality

in the theorem simply means as n —> 00, Du is expected to provide a better bound.

Proof. Since (A, f") is an expansive system and log ||Dxfn|| is continuous, there is

an equilibrium state a E 8 (A, f") satisfying
htIr) — 5. ] log Harman) = P(f”|m in / log IIDmII) = o.
By part (b) of Lemma 2.5 and Proposition 2.1 in [37]

dimHA 2 4,901)

35

h2(f")
— IA 108 llef"||dIt($)
= l—jna

 

which proves the inequality in Theorem 2.3. Since

llef"+’|| S llkaxf’llllefkll

using [36, Theorem 9.8] we obtain for any d

Pn(d) = P(f"|A,-d108||Dxf"ll)

n-l

Punt; —dlogIID,.-.fll)

i=0

= nP(f|A, —dlos llefll)

IV

Along with the fact that Pn(d) is strictly decreasing in d we have Du 2 D1. Similarly,
for all k, n 2 1

En): 2 Uk-

This shows lim sup,,_+00 D” = supn D... B

Our second lower bound is formed by metric entropy and the largest Lyapunov

exponent.

Theorem 2.4

dimH A Z sup hp”)
macaw) A101)

 

where A1(u) is the largest Lyapunov exponent ofp.

Proof. We follow an idea in [17]. If p E £(A, f), then for any N, p is an invariant

measure for f”, which may not be ergodic. The ergodic decomposition of u with

36

respect to fN must be of the form

1 8
I‘ - ‘3‘ Z ”I
3:1
where s is a factor of N, V1,...,V, are mutually singular ergodic measure of f”,
f...VJ- = V12“, for j = 1,...,s — 1 and f...V, = V1. By part (a) of Lemma 2.5, each V,-
has a well-defined lower pointwise dimension QP(VJ-). Since f. permutes VJ’ around,

repeating the proof of Lemma 2.5 (a), yields
ip(l/1) = = 47,049).

It is easy to show by the definition of lower pointwise dimension that the common

value equals 4pm). Therefore, applying part (b) of Lemma 2.5 to U”, l/j) we obtain

max MUN)
1325s f log IleleldVACB)

23:1 hu,‘(fN)
23:1 f 108 lleleldI/Aa?)

h2U”)
f 10s lleleldev)

hid) .
M102 lleleldIIIx)

By the definition of Lyapunov exponents, Emma“,o ~11V f logllefN ||dp(x) = /\1(p)

 

4.01) 2

 

IV

 

 

(Section 2.4) and thus we have

 

2.0a) 2 '18:] (2.26)

Since dimH A 2 dp(p), this proves the theorem. D

It is actually not hard to derive Theorem 2.4 from Theorem 2.3; namely, to show

supn Du 2 sup” §f§fiA But the by-product in this proof, inequality (2.26), is

37

interesting in its own right. Several inequality of the type

 

(dimension of ,u) 2 I

are already known. But our statement is stronger. In [17] it is proved that QLUI) Z
9&3, where Q1. (,u) is sometimes called the lower Ledrappier capacity, which according
to [37] is 2 41,02). (2.26) also appears in [14], but an extra C2 smoothness is assumed

due to the use of Lyapunov charts.

Our results have an interesting consequence for conformal systems. If f is confor-

mal, then

Fn(d) = P(fnlAa-dzlogllDf'xfll

= nP(fIA, -—dlog HDxflll-

So Du = D1 = the solution of the equation
P(fIA, -dlog llefll) = 0- (2-27)
On the other hand, we also see in Section 2.3 that D( f, A) satisfies the same equation.

Therefore, dim” A is the solution of (2.27) as well.

Moreover, denote the solution of (2.27) by d and let ,u be an equilibrium state of

the function —dlog ”DI f I] (There may be more than one equilibrium states.) Then

htIf) — d / log Ila/Imam) = 0.

So
2 h2(f) .
L, 10s Ilefllde)

Lemma 2.5 (b) hence implies that dimH p = dim” A = d. We summarize the discus-

 

sion here as a corollary.

38

Corollary 2.2 Iff is a conformal expanding map, then
(a) dimHA equals the unique solution d of equation (2.27),

(b) there exists an ergodic measure )1 on A with full dimension; namely,

dimH u = dim” A.

The corollary is the well-known result of Bowen-Ruelle (see Section 1.1.2). Our
assumption is weaker since f is only required to be C1. As mentioned in Section 1.1.2,
this weaker condition also appears in [13]. But our approach is more straightforward.
It does not use the Markov partition and symbolic dynamics. (Therefore there is no
need to assume A is isolated or is topologically transitive.) The crucial tool here is

the Brin-Katok entropy formula (2.2).

Chapter 3

Hausdorff Dimension of Hyperbolic
Sets on Unstable Manifolds

In this chapter, we show how to adapt the techniques in Chapter 2 to study hyperbolic
sets. A hyperbolic set A is a compact invariant set for a diffeomorphism f, where
f is uniformly contracting in some directions and expanding in others. Through
each point of A there passes a stable manifold and an unstable manifold. Since f is
expanding on the unstable manifolds, our method can be used to construct bounds
for Hausdorff dimension of the intersection of A with an unstable manifold. Let us

begin with a review of hyperbolic sets.

3.1 Hyperbolic Sets

Let M be a compact Riemannian manifold and let f be a C 1 diffeomorphism on M.
A compact set A is called a hyperbolic set for f, if fA = A and there is a continuous

splitting TxM = E’(x) + E"(x) on A satisfying the following conditions.
(i) (Dxf)E“’(x) = E’(fx), (Dxf)E“(x) = E“(fx), x E A.
(ii) There is a constant k > 1, such that

llefIE’($)ll S k'1
39

40
Ilef“|E“(x)l| s H

For a small 6 > 0 and x E A the set
W:(x) = {y E M : f"y E B(f"x,e),n = 0,1,...}

contains a Cl disk around x. Wc’(x) is tangent to E’(x) at x and varies continuously

with x. The set

W’(:c) = U f‘"W.’(f"a:)

1120

is a C1 injectively immersed copy of E’(x), which is called the ( global) stable manifold
at x. Two stable manifolds W’(x) and W’(y) either coincide or are disjoint. The
stable foliation {W’(x) : x E A} is invariant: fW’(x) = W’(fx). The set Wc“(x) and
the (global) unstable manifold W“(x) are defined by replacing f with f‘l.

Wc“(x) = {y E M : f'"y E B(f_"x,e),n = 0,1,...},

Wqu) = U f"W.“(f"":r)-

1120

We say that a hyperbolic set A has a local product structure if there is an e > 0
such that the set W:(x) fl W¢”(y), if not empty, consists of a single point of A. A is
topologically transitive if some point of A has a dense orbit in A. A hyperbolic set that
has a local product structure and is topologically transitive has many nice properties.
In this thesis we do not need these assumptions. For simplicity, however, we assume

that the splitting has a constant dimension dim E’(x) = s and dim E“(x) = u.

Two general references on hyperbolic sets are [23] and [31].

3.2 Upper Bound by Topological Pressure

Let x E A and let W“(x) be the unstable manifold of x with dimension u. View

W“(x) I) A as a subset of W“(x). We will find upper bounds for dimH(W“(x) D A).

41

Denote the metric on M by p and the induced metric on W“(x) by p“. Since W“(x) is
a C 1 injective immersion of IR“, using either metric yields the same value of Hausdorff

dimension. It is convenient for us to use the metric p“.

By the definition of a hyperbolic set, for every integer n > 0, Dxfn|E“(x) is an

expanding linear map from E“(x) onto E“(f”x). Denote by

A1(x,n) > Z Au(x,n) (Z nlog k)

the logarithms of the singular values of 0,, f "|E“(x), and for d E [0, u] set

ngm) = Z Mm) + (d — [dun—mm).
i=u—[d]+l
(The notation A,(x, n) and gd(x, n) are used in both expanding and hyperbolic situa-
tions. Their meaning should be clear from the context.) It follows from the continuity
of the splitting that /\,-(x,n) and gd(x,n) are continuous in x. Consider topological

pressure functions

Pi.‘(d) = P(fIA, —%ng-.n))

d E [0, u]. Clearly, the functions gd(-, n) also satisfy the superadditive condition
gd(x,n +1) 2 gd(x,n) + gd(f"x,l). (3.1)
Using the same technique as in Section 2.2 we can show that the limit function

P“*(d) 5 lim P:(d) = inf P:(d)

n-—)oo

exists. Furthermore, P” is continuous and strictly decreasing in d E [0, u], P“*(0) =
P::I0) = htoprIA) 2 0 and P“*(u) = Pqu) = P(fIA, -Iog IJac(D.f|E”(x))l)- If f is
C2 and A is a hyperbolic set with the local product structure, then it is known that
P(flA, - log |Jac(D,,f|E“(x))|) S 0. So the equation

P"*(d) = o

42

has a unique solution d E [0, u]. In general, however, the sign of P(fIA, — log |Jac(D,,f|E“(x))|)

may be positive. Parallel to Theorem 2.1, we have the following result.

Theorem 3.1 Let
D“(f, A) = max{d E [0,u] : P“*(d) 2 0}.

Then for every x 6 A,
dimH(W“(x) I) A) S ’D"(f,A).

Let us compare the theorem with a classical result of McCluskey and Manning.
Assume f is a C 1 diffeomorphism on a two-dimensional manifold and /\ is a topolog-
ically transitive hyperbolic set with local product structure. In this case both E‘(x)
and E“(x) are one-dimensional, and W"(x) is a C1 curve on M. It is proved in [21]

that dimH(W“(x) n A) is independent of x and is given by the unique d satisfying

P(fIA, -d10g llDyflE“(y)||) = 0- (3-2)

We claim that if dim E“(x) = 1, then D“(f, A) is the solution of (3.2). In fact, in this

case there is only one direction with rate
Man) = log llef"lE“(:v)ll-
Therefore

gd($,n) = d108||1?.2f"‘|13“($)||

= dzlogllexflE“(f‘x)ll-

i=0

So by the definition of topological pressure

PIfI..—%ng-.n))

= P(flm-dlogIleflE“($)|l)-

P2561)

43

Hence the limit P“*(d) equals P(fIA, —dlog ||Dxf|E"(x)||) as well, proving the claim.

We once again come across the same scenario as in expanding systems. A one—
dimensional (or conformal) result can not be fully generalized to multidimensional
nonconformal situations. The expanding ratios might provide a good upper bound

for the Hausdorff dimension, but the dimension is not completely determined by them.

The proof of Theorem 3.1 relies on the following lemma. Remember that if a
set is contained in a piece of unstable manifold, its Hausdorff measure is calculated

using the induced metric p". We also use B“(x, r) to denote the u-dimensional ball

in WWII), B“($,r) = {y 6 WW) = p“(:v,y) < 1"l-

Lemma 3.1 Let A be a hyperbolic set of a Cl difleomorphism f. Fix d E [0,u].
Then for any 60 > 2J5 and Co > 2dud/2, there is r0 > 0, such that for all x E A,
A C B“(x,r0) and r < r0

Hid/1) S CHM/4) (33)

where b and C (depending on x} are given by

b = boexp{—Au_[d](x,1)}, (3.4)

C = C0 exp{—gd(x, 1)}. (3.5)

Proof. This is a direct consequence of Lemma 2.2. Choose r1 > 0, so that for
every x 6 A, expx maps the ball Bx(0,r1) diffeomorphically onto B(x,r1). From
standard properties of unstable manifolds [31], there is r2 < r1 such that for every
x E A, B“(x,r2) C fB“(f‘1x,r2) C B(x,r1), and exp;1(fB“(f‘1x,r2)) is the graph
of a C 1 function from an open set of E“(x) to E"(x). Use pm. to denote the projection
of TxM onto E“(x) along E’(x). Let €12 be the inverse map of p2, restricted to
exp;1( f B“( f‘lx,r2)). Then q, is a C1 diffeomorphism and Doqx may be identified

with the identity map iniE“(x). Set f; = (If; 0 exp]; of 0 expI oqx. Then Def: =

44

D; f |E“(x) As r —> 0, all expz and q, uniformly C1 approach isometries on Bx(0, r)
and Bx(0, r) O E“(x), respectively. Thus there is r0 < r; such that Lemma 2.2 applies

to f; uniformly for all x E A. So ifx 6 A and A C Bx(0,r0) fl E“(x), then
HtIZ) s cum/1)) (3.6)
for all r < r0, where b and C satisfy

b = fax/Jexpi-Au—Iddxa 1)},

C = 2dud/2 exp{—gd(x,1)}.

Now set A = exp,, oq,,(A). Then f;(A’) = qf‘xl 0 exp}; of(A). Further reducing
r0 to make all exp,c and q:c closer to isometries, we can replace (3.6) by (3.3) with

slightly large b and C satisfying (3.4) and (3.5), completing the proof. CI

The following lemma plays the same role as Lemma 2.3.

Lemma 3.2 Set

D? = max{d E [0,u] : P1"(d) Z 0}.
Then for any x E A,
dimH(W“(x) (I A) S Di‘.
Proof. Fix :7: E A. Let W be any compact piece of W“(x). We show

dimH(W I) A) g 01‘. (3.7)

This obviously implies Lemma 3.2.
Without loss of generality, assume D; < u, i.e., P(fIA, — log |Jac(D,,f|E“(x))|) <

0 and P1“(D‘l‘) = 0. Take any (1 > D? and set 6 = —%Pl“(d). Since P1“ is strictly

decreasing, e > 0. Choose b0 > Zfi, Co > 2dud/2, n > 0 and an integer N > 0

45
satisfying
flsbok’N < 1, (3-8)

Coe’7 exp{——Ne} < 1. (3.9)

Consider A as a hyperbolic set for fN. Clearly, [IDfolE’(x)|| S If” and
||Dxf‘NlE“(x)|| S k‘N. By Lemma 3.1, there is r’ > 0, such that for all x E A,

A C B“(x,r’) and r < r’ we have

711%.(4) S Co exp{-gd($, N )Wi’ U” A) (310)

Choose small positive numbers p, re and r1 (in this order) with ro << p, re S r’

and r1 < r0, so that the following hold.

(i) If x E A and y E B(x,r1) I) A, then B“(y,p) fl B(x,r1) C B“(y,ro).
(ii) If x E A and y E W (1 B(x,r1) I) A, then W I) B(x,r1) C B“(y,r0).
(iii) For any x E A, fNB“(x,ro) C B“(fo,p).

(iv) If x E A and y E B(x,r1) F) A, then [gd(x, N) — gd(y, N)| < n.

(v) For any r < r1 and any continuous function d

PIfIA, 45) = lim sup-3,102 Q(<I>,n,r)-

71—)00

Here, (i) is due to some well-known facts about the unstable manifolds (see [31]). (ii)
follows from the compactness of W. It roughly means that W intersects any B (x, r1)

at most once. (iii), (iv) and (v) are obviously true for all small re and r1.

Fix r < r1. By the definition of 6,

lim sup 1 log Q(-gd(-,1), n, r) 2 —2e.

n—ioo n

46
So for all large n,
Q(_gd('a 1)) ”Na 7‘) < exp{-—nNe}.

Thus, there are points {xJ} C A, U]. BnN(xJ,r) D A, such that

nN-l

Zexp{- z: gd(fixJ,1)} < exp{—nNe}. (3.11)
j i=0

The sets {BnN(xJ-,r)} covers W I) A. Let AJ- = W (I A I) BnN(xJ-,r), and consider the

iterates fiNAj, i = 0,1,-~ ,n, if AJ- # 0.

By (ii), AJ- C W n B(xJ,r) C B“(yJ-,ro), for some y,- E A,-. We use induction to

show

 

fiNAj C B"(fiNyj,7‘o) (3.12)

i = 0,1,---,n — 1. Assume we already have f(i_l)NAj C B“(f(i'1)NyJ,ro). Then

by (iii), f’NAJ- C B”(f‘NyJ,p). Moreover, f‘NAJ' C B(f‘NxJ,r) by the fact that

At C BnN($jaT’)- 30 by (i), fiNAj C -l53"(f"N.1/2'2I0)n B(f‘ijaI‘) C B“(f‘Nyjaro),
proving (3.12).

(iv) implies E:

Igd(f‘Nyj,N) - 9d(f‘ij, N)l < II-

From this and applying (3.10) n-times, yields

 

n—l
Hg.,(A,-) S CS epr- Z gd(f‘N 3),, N )Wf-‘(fm A2)
i=0 .3-
n—l '
_<_ 0105‘ exp{nn} epr— Z 9d(sz$ja N)}a
i=0

where C1 = sup{’Hf.’(B“(x, p)) : x E A}, which is clearly finite. By the superadditivity

condition (3.1)
nN-l

n—l
ngU‘ij, N) 2 Z gut-,1).
i=0

i=0

47

Since UJ- A,- D W 0 A, we get
Hg.,(w n A) g ZHg.,(A,-)
,-

nN—l

S 0106' exp{nn}Zexp{- Z gd(f‘xj,1)}
j i=0
< CICS exp{nn} exp{—nN6}

where the last inequality is from (3.11). Using (3.8) and (3.9) and letting n ——> 00,
yields
H"( W n A) = 0.
This gives dimH(W n A) S d. Taking d -—> Di proves (3.7). D
Now we can complete the proof of Theorem 3.1 by just repeating the proof of The-
orem 2.1 (Section 2.3), almost word for word. This is because the proof of Theorem
2.1 is based on the superadditivity of functions gd(:c, n) and the inequality dimH S D1 ,

and these two facts were just verified in the hyperbolic case as well. So the details

are omitted.

3.3 Upper Bound by Lyapunov Exponents

Again as in Section 2.4, we derive another upper bound for dimH(W“(a:) (1 A) by
Lyapunov exponents and topological entropy. If p 6 8 (A, f), then [1 has exactly u

positive Lyapunov exponents. Write them in the usual order

and let

48
Then define for each u E 8 (A, f)

dim!) = max{d 6 [0w] = 9‘10!) S htop(flA)}.

This number takes the following form if 2;, x\,-(p) > htop(f|A).

 

dXUt) = [+- htop(flA) —' Auizilgfl”) ' _ )‘u—H-l (I!)

where l = max{i : Au(p) + - - - + Au_,+1(,u) S htop(f|A)}.
Theorem 3.2 For any a: E A,

dimH(W“(a:)flA)S sup dX(,u).
I‘€£(A9f)

Again the proof is omitted, since we can simply repeat the proof of Lemma 2.4 and
Theorem 2.2. It can also be shown that ’D“(f, A) S SUPuest) dX(/1). Thus ’D“(f,A)

is a better upper bound.

When A is a topologically transitive hyperbolic set with the product structure for
a two-dimensional diffeomorphism, the stable foliation and the unstable foliation are
both Lipschitz. Consequently, it is shown in [21] that dim” A = dimH(W"(a:) 0 A) +
dimH(W“(x) D A). So a formula for dim” A would directly follow from a calculation
of dimH(W“(:c) O A). In the higher dimensional case, however, the relation between
dimH A, dimH(W’(a:) (‘1 A) and dimH(W“(:r) n A) is not clear. So our results can not
immediately provide upper bounds for dimH A. We mension that some upper bounds

similar to supyegm’n dxm) have been obtained before for dimH A [17, 12].

3.4 Lower Bounds

Lower bounds for dimH(W“(a:) 0 A) may be obtained by mimicking Section 2.5.
However, we will use a result of Ledrappier and Young [18] to derive sharper lower

bounds. To quote their paper we will assume the diffeomorphism f is C2.

49

As usual, to study lower bounds we begin with the dimension of invariant mea-
sures. For an ergodic measure we will find a lower bound for the pointwise dimension
of its conditional measures on unstable manifolds. Let us first recall the definition
of conditional measures. Our treatment does not intend to cover the most general

situation, yet is sufficient for later use. See [26, 18] for more details.

Let p be an invariant measure on A. If U C M is an open set, we use {“(U) to
denote the partition of U 0 A by the connected components of unstable manifolds.
Use the notation W“(:z:,U) for a general element of (”(U), where W“(:r,U) is the
connected component of W"(a:) flU containing :1: E U flA. Since the unstable foliation
is continuous with C 1 leaves, if the diameter of U is small enough, then {“(U) is a
measurable partition with respect to a. An important consequence is that there is a

family of conditional measures
{Vi} : :1: E U n A}

where Vi]

14c:LI

is a Borel probability measure on W“(a:, U), such that for every Borel set

,u(A) 2/ Vij(A fl W“(a:, U))d/t($). (3.13)
MM
This family of conditional measures is unique up to a null set.

The conditional measures are essentially determined by the invariant measure
a. If U and U’ are two small open sets, then there are two families of conditional
measures {V3} and {113'}. It is easy to check that for :1: E U n U'flA, V5,] is a constant
multiple of 113' on W“(:1:,U) fl W“(.r,U’). In particular, dimH V5,] = dim” Vi” on

W“(a:, U) 0 W“(:r, U’). And the lower and upper pointwise dimensions at a:

U u
4,,(13 VU) = liminflogux”? (:13,r))

’ I r-)0 log r

 

 

— l U B“ .
dp(:c, V?) = lim sup 0g VI( (1" r))
raO logr

 

50

are independent of the open neighborhood U. So it makes sense to leave out U and

simply write §p(:1:, xxx) and 3,,(33, 14,).

Moreover, since f is a diffeomorphism and the unstable foliation is invariant,

{“( f U ) consists of exactly the f-images of elements of (“(U), i.e.,
W“(f:r,fU) = fW“(:v,U).

And since a is an invariant measure, the conditional measures corresponding to

{“( f U ) are exactly the images of those corresponding to €“(U)

U U
fox = flux.

This implies Qp(x,ux) = dp(f:c,ufx) and 3423,11,.) = dp(fx,ujx). Therefore if p is an
ergodic measure, 41,013, V13) and 3,,(x, V3,.) are constant almost everywhere. So we write

them as 4201) and dp ()1), respectively. In fact, more is proved in [18].

Lemma 3.3 [18, Proposition 7.3.1] Iff is a C2 diffeomorphism, A is a hyperbolic
set and [1 E £(A, f), then

The common value is called the pointwise dimension of p on unstable manifolds, and
is denoted by d;(p).

[18] also shows a remarkable relation between d;(p) and the Lyapunov exponents.
Collect the expanding Lyapunov exponents according to their magnitudes, say m1 of

them equal to MIN/1), , mg of them equal to A(")(]u), where
Mimi) > > AWN/1)

q
mg>0and ngzu.

i=1

m,- is called the multiplicity of A(‘)(p).

51

Lemma 3.4 [18] There are partial dimensions 0 S 7,- S m,, such that

hulf) = Z’WMOW) (3-14)
([201) = 2% (3.15)

Return to the previous notation. The Lyapunov exponents of u, counting multi-

plicity, are denoted by

My) 2 2 Mn)-

For d E [0,u] let

By Ruelle’s inequality [29]
2 AM) 2 hu(f)°
i=1
Therefore, there is a unique number at satisfying
TU!) = h”(f)-
Use 6”(p) to denote this number. From (3.14) and (3.15) we clearly have

0{$01) 2 My).

Hence, if U is open and :c E U is a typical point with respect to a family of conditional

measures {V351}, meaning that for 12;] almost every y E W“(:z:, U)
4.41/3) = 341/5) = dzm).

then from [37]

dimH(W“(:c) D A) Z dimH 12;} Z 5“(p).

To summarize

52
Lemma 3.5 Up 6 8(A, f), then for p-a.e. a: E A

dimH(W“(:c) 0 A) Z 6“(p).

Corollary 3.1

sup dimH(W“(:r) 0 A) Z sup 6“(p) (3.16)
136A [166(A9f)

Roughly, (3.16) says that supuegm’f) 6“(p) is a lower bound of dimH(W“(:r) n A)
for some as E A. It is usually not a lower bound for all :1: E A, since the whole
of A may be decomposed into several separate components and on some of them
dimH(W“(:c) 0 A) is possibly much less. Even if assuming topological transitivity,
we still do not know if dimH(W“(:c) F] A) is the same for all :3, because the stable
foliation of a multidimensional hyperbolic set is not Lipschitz. Thus the supxeA in

(3.16) cannot be dropped.

Now we formulate a similar lower bound using topological pressure. If d E [0, u]

and n > 0, set

[d]
F1033") = Z )‘i($an) + (d _ idl))‘[d]+l($an)'
i=1
Then again TUB, n) = log |Jac(Dx f |E"(a:))| Consider the topological pressure

—u

Pn(d) : P(fnlAa —§d(3n))'

Let T7,: be the unique solution d of the equation

if F:(u) S 0; Otherwise, let 5: = u. The first case must be true if A has the local

product structure.

Theorem 3.3

sup dimH(W“(a:) O A) Z supE: = lim supm. (3.17)

xEA n—roo

53

Proof. To prove the first inequality, it suffices to show

sup 6%) 2 '5';- (3-18)
u€£(A,f)

In fact, if (3.18) holds for any f, then it must hold for f”, i.e.,

SUP 5‘01) 2 D:-
1168(AJ")

But (3.16) obviously implies for any n

sup dimH(W"(a:) n A) Z sup 6“(p),
{BEA uES(A,f")

which proves the first part of (3.17).

For simplicity let D = 5:. Since FWD) 2 0, there is an ergodic measure [1, called

an equilibrium state, such that

W) — j W. max) = o. (3.19)
It is easy to check that §D(x, n) satisfy the subadditive property
§D(x, n +1) 3 §D(:c, n) + §D(f"x, z).

This implies that

By the invariance of [i

I/\

A§D($,n)dfl(x) Z §D(rx.1)dfl(x)

Along with (3.19) we get for any n

54

From the Multiplicative Ergodic Theorem

@002) = lim1 gamma)

n—ioo n A

S (WU)-

'-3
:3"
(D
H
9...
O
'1
“(D
V
t<
H
{3"
CD
0..
CD
:13
E.
5.".
O
:3
0
H1
12>.
g:
A
tn
V

proving (3.18).

The equality in (3.17) can be established the same way as in Theorem 2.3. Thus

the details are omitted. D

Chapter 4

Expanding Systems in Hilbert
Spaces

In this chapter we extend the results in Chapter 2 to expanding systems on Hilbert
spaces. Since our invariant set now sits in an infinite-dimensional space, the first
concern is whether the set has finite Hausdorff dimension. Questions like this have
appeared repeatedly in many different settings. The usual way to establish the finite-
ness of the dimension is to find a finite upper bound by means of metric properties
of the map [19, 20, 6, 34]. In our case, a finite upper bound will be provided by the
expanding ratios and topological pressure. In view of the finite—dimensional examples
presented in Chapter 1, we expect this one to be a close upper bound. Other upper
and lower bounds will also be obtained using other dynamical quantities. Because
the operators we are dealing with are infinite-dimensional and noncompact, a careful
definition of the expanding ratios and Lyapunov exponents is needed. Otherwise, the

methods here are the same as in Chapter 2.

4.1 Expanding Ratios and Lyapunov Exponents

Assume H is a real Hilbert space with inner product < -, ' > and norm [I ' I], and let

U C H be an open set. A map f : U —> H is said to be Fréchet differentiable at

55

56

:c E U, if there is a bounded linear operator L : H -—> H such that

llfy - fx - L(y - xlll = 0(lly - fell)

as ”y — at“ —> 0. The L that satisfies the above condition is unique. It is called
the Fre’chet derivative of f at x, and denoted by Dz f. A Fréchet differentiable map
f must be continuous. f is C 1 if D,c f is continuous under the operator norm. Let
f : U —-) H be a (71 map and let A C U be a compact invariant set, fA = A. f is said

to be an expanding map on A if its derivative DI f satisfies the following conditions.

(i) 0,, f is uniformly expanding; that is, there is k > 1 such that for every :1: E A
and v E H
llefvll Z lCllvll- (4-1)

(ii) Dxf is onto; that is, for all a: E A,
D, f (H ) = H.
By (i), 0,, f is injective. It then follows from (ii) and the Inverse Mapping Theorem [7,

p. 94] that there is a bounded inverse operator (D; f )‘l : H —> H with ”(Dr f )’1 I] S

k'1 for all .7: E H. Furthermore, (Dxf)‘l is continuous in :1: [2, p. 31].

Let the open unit ball centered at 0 E H be denoted by B(0,1). We review the

metric property of the set (D, f )'IB(0, 1). Our treatment follows [33, Chapter V].

Use A" to denote the adjoint operator of any bounded operator A. Set

T. = [(Dxf)‘(Dxf)li-

T3 is a positive self-adjoint operator. Obviously T3 2 kl for .7: E A, where I is the

identity operator in H. Define a sequence of numbers {X1(:r)} as follows

X101?) = sup inf < T v,v >
dim Egl—l Ilv|I=1.v<-:Ei x

57

where the sup is taken over all subspaces E C H with dimension S l — 1, and

EJ' = {w E H : < w,v > = 0, all v E E}. The sequence is increasing

k S X1(:z:) S X2(:1:) S

Set
Xoo($) = [13:10 X1013) = SUP X100)-
Note that if Xz(a:) = Xoo(1‘) for some I, then Xl+1(-’17) = Xl+2($) = = X00013)-

Lemma 4.1 [33, p. 262] (a) If X1(a:) < xoo(:c) for some I, then X1(:I:),...,X1(a:)
are eigenvalues of T3,. In fact, they are the n smallest eigenvalues of T1, counting
multiplicity.

(b) Let Ex be the closed subspace spanned by an orthonormal family of eigenvectors
e1(a:) corresponding to the eigenvalues xz(a:) < Xoo(:1:). Then E, and E: are invariant

subspaces, namely, TzEx 2 Ex and TxEj: 2 Bi. Moreover, ifv E E5}, then
< Txv,v >_>_ Xoo(:r)||v]|2.

In particular, either Er or E: may be all of H. Ea, = H only if H is a seperable

Hilbert space.
The next result gives a geometric description of the set ( D3,. f )‘IB(0, 1).

Lemma 4.2 [33, p. 266] (a) If Ex = H, then (Dxf)"lB(0,1) is an ellipsoid, whose
axes are along the vectors e;(:c) of lengths X1(:c)‘1.
(b) If E1. 7E H, then (Dxf)‘lB(0, 1) is contained in the direct sum of an ellipsoid in

Ex, as described in (a), and the ball in E: at 0 with radius Xoo(:r)“.

To construct bounds for the Hausdorff dimension of A, we also need to describe

the sets (Dxf")'lB(0, 1) for all n 2 1. The above notation is generalized as follows.

58

Write
T: = [(Drfn)*(Drfn)l§'
Define
Xl($,n) = SUp inf < Tgvyv >
dim Est-1 ||v||=1,veEi
and

Xoo(:1:,n) 2 11-1310 X1(a:,n).

Hence, X1(a:, 1) = X1($). From Lemma 4.2 the shape of (Dxf")‘lB(0, 1) is determined

by the sequence {X1(~’B, 12)}?2l in a similar way.

For any d E [0, 00), write

[‘1]
gd($vn) = 210g Xi($an) + (d - ldl) 10g X[d]+l(xan)'

We again have the following superadditive property.

Lemma 4.3 [33, p. 267] Ifn,k Z 1 and d E [0,oo), then

gd($, n + l) 2 9d(1‘, n) + gd(f"$, l)- (4.2)

Now consider the asymptotic growth rate of X1(:v,n) as n —-> 00. Given a: E A. If
the limit

/\',"(a:) = lim llog)(1(:r,n)

n-+oo n
exists for all I _>_ 1, then they are called the (bottom) Lyapunov exponents at 3:.
Obviously logk S z\’{(:c) S A;(:::) S This definition of Lyapunov exponents is
different from that in [33]. For expanding maps it is useful to define the sequence of
Lyapunov exponents starting from the least expanding rate, while in [33] the most
expanding rates play a major role. The Lyapunov exponents there may be more

specifically called the top Lyapunov exponents. The largest top Lyapunov exponent

59

(See the last part of the section.) will be used to construct a lower bound in Section

4.3.

The (bottom) Lyapunov exponents may not exist for a given x. However, by

Lemma 4.3 and the Subadditive Ergodic Theorem (see, e.g., [36]) we have

Lemma 4.4 Ifu is an f-invariant ergodic measure with support in A, then for p—a.e.
x the Lyapunov exponents {A7(x)}}§, exist. This sequence of numbers is independent
ofx and thus may be called the Lyapunov exponents ofp and denoted by {A1(p)}}’:l.

Moreover,

, 1
11m - 10g Xl($an)d#($) = Mn)-

n—+oo 77,

Part (b) of Lemma 4.2 describes an ellipsoid-like set containing (Dxf")‘lB(0, 1),
which will be used to obtain upper bounds for dimH A in Section 4.2. To study lower
bounds, we need to estimate the volume contained in each (Dxf")‘lB(0, 1). By the

definition of norm we have
(Dxf")—IB(0,1)D B(O, llef"||_1)- (4-3)

And B(O, ||Dxf"||‘1) is obviously the largest inscribed ball in (Dxf")"lB(0,1). The
limit

1' 11 IID f"l|

n33, 5 0g " ’

if it exists, is called the largest (top) Lyapunov exponent at x and is denoted by A(x).

Since log ||Dxf"|| possesses the subadditive property
log ”DranH S log “Dxfnll + log ”DPxfllla (4'4)

the Subadditive Ergodic Theorem implies that for any ergodic measure a,

Mas) = lim 110g warn
n—)oo n

60

exists for almost all x, X(x) E- X(p) is independent of x and

1
n

. 1 n . n —
Jgrgo-g/losllef lld#(rv) = Igf-flogllef ||d#(:v) = MM)-

Although a sequence of top Lyapunov exponents for a point x or an ergodic
measure u may also be defined (see [33]), we will not use them in this thesis. So only

the largest one, T, was mentioned here.

Setting K = max,“ HDxfll we obviously have
k" S X1($,n) S X2(xan) S S Xoo(x,n) S llefnll S K" (4-5)
and when the Lyapunov exponents exist

logk g Xf(x) g A;(x) < g M) 3 log K. (4.6)

4.2 Upper Bounds

Fix a C1 map f : U —+ H and a compact invariant set A C U satisfying properties (i)
and (ii) in Section 4.1. Since Dxf is continuous in x, the functions x —> X1(x, n) and
x —> gd(x,n) are all continuous on A for given n, l and d. Consider the topological

pressure
1
Pn(d) = P(fl/h —;gd($,n))-
This function of d obviously has the following properties.

Lemma 4.5 Pn : [0,00) ——> R is a continuous and strictly decreasing function.

Pn(0) = htop(f]A) 2 0 and Pn(oo) = limd_,00 Pn(d) = —00.

Here Pn(oo) = ——00 is because gd(x,n) uniformly approaches 00 on A as d ——> 00.

The sequence of functions Pn : [0, 00) —> R converges pointwise as n —> oo.

61

Lemma 4.6 For any (1 E [0,oo), the following limit exists

lim Pn(d) = inf Pn(d).

71—)00

Set P*(d) = lim,,_,00 Pn(d). Then P" is a continuous and strictly decreasing function

on [0,oo), P"(0) = htop(f|A) Z 0 and P"'(oo) =li1rnd_,00 P‘(d) 2 —00.

The proof is the same as that of Lemma 2.1, except that the last part follows

Lemma 4.5.

Consequently, the equation

P*(d) = 0

has a unique solution d E [0, oo). Denote this solution by D( f, A). The following is

the main result in this section.

Theorem 4.1

Since D( f, A) is finite, dimH A is also finite. This fact is interesting considering
that A is a set in the infinite-dimensional space H. In studying an invariant set of
an infinite—dimensional dynamical system, the finiteness of its Hausdorff dimension
appears to be a first concern. Several other conditions have been found under which
an invariant set has finite dimension [19, 20, 6, 34]. In view of the discussion in Section
2.2, we may expect that D( f, A) is also a close upper bound for dimH A. ’D( f, A) can

be explicitly calculated in special cases (See Section 4.4).

A proof of Theorem 4.1 may follow the same track as that of Theorem 2.1. The
only significant change due to the infinite dimensionality is the following modification

of Lemma 2.2. Recall that a set A C H is precompact, if its closure is compact.

62

Lemma 4.7 Fix x E A and d E (0, 00). There is r0 > 0 such that for any precompact

set A C B(x,r0) and 0 < r < r0 we have
Him) 3 cum/1) (4.7)
where

b = 2\/[dl+1)<{d]+1(x)’1 (4-8)

0 = 2“([d1+1)%e><p{—gd(x,1)}. (4.9)

Proof Let c be a small positive number satisfying (1 + c)2+5 S 2. By the continuity

of X[d]+1 and gd(.,1), there is re > 0 such that

(i) if y E B(x,ro), then

lgd(y.1)-gd(x,1)l <10g(1+ C)

and

(1+c)_1<Z(-M—+l-(-y—)<1+c.

X1d1+1($)
Since f is C1 and D1 f is invertible, by the Local Inversion Theorem [2, p. 32] f
has a C1 inverse in a neighborhood of x. D1. f being expanding implies f being locally

so. Choosing a smaller r0, if necessary, we have

(ii) f maps B(x,2ro) diffeomorphically onto f B(x,2ro) and

[lfz — f9” > [[2 - 3!”: if y. 2 E B(x,2ro) and y 75 2.

Using the definition of Fréchet derivative and the continuity of 0,, f and (Dr f )—l

and reducing r0 once again, if necessary, yields

(iii) for all y E B(x,ro) and z E B(y,ro)

||z - y - (Dyf)'l(fz - fy)|| < Cllz - yll-

63

Let A C B(x,ro) be a precompact set. Then for any r < r0, a = HflfA) is finite.
Choose 6 > 0. There are finitely many balls B(x,-,rj), with Zj E fB(x, r0) and r,- S r,
such that U]. B(Zj,7'j) D fA and 2’- r3? < a + e. Let B;- = B(x,ro)flf'lB(z,-,rj).
Then U1. B;- D A. By (ii) and (iii) B;- C 31,- + (1 + c)(DyJ.f)‘lB(0,r,-), where y,- E
B(x,ro) and fy, = 21-. From Lemma 4.2, B] is contained in a direct sum of a
rectangular box with sides 2(1 + c)r,'X1(y,-)‘1, . . . ,2(1 + C)rjX[d](yj)—l and a ball with
radius 2(1 + C)rjX[d]+1(yj)—l. Lemma V.3.1 of [33] says this set can be covered by

2”] eXP{-9M(yj,1)}X[d]+1(yj)[d] balls of radius Vldl + 1(1 + C)T1X[d]+1(yj)'l- Using

(i), the assumption (1 + c)2+% S 2 and the definition of Hausdorff measure, yield

7121(4) S 2 2“” exp{—g[‘”(yj.1)}de1+1(yj)[d](\/[dl +1(1 + C)rjX[d1+1(yj)")d

J

= Z 2ld1([d] +1)i(1+ c)“exp{-g“'(yj,1)}7"§-i

s 2qu] +1)%(1+ c)d+‘exp{—gd(w,1)}ZT3-‘

J

4
2

S 22d([dl+1) exp{-gd(x,1)}Z"i

< C(a+e).

Since 6 > 0 is arbitrary, this proves the lemma. D

We now present an upper bound by topological pressure and Lyapunov exponents.
It generalizes Theorem 2.2 to infinite—dimensional systems. Having made appropriate
definitions, such a generalization is straightforward. The inequality of Ledrappier

(2.20) also has Hilbert space and Banach space versions [6, 34].

Recall that £(A, f) is the set of ergodic probability measures supported on A.

If p 6 8(A,f) has (bottom) Lyapunov exponents A101) S A2()u) S ..., then each

64

N(p) Z logk > 0. Since the topological entropy htop( f IA) < 00, there is a unique

number 0 S d < oo solving

htop(f|A) = Mu) + + M1101) + (d - ldll)‘[d]+1(#)-

Denote this d by dA (p). Again we may rewrite the definition as

htop(f|A) — N(fl) — — Ml!)
AI+1(H)

where l = max{i : A101) + + /\,-(/x) S htop(f|A)}.

(Ml!) =l+

 

Theorem 4.2

dimHAS sup dA(/t).
u6£(A.f)

The proof is identical to that of Theorem 2.2, or it may follow from Theorem 4.1

since the inequality P(f, A) S suPue£(A.f) dA (u) can be easily verified.

4.3 Lower Bounds

Some lower bounds for dimH A can be derived using the largest expanding ratio. The
following lemma provides a lower bound for the pointwise dimension of an ergodic

measure. It extends Lemma 2.5 to the Hilbert space setting.

Lemma 4.8 (a) Ifu E €(A, f), then the lower pointwise dimension ofp at x

_. . log/13(scm)
dp(:v,u)—hgyglf 1ng

 

is constant almost everywhere. Use dp(u) to denote this value of dp(x,u).

(b)

 

W)
44“) 3 I. log llefllde)’

The lower bound below follows from Lemma 4.8. It is parallel to Theorem 2.3.

65

Theorem 4.3 Denote by H" the unique number d satisfying
7311(4) 5 P(f"|A, -d10g llef"||) = 0-

Then

dimH A Z sup 5,, = lim sup l—jn.

n—ioo

Another lower bound for dim” A uses the largest (top) Lyapunov exponent.

Theorem 4.4

dimHA Z sup -h_”—(f-l.
pea/u) Mu)

Applying Theorem 4.3 and Lemma 4.8 (b) to conformal maps, we obtain a Hilbert

space version of the formula of Bowen and Ruelle.

Corollary 4.1 If f is a C l conformal expanding map, then

(a) dimHA equals the unique number d satisfying

P(flA,—d108llefll) = 0-
(b) There exists an ergodic measure [1 on A with full dimension, i.e.,

dim}; p = dim” A.

4.4 Example: A Hilbert Space Self-afline Set

Self-affine sets in a Hilbert space can be defined in the same way as in an Euclidean
space. They may be viewed as invariant sets for some expanding systems. We will
examine a special self-affine set where all the upper and lower bounds studied in this
chapter can be easily calculated. This set is an immediate generalization of a well-
known two-dimensional example, Example 1.3. Note that the Hausdorff dimension

itself is very difficult to compute, even in the two-dimensional case.

66

Let H be a real separable Hilbert space and let {e1}l°:l be an orthonormal basis.
Suppose {a,};>;, is a sequence of positive numbers satisfying 2:, a? < 00. The set
R = closure({z:[:1 xze) : 0 S x) S a)}) is totally bounded and hence, is compact. R

is an (infinite-dimensional) rectangle with side lengths a), l = 1, 2,

Fix a sequence of numbers 1 < X1 S X2 S and assume 7 = limHoo XI =

sup, )0 < oo. Define a linear expanding operator T on H by setting
T61: X16), l: 1,2,

Then T'IR is a rectangle with side lengths xfla). Assume Y > 2. Then it is
possible to find b1, b2 6 H and construct expanding affine maps Slx 2 Tx + b1 and
52x = Tx + b2, so that SflR and 517112 are disjoint rectangles contained in R. Set

R1 = STIRU 52-112.
Inductively, if RM; is defined, let
R. = 3:112.-. u $112.4.

Then R1, is a compact set, Rn C Rfl_1 and Rn consists of 2" identical rectangles with

sides xfna), l = 1,2, . . .. The limit set

A = .132. R» = Q Rn
is a self-affine set. It is compact and homeomorphic to the middle-third Cantor set.
Let us calculate the bounds for dimH A.

Define a piecewise linear map f : R1 —+ H in a natural way: fls;‘R = Sj, j = 1,2.
Then f is expanding, fA = A and f | A is topologically conjugate to an one-sided shift
on two symbols. The last assertion tells us htop( f | A) = log 2.

For any x E A, X)(x,n) == )6, so gd(x,n) = ngd(x, 1) = nlog(X1...X[d]Xg[d]). This

gives

W) = P(flm —%gd(x.n)) = P(fIA, —gd(:c. 1)) = logz — Iog<xl...xwxi‘.§1‘l).

67

Therefore, 'D( f, A) is the unique solution d of the equation
X1---X[d]X[1,)-]_[d] = 2. (4.10)

For any x E A, the bottom Lyapunov exponents are log X1 S log X2 S .. .. Thus

for any a E €(A, f), dA(u) is the unique number d satisfying

log2 = log X1 + + log deI + (d — [d]) log X[d]+l-

This equation is equivalent to (4.10). So supp dA(p) = P(f, A) = the solution of

(4.10). Note that the two upper bounds are not equal in general.

Now we look at the lower bounds. For any x E A, [IDxfnll = 7". Therefore
Eu) = P(rli. —dlog llef"|l) = mopvnli) — dlogr" = naogz — dlogy).

Its zero d = in is independent of n and

supDn = H" 2' log-2.
n losx

 

(4.11)

Every x E A has the same largest Lyapunov exponent log 7. So X01) = log? for
all p 6 8(A,f). By the Variational Principle supuegm,” h,,(f) = htop(f|A) = log 2.

Thus
hu(f) _ 1032

#1 A01) _ logy’

 

which coincides with (4.11).

To sum up, we have

log 2

_ S dimHA S solution of (4.10).
logx

 

The two bounds are determined by the expanding ratios XI’S only and are inde-

pendent of the placement of the rectangles STIR and 527112, i.e., independent of b1,

68

b2. However, dimH A relies on both the XI’S and b1, b2. Let us examine the accuracy

of the bounds as b1, b2 vary.

First, when b1 and b2 have certain special relations, dimHA can be arbitrarily
close to [SE—522“ In fact, if the first L components of b1 and b2, with respect to the basis
{e1}, are identical, then it is easy to show dimH A S %, where the right—hand
side approaches [fig—é- as L —> 00.

On the other hand, how well the solution of (4.10) approximates dim” A is a more
subtle and interesting question. The discussion in Example 1.3 seems to suggest that,
in typical situations, the solution of (4.10) precisely equals dimH A. Here the typical
situations should exclude those b1, b2 satisfying the above relations and some XI’S

possessing certain number theoretical properties. To our knowledge, there is little

result even in three-dimensional case.

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