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# Hausdorff dimension of some invariant sets

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Title:
Hausdorff dimension of some invariant sets
Physical Description:
vi, 81 leaves : ; 29 cm.
Language:
English
Creator:
Gu, Xiao-Ping, 1963-
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Mathematics thesis Ph.D
Dissertations, Academic -- Mathematics -- UF
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## Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 77-80).
Statement of Responsibility:
by Xiao-Ping Gu.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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Full Text

HAUSDORFF DIMENSION OF SOME INVARIANT SETS

BY

XIAO-PING GU

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1992

I 531UVUSII VGIUO'U 10 )a'SM'Ko

ACKNOWLEDGEMENTS

It is a pleasure to thank Professor Albert Fathi for his supervision and valu-

able advice. The author would also like to thank the Mathematics Department for

its support and the National Science Foundation for its partial summer financial

support.

- ii -

ACKNOWLEDGEMENTS ................... .ii

ABSTRACT . . . v

CHAPTERS

1. INTRODUCTION ................. ..... 1

2. HAUSDORFF DIMENSION AND CAPACITY . 7

2. 1. Hausdorff Measure and Dimension . 7

2. 2. Frostman's Lemma ................... ...... 9

2. 3. The Capacity . . .. 13

3. THE HYPERBOLIC SET AND LYAPUNOV EXPONENTS .. 16

3. 1. The Hyperbolic Set ...... ........... 16

3. 2. The Topological Entropy . . 19

3. 3. The Uniform Lyapunov Exponents . ... 21

3. 4. The Distortion of a Ball under a Diffeomorphism .. 23

3. 5. The Proof of Theorem I ................. 29

3. 6. The Case of a C2 Flow ................. 34

4. THE HYPERBOLIC SET AND TOPOLOGICAL PRESSURE 37

4. 1. The Hyperbolic Set Revisited . .... .37

111 -

4. 2. Topological Pressure . ... 40

4. 3. A Rough Estimate on Hausdorff Dimension .. 43

4. 4. Under the Pinching Condition . ... 46

4. 5. The Transversals .. .. .. .. 49

5. THE SELF SIMILAR SETS . ... 52

5. 1. The Self Similar Sets . . .. 52

5. 2. The Upper Bound ................... 56

5. 3. The Lower Bound ............ ....... .62

5. 4. Some Continuity in the C1 Topology . .. 64

6. THE HENON ATTRACTOR ................. 69

7. SUMMARY AND CONCLUSIONS . ... 75

REFERENCES . . ... ... 77

BIOGRAPHICAL SKETCH .................... 81

- iv -

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE HAUSDORFF DIMENSION OF SOME INVARIANT SETS

By

Xiao-Ping Gu

May, 1992

Chairman: Albert Fathi
Major Department: Mathematics

We study the Hausdorff dimension for some invariant sets. For a hyperbolic

set of a diffeomorphism or a flow of diffeomorphisms, the uniform stable and

unstable Lyapunov exponents have been defined. Using the uniform Lyapunov

exponents we define a characteristic function. At some real positive number

t the characteristic function equals the value of the topological entropy of the

diffeomorphism on the hyperbolic set. We prove that t is an upper bound for

the Hausdorff dimension of this hyperbolic set. Technically, we use a pinching

condition to cope with the nonlinearity of the diffeomorphism, to prove that the

iterated image of a ball is somehow contained in the image of the ball under the

derivative of the iterated diffeomorphism. Some earlier results are thus improved.

In Chapter 4 we have also given upper bounds for the Hausdorff dimension of

the transverse of a hyperbolic set with stable and unstable manifolds, using the

topological pressure on the hyperbolic set.

v -

Chapter 5 deals with the self similar sets. The self similar sets of iterated

mapping systems have been studied thoroughly in the cases where construction

diffeomorphisms are conformal. We study the case where the construction diffeo-

morphisms are not necessarily conformal. We give a new distortion lemma for the

construction mappings. Using topological pressure on a shift space we give an up-

per estimate of the Hausdorff dimension, when the construction diffeomorphisms

are C1+" and satisfy a K pinching condition for some K < 1. Moreover, if the

construction diffeomorphisms also satisfy the disjoint open set condition we then

give a lower bound for the Hausdorff dimension. We also obtain the continuity

of the Hausdorff dimension in the C1 topology at conformal constructions.

We study the image of a disc under the iteration of a Henon map, which

enables us to find an upper bound for the Hausdorff dimension and capacity of

the Henon attractor. That improves some earlier estimates when applied to the

case of a Henon attractor.

- vi -

CHAPTER 1

INTRODUCTION

For a topological space, especially a subset of an Euclidean space, a variety

of "dimensions" have been introduced and studied. Among them it has been

widely agreed that "Hausdorff dimension" is the most important one to use. In

the case of Euclidean spaces, the Hausdorff dimension agrees with their topo-

logical dimension which are integers. Those subsets with non-integer Hausdorff

dimensions are called "fractals." The sets we study here, i.e., the hyperbolic sets,

the self similar sets, and the Henon attractors are mostly fractals. Understanding

their structures, in particular finding their dimensions, is one of the interesting

open problems.

In 1914, Carath6odory introduced the classical idea of using covers of sets to

define measures. That idea was adopted by Hausdorff in 1919 to define a special

measure which is now called a Hausdorff measure for his study of the Cantor

set. This special measure can be defined for any dimension. However, it either

vanishes or equals infinity in all but at most one dimension, which is now called

the Hausdorff dimension. In his 1919 paper Hausdorff found the dimension of

the Cantor set is log3 2, with a positive Hausdorff measure at that dimension, see

Hausdorff (1919). The properties of Hausdorff measure and Hausdorff dimension

have been thoroughly studied by Besicovitch et al.

In this paper, we study the Hausdorff dimension of the hyperbolic sets, the

self similar sets, and the H6non attractors.

-1-

-2-
Generalizing the concept of a hyperbolic fixed point for a differentiable self

map of a smooth manifold, people introduced the hyperbolic sets. For con-

venience, we call a differentiable one to one self map of a smooth manifold a

diffeomorphism. The hyperbolic sets have been studied by Smale (1967), Mather

(1968), Hirsch and Pugh (1970) among other dynamists. After the introduction

of Pesin's theory, it was realized that Hausdorff dimension of invariant measures

was related to dynamics, and the relationship was studied in its final form by

Ledrappier and Young (1985). It is still an open problem to find the Haus-

dorff dimension of an invariant set. Fathi (1989) gave an upper bound for the

Hausdorff dimension of the hyperbolic sets, using the topological entropy. His

result shows that the Hausdorff dimension can be upper bounded by a fraction of

the topological entropy over the skewness of the hyperbolic set. We use Fathi's

idea. Technically, we use a pinching condition to cope with the nonlinearity

of the diffeomorphism, to prove that the iterated image of a ball is somehow

contained in the image of the ball under the derivative of the iterated diffeomor-

phism. For a hyperbolic set of a diffeomorphism or a flow of diffeomorphisms,

the uniform stable and unstable Lyapunov exponents have been defined. Using

the uniform Lyapunov exponents we define a characteristic function. At some

real positive number t the characteristic function equals the value of the topo-

logical entropy of the diffeomorphism on the hyperbolic set. We prove that t is

an upper bound for the Hausdorff dimension of this hyperbolic set. Some earlier

results are thus improved. Moreover, under the pinching condition, we use the

topological pressure to upper bound the Hausdorff dimensions of a hyperbolic set

and the transversals of the hyperbolic set with stable and unstable manifolds.

For the two-dimensional diffeomorphism with a horeseshoe as a basic set, our

upper bounds for the transversals agree with the identities given by McCluskey

-3-
and Manning (1983). However, due to the increase of dimensions, one does not

expect as explicit an identity for the Hausdorff dimension as the one given by

McCluskey and Manning (1983).

Self similar sets of iterated mapping systems are a special kind of limit

sets. They have been studied thoroughly in the cases where construction dif-

feomorphisms are conformal. Essentially, the conformal constructions are one-

dimensional and one has a simple formula for the Hausdorff dimension. We study

the case where the construction diffeomorphisms are not necessarily conformal.

We give a new version of distortion lemma. Using topological pressure on a shift

space we give an upper estimate of the Hausdorff dimension, when the construc-

tion diffeomorphisms are C1+' and satisfy a i pinching condition for some X < 1.

Moreover, if the construction diffeomorphisms also satisfy the disjoint open set

condition we then give a lower bound for the Hausdorff dimension. We also ob-

tain the continuity of the Hausdorff dimension in the C1 topology at conformal

constructions.

We study the image of a disc under the iteration of a Henon map, which

enables us to find an upper bound for the Hausdorff dimension and capacity of

the H6non attractor. That improves some earlier estimates when applied to the

case of a Henon attractor.

Our main results are the following four theorems. For the upper bound of

Hausdorff dimension of a hyperbolic set using uniform Lyapunov exponents, we

have

THEOREM I. Let f: U -- X be a C2 diffeomorphism with a compact hyperbolic

set K C U, where U is an open subset in the n-dimensional Riemannian manifold

X. Let ps, pI,. p be the stable uniform Lyapunov exponents of f; and let

-4-
", 2 be the unstable uniform Lyapunov exponents of f. Relabel the
numbers A, P and f, pI, as pI, /n with the order

0 > #1 #2 > "' > n.

Define r : [0, n] R the characteristic function by

r(s) = #1 + + [1s] + (s [sI)8[s]+1.

Let A = ent(f IK) be the topological entropy of f on K. Let D = Tr-(-2A), or
D = n if#1 +2+ *+P.+n > -2A. IfAB2 < 1, where A = limAk, B =limBk,
and

Ak = max{max lTzfk l max IITzf -k ,
zEK zEK fk },
Bk = max{maxllTxfk IE- I1, maxlTzf-k IE- I,
zEK zEK

then the Hausdorff dimension HD(K) < D.

Theorem I is proved in Chapter 3. For the upper bound of Hausdorff di-
mension of a C2 hyperbolic set and its transverse using topological pressure we
have

THEOREM II. If f is C2, pinched, then HD(K) < t, where t is uniquely such
that

max{P(ff,,) + P(f K,A) = 01 t + t2 = t} = 0.

f f is C2, pinched, and P(f K,Af) = 0, then

HD(K W(x, f)) < t.

-5-
If f is C2, pinched, and P(flK,tA) = 0, then

HD(K n WU(x,f)) < t.

Theorem II is proved in Chapter 4. For the self similar sets, we give upper

and lower bounds for the Hausdorff dimension. We have

THEOREM III. Let {Qij} be the C1 construction diffeomorphisms for the self
similar set E, satisfying the K pinching condition for some positive number K < 1.

Suppose the derivatives of all {pOij} are Hiilder continuous of order K. If t is the

unique positive number such that the topological pressure P(a, At) = 0, then the

Hausdorff dimension HD(E) < t.

Let {('ij} be the C1 construction diffeomorphisms for the self similar set
E in an Euclidean space R1, satisfying both the K pinching condition for some

positive number K < 1, and in addition the disjoint open set condition. Suppose

the derivatives of all {y ij} are Hilder continuous of order x. If t is the unique

positive number such that the topological pressure P(a, At) = 0, then the Hausdorff

dimension

HD(E) > l.
1+K
Theorem III is proved in Chapter 5. For a Henon attractor, we have

THEOREM IV. Let 0 < b < 1/2. Suppose that A is a compact invariant attractor

under the Henon map Ha,b and an orbit {zn = (xn,yn) = Hn(zo)} is dense in

A. Let R be the bound for A such that for every (x,y) E A we have Ixl, ly < R.
2R+ 1
Denote m = and
b

n-1
A = limsup1 n log(b+ 21 + < 0.
n-oo n m m2
i= 0

-6-

Then the upper capacity

A
C(A) 2 + < 2.
log m

Theorem IV is proved in Chapter 6. Finally, we conclude this work by

discussing the difficulty of relating the Hausdorff dimension of the Henon attractor

to the topological entropy of the Henon map.

CHAPTER 2

HAUSDORFF DIMENSION AND CAPACITY

2.1. Hausdorff Measure and Dimension

Let (X, d) be a metric space and let s be any non-negative real number. Let

K C X be a subspace of X. Let a be a collection of balls, with the union of all

balls containing K. Such a collection a is called a covering of K. For any real

number e > 0, if every ball in a has radius less than or equal to e, then a is called

an e covering of K. Suppose a = {Bi : i E I} and each ball Bi has a radius ri.

We denote by 9iJ(K) the quantity

inf rf
iEI

where the infimum is taken over all e coverings of K. If another real positive

number S < e, then any b covering is also an e covering. Thus 'Hj(K) is no larger

than HK(K). One can define the quantity

WS(K) = lim -Hi(K) = sup Hn(K).
e-*0 >0

We shall call the quantity W(K) the Hausdorff s dimensional measure of K. It

can be easily verified that Hausdorff measure 8 is an outer measure with the

-7-

-8-

following properties:

=,(0) = 0,

N'(UnAn) _5 NS(An),
n
H8(A U B) = -8(A) + iS(B),

where A and B have disjoint closures. Moreover, for any real positive numbers

s, t, e, by the definition,

H,(K) = inf ri
e coverings .i

< et- inf rf
e coverings .
= E-"H^(K).

Therefore, if s < t and VI(K) < +oo, then

Ht(K) < lim et-s (K) = 0 Hs(K) = 0.

This also implies that if t < s and V-8(K) > 0, then

't(K) = +oo.

Hence we have the following identity:

inf{s : H8(K) = 0} = sup{s : "(K) = oo}.

The Hausdorff dimension of the set K is defined by the above number, denoted

by HD(K). It is worth to notice that for an Euclidean space the Hausdorff

-9-

dimension equals the topological dimension, and the Hausdorff measure is equal

to the Lebesgue measure multiplied by a constant factor. Also, it is noticed that

many subsets of Euclidean spaces have non-integer Hausdorff dimensions. Such

sets are called fractals.

The Hausdorff measure was first introduced by Hausdorff in 1919 based on

Caratheodory's idea of using covers to define measures. Since then many research

works have been done. Here we give a well-known Lemma of Frostman which will

be used later when estimating the Hausdorff dimension of the self similar sets.

See Temam (1988) or Hurewicz and Wallman (1948) for other properties of the

Hausdorff dimensions and Hausdorff measures, and examples of fractal sets.

2.2. Frostman's Lemma

The following lemma, Frostman's lemma, is widely used when estimating

Hausdorff dimensions. It gives a necessary and sufficient condition for a subset

of an Euclidean space to allow a non trivial Hausdorff measure. For convenience

we use IBI to denote the diameter of a set B. If d is the metric then

IBI = sup{d(x,y) : x,y E B}.

LEMMA 2.2.1. Let (X,d) be an I dimensional Euclidean space or an I dimen-

sional manifold, with the metric denoted by d. Suppose K C X is a compact

subset. Then K has a positive s dimensional Hausdorff measure if and only if

there can be defined a probability measure p on K, with the property that for all

balls B in (K, d) and some positive constant C, the measure satisfies

jz(B) < C. -B18.

10 -
PROOF: The "if" part is obvious. Given a probability measure 1p on K with

p(B) C IB

for all balls B, we see that probability measure is close to a Hausdorff measure.
In fact, for any given e covering {Bi : i E I} where the radius ri = IBil/2, we
have

Z rf > IBil8/28 > E p(Bi)/28C > p(K)/28C = 1/28C.
iEI iEI iEI
Thus the Hausdorff measure tS(K) > 1/2SC > 0. That proves the "if" part.
To prove the "only if" part, let s > 0 be given such that S(K) > 0, and
let {Bi : i E I} be any covering of balls. For any e > 0, one must have either

sup{IBi : i E I} < e or sup{|Bi| : i E I} > e. Thus it follows that

SIBil' > 7, where 7 = min{es,-' (K)}.
iEI

If e is small enough, then H, (K) > 0, and also 7 will be strictly positive. There-
fore we can fix some > 0 such that for all coverings of balls of K, the sum

EIBil8 > .
iEI

The same conclusion will be true if the balls are replaced by cubes.
For a positive integer m we define a dyadic interval of order m to be any
interval of the form [p2-m, (p+ 1)2-m], where p is any integer. A dyadic cube of
order m is defined as a Cartesian product of dyadic intervals of order m. Consider
now a covering of K using dyadic cubes of order m. In order to introduce a
probability measure on K, let us distribute the mass 2-ms uniformly on each

11 -

dyadic cube of order m that intersects K. We then get a measure pm. On each

dyadic cube of order m 1 we keep the measure pm if its mass does not exceed

2-s(m-1), or we multiply pim by a constant less than 1 in such a way that the

resulting measure has mass 2-s(m-1); that way we get a measure Pm-1. For

each k > 1, the measure 1L'k- is obtained from m_ in the same way, that

is

m-k-1 =- (D) m-k

on each dyadic cube D of order m k 1, with

A(D) = min{1,2-s(m-k-1)tmk(D)-1}.

Clearly p'k is independent of k when k is large enough. Then we can write

mmk

Pm-k = Pm-

Notice the measure pm is carried by the dyadic cubes of order m which intersect

K and we have

pm(D) < 2-s(m-k)

for each dyadic cube D of order m k where k = 0, 1, Each point of K is

contained in some dyadic cube D such that the following equality holds,

pm(D) = clDIS, c = 1-/2,

where I is the topological dimension of X. For each point of K let us consider

the largest such cube D: we get a covering of K by disjoint dyadic cubes, Dn.

12-
Then using the property of 7, one has the following estimate for the norm of the

measure pm:

IIpmll = pm(Dn) = c j~E Dnl > cy
n n
We normalize the measure pm through pm/1 pmll, still denoted as pm. We thus

get a probability measure am with

Pm(D) 5 7-11DI

for each dyadic cube D of order less than or equal to m, and pm is carried by

a given neighborhood of K if m is large enough. From the sequence {pm, m =

1, 2, } we can extract a subsequence ptmi which converges weakly. The limit,

denoted by p, is a probability measure carried by K and

p(D)< 7- lIDl

Now, given an interval of length b, find k such that 2-k-1 < b < 2-k; then

the interval is covered by two dyadic intervals of length 2-k. Given a ball of

diameter b, it is covered by 21 dyadic cubes of order k. Therefore,

pi(B) 5 217-11B1

for each ball B. Let C = 21'-1 and then the "only if" part is complete. I

13-

2.3. The Capacity

There have been many other dimensions defined for a topological space be-

sides the Hausdorff dimension. Among them is the capacity which some people

call fractal dimension and others call entropy dimension. Let (X, d) be any met-

ric space. Let N(X, e) denote the minimal cardinality of an e covering of balls

of X. Note that N(X, e) is the minimum number of radii e balls needed to cover

X. When X is given, N(X, e) is a decreasing function of e. The capacity of X,

denoted by C(X), is given by

S log N(X, e)
C(X) = lim suplo
e-0 log e

The capacity C(X) is also called the upper capacity. In the same way we define

the lower capacity C(X) by

C(X) = lim inf log N(X,e)
e-0 log e

Mandelbrot has suggested an alternative way of defining capacity. Let

Cs(X) = limsup eN(X, e).

The upper capacity is given by

C(X) = inf{s > 0 : C(X) = 0}.

As for lower capacity, if we let

Cs(X) = lim infeN(X,e).
E--0

- 14-

Then the lower capacity is given by

C(X) = inf{s > 0: C(X) =0}.

It is easy to verify the two definitions for upper and lower capacity agree
1
with each other. A covering using diameter e balls is indeed an e covering. So

C8(X) = lim inf eN(X, e)

> lim inf r
E- 0 E coverings E
iEI
= 1"(X).

We have

HD(X) < C(X) < C(X).

It is worth mentioning that the inequality can indeed be a strict one. See Temam

(1988) for such examples. The following fact due to Fathi (1988) about capacity

is very useful. It rewrites the capacities as the limits of sequences.

LEMMA 2.3.1. If 0 < < 1 and 6 > 0, then

log N(X, e) log N(X, Onb)
C(X) = lim sup = lim sup ,
e-0 log e n--oo -n log 0

log N(X, e) log N(X, On6)
C(X) l=m inf = lim inf
e-*0 log E n-+oo -n log 0

PROOF: To simplify notations let 6 = 1 and define

n(e) = [loge/log 0].

- 15-

We have
log e
lim n(e) = oo, and lim = 1.
e-0O e-+0 log on(e)

Since N(X, e) decreasing of e,

N(X, n()) < N(X,e) e N(X, On(c)+1).

It is reasonable to assume e < 1. We have

log N(X, On(e)) log N(X, e) log N(X, On(E)+1)
-loge -loge -loge

Now let e -- 0, and we can obtain the qualities desired. I

For other properties of capacity, see Temam (1988).

CHAPTER 3

THE HYPERBOLIC SET AND LYAPUNOV EXPONENTS

3.1. The Hyperbolic Set

In this chapter we give an upper bound for the Hausdorff dimension of

a hyperbolic set using the uniform Lyapunov exponents. By diffeomorphism

we mean a differentiable one to one map. Let U be an open set in a smooth

Riemannian manifold X and f : U -+ X a C1 diffeomorphism from U onto an

open subset of X. If x is in X, we denote by TzX the tangent space of X at x,

and by Txf : TzX -+ Tf(x)X the derivative of f at x. More generally we denote

by TKX the restriction of the tangent bundle of X to a subset K of X.

DEFINITION 3.1.1. We say that a compact set K C U is hyperbolic for f if

f(K) = K and if there is a Tf invariant splitting of the tangent bundle E =
TKX = E8 Eu such that

lim log(max JTafk IE, II)< 0,
k-++oo k zEK

and

lim 1 log(max ITlf-k IEu 1) <0,
k--++oo k zEK
where the norm is obtained from a Riemannian metric on X.

In the above definition we used the subadditivity of the two sequences

log(max lTzf-k IEu II), and log(max IlTfk IEs II)
xzK zEK

- 16-

-17-
to obtain the existence of the limits. The number r given by

r = max{ lim max lTlfk IE lri, lim max JTzf-k IEU II}
k-++oo zEK k-^+oo zEK

is called the skewness of K. Note that the notion of hyperbolicity does not

depend on the choice of the Riemannian metric. We will need the notion of
topological entropy, which is defined in the following section. Refer to Walters

(1982) for different definitions and properties. The definition of the Hausdorff

dimension is in Chapter 2 and the theory of Hausdorff dimension can be found

in Hurewicz and Wallman (1948). One can also use Temam (1988) as a reference

for the definition and properties of the Hausdorff dimension of a metric space.

The following theorem gives a better upper bound for Hausdorff dimension than

the one obtained by Fathi (1989), also see Fathi (1988) for similar results in

the case of hyperbolic linear toral maps. For related results in dimension 2, see

McCluskey and Manning (1983); and see also Ledrappier and Young (1985) for

Hausdorff dimension of invariant measures. Some earlier related works can be

found in Ledrappier (1981) and Frederickson, Kaplan, York and York (1983).

For the definition of Lyapunov exponents, see Section 3.3.

THEOREM I. Let f: U -+ X be a C2 diffeomorphism with a compact hyperbolic

set K C U, where U is an open subset in the n-dimensional Riemannian manifold

X. Let p', p',-- ps be the stable uniform Lyapunov exponents of f; and let
4I,C ,-- p2 be the unstable uniform Lyapunov exponents of f. Relabel the

numbers p., ... ,p s and /, .. pu2 as p,, Pn with the order

18-

Define r : [0, n] -+ R the characteristic function by

r(s) = 1t + + [s] + (s [sI)p[s]+l.

Let A = ent(f IK) be the topological entropy of f on K. Let D = 7-1(-2A), or

D = n if #1f +2+""'+Pn > -2A. If AB2 < 1, where A = limAk, B = limBk,

and

Ak = max{max ITf Imax IITrf-k },
zEK zEK
Bk = max{max llTfk A IEI max IITzf- IEU II~},
zEK zEK

then the Hausdorff dimension HD(K) < D.

The condition AB2 < 1 is called the pinching condition.

REMARK 1: In fact, after reordering, B = exp p. So B is the largest of these

Lyapunov numbers. Neither A nor B depends on the choice of the Riemannian

metric. Some people would prefer to define A = A1, B = B1, as Hirsch and Pugh
(1970). In the case when K = X, the condition AB2 < 1 implies that the stable

and unstable foliations of f are C1, see Hirsch and Pugh (1970). Remark also

that the condition AB2 < 1 is open in the space of C1 diffeomorphisms with C1

topology.

REMARK 2: We choose sets of the form nfl"f-'B(f'(x), a) to cover K where

x is in a (2m + 1, a) spanning subset (of K) whose cardinality grows according to
2A. The lifting of n imf B-'B(f'(), a) to the tangent space TzX looks roughly

like an ellipsoid whose axes are given by the uniform Lyapunov exponents, and

now the Hausdorff measure of K can be estimated by covering these ellipsoids by

balls. The pinching condition AB2 < 1 is needed in Lemma 3.4.2 to cope with

the effects of nonlinearity on these ellipsoids.

19-

3.2. The Topological Entropy

Let X be a compact topological space and let f : X X be a continuous
self map on X. We shall use open covers of X, denoted by a, /, **, to define

the topological entropy of f which we denote by ent(f).
If a, / are open covers of X, then denote by a V / their join, which is the

open cover by all sets of the form A n B where A E a, B E /. The join of any

finite collection of open covers of X is defined similarly. For the open cover a,
let N(a) denote the number of open sets in a, and let H(a) = log N(a). Since f
is continuous f-1a is also an open cover, with N(f-la) < N(a). Thus we can
introduce the open cover V-1 f-ia and let an = H(Vlf-'Ia).

Now N(a V /) < N(a) N(/), and hence H(a V f) < H(a) + H(/). Thus

the sequence {an} has the subadditivity as follows:

ak+n = H(V n-If a) < H(V-1f-ia) + H(f-k vn-1 f-ia)

< ak + an.

lim lan= lim 1H(f- Vn-1 f-a)
n-oo n n-oo n i=-

exists. We call it the entropy of f relative to the cover a, denoted by h(f, a).
For the continuous self map f of X, the topological entropy is given by

ent(f) = sup h(f, a)
a

where the supremum is taken over all the open covers of X.

20 -

For (X, d) a metric space, Bowen introduced the following alternative way

of defining the topological entropy. Bowen's definition relates the topological en-

tropy to the spanning set. This is used by Fathi (1989) to relate the upper bound

of the Hausdorff dimension of a hyperbolic set to the topological entropy of the

diffeomorphism. We also use this definition when relating the Lyapunov expo-

nents and topological entropy to the Hausdorff dimension. See Walters (1982)

for a proof that in case of a metric space Bowen's definition is equivalent to the

one using covers given above.

For each positive integer define a new metric on X by

dn(x,y)= max d(fi(x),f(y)),
O

where x,y E X. Let K be a compact subset of X and let F C K. For a real

number e > 0 we say that F is an (n, e) spanning set of K if for every point x E K

there exists y E F such that dn(x,y) < e. A subset E of K is said to be (n,e)

separated if dn(x, y) > e for any distinct x, y E E. Denote the smallest cardinality

of any (n, e) spanning set by rn(e, K, f); and denote the largest cardinality of any

(n, e) separated set by Sn(e, K, f). It is easy to check that

rn(e,K,f) sn(e,K, f) < rn(e/2, K,f).

Now let

r(e, K, f) = lim sup rn(e, K, f),
n-.oo

and

s(e, K, f) = lim sup Sn(e, K, f).
n-4-oo

21 -

Notice that r(e, K, f) and s(e, K, f) have a common limit when e -+ 0. The
topological entropy of f, denoted by ent(f), is given by the common limit:

ent(f) = sup limr(e,K,f) = sup lim s(e,K,f),
KCX 6-0 KCX -+0

where the supremum is taken over the collection of all compact subsets of X.

3.3. The Uniform Lvapunov Exponents

Let L : E -- E' be a linear map between n-dimensional Euclidean spaces.
We write

ai(L)= sup inf IIL(c)IIE',
FCE,dimF=i (pF, I~llE=1
and wi(L) = a1(L) .a ai(L) for i = 1,2, ,n. One can verify that

ai(L) 2 a2(L) > ... > an(L),

and that wi(L) is the norm of the map Ai(L) : AiE -. AiE' obtained from L.
It follows that if L' : E' -- E" is another linear map between n-dimensional
Euclidean spaces, then

wi(L'L) < wi(L') wi(L) for i = 1, .. n.

Eu As in Temam (1988),
Consider Tzf : E, Ef(x) and Txf-1 : Ex --+ E-) As in Temam (1988),

we are going to define the "stable" and "unstable" uniform Lyapunov exponents

of f on K, which can be understood as the uniform Lyapunov exponents on the
stable bundle E8, and the unstable bundle E", respectively.
Let K be a compact hyperbolic set for the C1 diffeomorphism f with TKX =
ES*eE as an Tf invariant splitting. Denote that nl = dim E8, and n2 = dim E".
Thus the dimension of X is n1 +n2 = n. The uniform stable Lyapunov exponents
and Lyapunov exponents are defined as follows.

22 -
DEFINITION 3.3.1. For 1 < i < ni define of,(f) = supzeK i(Txf E(). The
definition of oi implies that

ofs (fP+q) < fJ (f P) -o (f).

lim log Cf(
k--oo k

exists, and is denoted by vi. Let p8 = vf, and pf = vf vs_ for 1 < i < nI.
We call those pf the uniform stable Lyapunov exponents off.

A similar argument to above leads to the uniform unstable Lyapunov expo-
nents, which are defined as follows.

DEFINITION 3.3.2. For 1 < i < n2 define Cfu(f) = sup-zKwi(Tzf-1'I E). The
definition of wi implies that

ou(fp+q) < fyu(fp) (Zu((fq).

lim 1 log u(f )
k--+oo
exists, and is denoted by v.. Let y = vU, and pY = v~ vi_ for 1 < i n2.

We call those py the uniform unstable Lyapunov exponents off.

REMARK: It is worth to notice that the unstable uniform Lyapunov exponents
of f are equal to the stable uniform Lyapunov exponents of f-1.

As a matter of fact, the uniform Lyapunov exponents pf and py are negative
numbers.

23 -
3.4. The Distortion of a Ball under a Diffeomorphism

At each x E X, the tangent space TxX is a linear Euclidean space. As in
Fathi (1989), define wrs : E -+ E8, and ru : E -+ EU to be two projections.

Let I|| IE', and 1 || IIE be norms on E8 and Eu respectively derived from the
Euclidean structure of the tangent bundle E. For a vector v E = TxX, our

max norm II||v is defined by

IlvlIE = max{JIlrs(v)IIE., I17u(v)II Eu.

We adopt this max norm throughout this chapter.

Locally around x in X is a neighborhood homeomorphic to an open subset in
Rnl+n2 = Rn. Let Ox denote the origin of the linear space Ez = TxX = E&EEx.

Using the exponential map of some Riemannian metric on X, we can find a CO
map 0 : U X, where U is an open neighborhood of the zero section in TX,
such that for each x in X, the map Oz = 0yInTX is a diffeomorphism onto some
open subset in X. Also Ox(Oz) = x and the derivative of Ox at the origin Oz of

TzX is equal to the identity. Of course, the tangent space of Ez at the origin Ox
is identified to Ez itself. Since K is compact, we can fix a real number 6 > 0 for
all x in K that 0O : B(Oz, 6) X is well defined, and

1
Slvll < IITOr (v)ll < 4|11Ji

when v E B(Oz, S). Choose 6bx > 0 for each x in X, such that

f(0x(B(Ox, b))) C Of(x)(B(Of(x), )).

24 -
Since K is compact, we can fix a small 6' > 0 meeting the needs of all x E K
as 6b. If 6' < 6, write 6 for min{6',6}; noticing that 0z is still well-defined on
B(Oz, 6) for each x E K. Define for v E B(Oz, 6) the map

fz : B(OZ,6) -- Ef() by fx(v) = 0() o f o 0(v).

Since the derivative of Ox at O, is equal to the identity, the derivative of fz at
the origin Ox is equal to Txf, thus T~fx(Oz) = Tzf : Ezx Ef(x)
Now assume that f : U -+ X is a C2 diffeomorphism. Hence fx : B(Oz, 6) -b
Ef(x) is also C2. Using the compactness of K and Taylor's formula, we can write
for v near Ox in TxX,

fx(v) = fx(Oz) + Tfx(Oz)(v) + gx(v) = TZf(v) + gx(v),

where gIgz(v)|J < AIv0 2 for some positive number A not depending on x. Again,
since K is compact, by making 6 smaller if necessary, and A big enough, then
for any v E B(Oz, 6),

fx(v) = Txf(v) + gx(v),

where Ilgz(v)lI <5 Ajv 12, and A > 0 is a constant not depending on x. Repeat
the above argument for f-1 to get

fX-1(v) = Txf-l(v) + x(v),

LEMMA 3.4.1. Let 6 be as above. For any positive number e, there exists a
positive number o10 < 6, such that at each x E K under the maps fx and f-1,

25 -
when 0 < q7 < r0, the image of a ball centered at Ox in Ez satisfies

fz(B(Oz, r)) C (1 + e)Tzf(B(Oz, i)),

fZ'(B(Ox, 9)) C (1 + e)T f-l(B(Ozx, )),

PROOF: This follows easily from the compactness of K and the fact that the
map

(T.f) fj : B(O, 6) TzX

has the derivative identity at Oz.
In order to prove Theorem I, we notice that Np1, ...,Npn are the uni-
form Lyapunov exponents of fN and the topological entropy ent(fN1K) = N
ent(f IK). So if we can show that Theorem I is true for fN, then it is also true
for f. Since AB2 < 1, we can let N be large enough that ANB2 < 1. Without
loss of generality simply assume AiB2 < 1. The next lemma tells the distortion
of balls under iterated maps. The pinching condition is used.

LEMMA 3.4.2. Define a subset in Ex by

Bm(Oz,a) = {v E Ex\\fi()(...fx(v)lI < a,

j1-i()1-f ()I < ai= 0,...,m- 1}

where a > 0 and m is a positive integer. Let f : U -+ X be a C2 diffeomorphism
with a compact hyperbolic set K, and A1B2 < 1. Then for any e > 0, there exists
q > 0, such that

Bm(Oz,a) C (1 + e)m{Tf m(z)(B(Of-m(z), a)) n Tfim()(B(Ofm(x), a))}

26 -
for all m = 1,2,..., and all x in K, when 0 < a < l.

PROOF: Make e smaller if necessary so that

A1B (1 + ) < 1, and B1(1 + e) < 1.

Let 7 be the same as in Lemma 3.4.1, making it smaller if necessary so that
7 < eAl A-1. Lemma 3.4.1 gives

Bi (O, a) C (1 + e)Tf(B(Of-(,), a)) n (1 + e)Tf- '(B(Of(), a)).

So the case m = 1 follows. Suppose that our conclusion is true for m at all x E K.
We show that it is also true for m + 1 at all x E K and complete the induction
process.
It is clear that fx(Bm+l(Ox,a)) C Bm(Of(z),a), and so

Bm+1(Ox, a) C (fx)- (Bm(Of(z),a)) = fJ((Bm(Of(),a)).

Thus for any v E Bm+1(Ox,a), we can have
~-1
ff()(u) = v, where u E Bm(Of(z),a) exists.

Applying the induction hypothesis at the point f(x) for m, we obtain that

u E (1 + e)m{Tfm(B(Of-m+(I), a)) n Tf-m(B(O f +1 (),a))},

and in particular we can pick some

W1 E B(Ofm+1(W), a)

- 27 -

with (1 + e)mTf-m(wl) = u. Moreover,

IPr(u)I I5 (1 + e)mBgma,

and

1ru(u)11 5 (1 + e)mBma

(because IITfm IE, II < Bfm and IITf-m IEu II < Blm). It follows that

lull (1 + e)mBma < a < 6,

and thus

ff)(u) = Tf(x)f-l(u) + gf(x)(u)

is well-defined, with

II1f() (u) I < Alu112 < A (1 + e)2mB2ma2 < e(1 + e)". aA-m-1.

Since maxzEK IITzf Jll A1, we have IlTxf-1(v)ll > Al1 vlj for all x E K and
v Ex. Thus

Tzf-1(B(O, a)) Al B(Of-1(xa),

and

follows. Hence

Yf(z)(u) e(1 + e)mAlm-l(B(Oz, a)) C e(1 + e)mTf-m-l(B(Ofm+1(), a)),

Tf -n(B(Ox, a)) D A nB(Of-,,(Z),a)

28 -
which allows us to pick some w2 E B(Ofm+l(,), a) satisfying

e(1 + e)mTf-(m+l)(w2) = f(z)(u).

Setting w = (1 + E)-1(wl + ew2) E B(Ofm+,l(),a), then

v = Tf-1((1 + e)mTf-m(wi)) + e(1 + e)mTf-(m+l)(w2)

= (1 + )m+Tf-m-l(w),

showing that

v E (1 +e)m+lTf-(m+l)(B(Ofm+l(z),a)).

Hence we conclude that at x E K, one has

Bm+1(Oz, a) C (1 + e)m+lTf-(m+l)(B(Ofm+1(x), a)).

Similarly, applying induction hypothesis at f-l(x) and considering the re-
lation

(fz)- (Bm+l(Ox,a)) C Bm(Of-,(x),a),

one obtains another relation

Bm+1(O, a) C (1 + e)m+lTfm+l(B(O f(m+)(), a)).

So the induction hypothesis holds for m +1, finishing the proof of Lemma 3.4.2. |
In the proof of our Theorem I, we use the following fact:

29 -
LEMMA 3.4.3. IfL : E -+ F is a linear isomorphism between two n dimensional
linear Euclidean spaces, and B = B(O, a) is a ball of radius a, centered 0 in E,
then the image L(B) of B under L is an ellipsoid of axes (a-ail(L),...,a.an(L))
in F.

PROOF: See Temam (1989).
Lemma 3.4.3 shows that rsTfmB(O,,a) and ruTf-mB(O, a) are ellip-
soids with certain axes in E8f and E-m respectively.
fm(z) f- (.), respectively.

3.5. The Proof of Theorem I

First, for any e > 0 we can find some integer N > 0, such that when
k > N, AkB2 < 1 and

Slog o(fk) < ( + e) +.. (+ + )

for i = 1, .. n as well as

log C < ( + +.. + +

for i = 1,... ,n2.
It suffices to prove the theorem for fN since the entropy and the uniform
Lyapunov exponents of fN are those of f multiplied by N. So without loss of
generality we simply assume that

log0f((f)< (A + )+...+ (s + )

for i = 1, .. n and

log 1U(f) < (P 1 + ) + + (Pt +e)

30 -
for i = 1, ,n2, and AIB2 < 1. Relabel {/p7,p} to {pi,*..-,Un} with

0 > l> > /An.

Making e smaller if necessary assume that pi + e < 0 for i = 1,-.. ,n. Let
Pi = exp(pi +e) < 1 for i = 1,2,-- ,n, and let pf = exp(p + e) < 1 for
i = 1,2, ,ni1, as well as pU = exp(py + e) < 1 for i = 1,2, ,n2. Make e
even smaller if necessary so that (1 + e)pi < 1, for i = 1, n.
Define fz and Bm(Oz, a) as in Section 3.4. By Lemma 3.4.2, there exists
r7 > 0, such that when a < q7,

Bm(Oz, a) C (1 + e)m{Tfm(B(Of-m(x), a)) n Tf-m(B(Ofm((),a))}.

Using Lemma 3.4.3, we know that rxTfm(B(Of-m(z),a)) is an ellipsoid in EJ
with axes {aPl,m(x),- ,apnl,m(x)}, where pi,m(X) = ai(Tf"fm IE) for i =
1, n; and it follows that

Pl,m(x) ... Pi,m(X) = wi(TzfmlE') < (PA ... p)m.

In the mean time one can check that ruTfm(B(Of-m(z), a)) 3 ruB(Ox, a).
Similarly, we know that ruTf-m(B(Of-m(), a)) is an ellipsoid in EU of axes

{aPnn+l,m(x), -- apn,m(x)} where Pnl+i,m = ai(Tf-m I Ex), with

Pnl+l,m(x) D *pm+i,m(x) (p P' )m

and rsTf-m(B(O fm(z), a)) D rsB(Ox, a). Notice that we require

Pl,m(x) > ... > pni,m(x) and Pmn+l,m(x) > ... > Pn,m(x).

31 -
Therefore the set Tfm(B(Of-m (), a)) nTf -m(B(Ofm(x), a)) in Ez = E ED
is a product of two ellipsoids, one in E. and the other in Eu, with axes

{apl,m(x), apni,m(x)}, and {apnl+l,m(x), -" ,apn,m(x)}

respectively. By the following fact, if we renumber those Pi,m(x), such that

Pl,m(z) > '" > pn,m(x), and note that {P,--- ,pn} is actually a reshuffle
of these pl, py so that pl >- .. > pn, we can still get pl,m(x) Pi,m(X) <
(Pl--. pi)m for i = 1,.-. ,n.

FACT. Suppose that al > > anl and anx+1 > *.. > anl+n2, and that bI >
S. >_ bnl and bnl+l > *-* > bni+n Assume that aI .. ai < b 1. bi for

i = 1, ,ni and ani+1 ani+j < bnx+1 bni+j for j = 1, --- ,n2. If we
reshuffle those ai and bj to get a' > ... > a' +n and b > .. > bn+ we still
have a't 1...* a' i b for i = 1, n + n2-

As a product of two ellipsoids, the set

Tfm(B(O f-m(), a) n Tf -m(B(Ofm(x), a))

is covered by no more than

J Pl,m(x) ... Pj-l,m()/l(Pj,m(x))j-1

balls of radius apj,m(x), where J is a constant depending on the dimension n and
the compact set K. Hence by Lemma 3.4.2, Bm(Oz, a) is covered by no more
than

J P1,m(x) ... Pj-l,m(x)l(pj,m(x)Yj-1

32 -
balls of radius (1 + e)mapj,,m(). Since K is compact, we can find a constant

C > 0 such that for any x in K, a subset of the form Ox(B(v, l)) with P < 6 in

X can be covered by a ball of radius C38 of X for the metric on X obtained from

the Riemannian metric, where v E B(Oz,6). So Oz(Bm(Ox, a)) can be covered

by no more than J pI,m(x) pj.-l,m(z)/(pj,m(x))j-1 balls of radius

C(1 + e)mapjm(x) < C(1 + e)mpra < Ca.

Denote by N(m, a) the minimum number of sets of the form Ox(Bm(Oz, a))

needed to cover K. We let M be a subset of K such that

{Oz(Bm(Or,a)): x E M}

covers K and the cardinality of M is N(m, a). Since

m
Ox(Bm(Oz,a)) D n f-i[B(fi(x)a/C)],
i=-m

by taking f-m, we know

N(m,a) < r2m+l(a/C, K),

where r2m+1(a/C, K) is the smallest cardinality of a (2m + 1, a/C) spanning set

for f IK (see Section 3.2 or Walters (1982) for the definition of a spanning set).

When m is large enough, and a is small enough, we have

r2m+l(a/C, K) < exp m(2A + e),

33 -
where A is the topological entropy of f over K. Thus

N(m, a) < exp m(2A + e).

Pick the integer j such that j 1 < D < j. Let e' = e(n + 3)/(-pj) > 0
be fixed and let s = D + e'. Here the number e is chosen small enough that
j 1 < s < j. If {Bi : i E I} is an open cover for K, where Bi is a ball of radius
ri, we define

II = max Iri.
iEl
By definition (see chapter 2), the Hausdorff pre-measure 'H7 is given by

inf rf
III
and has been proved to be a non-increasing function of e's. Therefore,

sCa(K) = inf E rf < p(C(1 + e)mapT, s)
III JPl,m(x)...P -l,m(x)
< Pm(x- (pj,m(x))" [C(1 + e)ma]s
zEM [p,m()-i

< exp[m(2A + e)] expm(p1 + ... + pj-1 + (s j + 1)pj + se)

Sexp[m log(1 + e)]. C'Ja

= exp m[2A + pl + ... + lj-1 + (s j + 1)pj + (1 + s)e + log(1 + e)] C8Ja

= exp m[(s + 1)e + 2A + r(D) + e'pj + log(1 + e)] C8Ja8

< expm[(s + 1)e + e'pj + e] C'Ja

< expm[(n + 2)e + (-n 3)e] C'Jas

= exp(-me) CJas

0,

34 -
as m -+ oo. Thus 'H (K) = 0, if a is small enough. So

7S(K) = lim bCa(K) = 0.
a--0

Therefore

HD(K) < s = D + e(n + 3)/(-~j).

Let e -- 0, we get HD(K) < D as desired. The proof of Theorem I is thus

completed. |

3.6. The Case of a C2 Flow

Let {ft : t E R} be a C2 partial flow on the Riemannian manifold X, and

let K be a compact invariant hyperbolic subset of ft. See Fathi (1989) for the

definitions. We have results similar to Theorem I about flows. Let n1 and n2

be the dimensions of the stable and unstable foliations. Let us define stable and

unstable uniform Lyapunov exponents for a C2 partial flow ft. The subadditivity

is used for the existence of limits.

DEFINITION 3.6.1. For 1 < i < nl and 1 < j < n2, define

g(ft) = sup wi(TzftlE.),O(ft) = sup wj(Tzf-tlE).
zEK zEK

It follows that for all p > 0 and q > 0,

n(fpq) d (fP) (fq)

and

Coy(fp+q) < Coy(fP) Co(f).

- 35 -

lim log wO(ft) and lim 1 log (ft)
t-.+oo t t--+oo t

exist, and are denoted by 4i, vi. Let ps = vs (p = V'), and pi = vf vf I for
1 < i < n (respectively, p'j = vj vy1 for 1 < j < n2). We call p! and p the

uniform stable, or unstable Lyapunov exponents for ft.

THEOREM 3.6.2. Let ft be a C2 partial flow with a compact hyperbolic set K
in the n-dimensional Riemannian manifold X. Let it,, ..., be the stable
uniform Lyapunov exponents of ft; and let p,,u, ..., I2 be the unstable uni-
form Lyapunov exponents of ft. Relabel the set {iP ,-- -- i,; pn2} as

{ili,-- ,Pn} with

0 > P1 > P2 2 > ".. Pn-

Define r : [0, n] -+ R by

T(s) = p1 + ... + l[s] + (s [S])P[]+l.

Let A = ent(f1 IK) be the topological entropy of f on K. Let

D = r-1(-2A),

or D =n if P +p 2 + ... + pn > -2A.

If AB2 < 1, where A = limAt, B = limBt, and

At = max{max IITzft 1l max IITzf-t I 11,
zEK xEK
B = max{ma.lftx IT IE IImax ITf-t IEu I1t,
zEK xEK

36 -

then the Hausdorff dimension of the hyperbolic set K of the flow satisfies

HD(K) < D + 1.

The proof of the above Theorem, which is about the case of follows, is much

the same as that of our Theorem I.

REMARK: The geodesic flows of 1/4 pinched negatively curved Riemannian met-

rics on compact manifolds satisfy the hypothesis of our Theorem 3.6.2 for any

subset of the unit tangent bundle which is invariant under the geodesic flow.

CHAPTER 4

THE HYPERBOLIC SET AND TOPOLOGICAL PRESSURE

4.1. The Hyperbolic Set Revisited

Now let us use the topological pressures on a hyperbolic set to bound the

Hausdorff dimensions of the hyperbolic set and its transverse with the stable

and unstable manifolds. For convenience in this chapter, we use an alternative

definition of the hyperbolic set. Mather has pointed out that by choosing an

appropriate Riemann metric, any set that is hyperbolic according to the previous
chapter will satisfy the definition we give here. We prove Theorem II given in

the introduction here by proving Theorems 4.2 through 4.4. Our Theorem 4.1 in

this chapter gives a generally rough estimate, and we will see the introduction of

the pinching condition enables a better estimate.

Let U be an open set in a smooth Riemannian manifold X and f : U -- X

a diffeomorphism from U onto an open subset of X. If x E X, we denote by TxX

the tangent space of X at x, and by Txf : TxX -- Tf()X the derivative of f at

x. More generally we denote by TKX the restriction of the tangent bundle of X

to a subset K of X.

DEFINITION 4.1.1. Let K C U C X be a compact subset of a Riemannian

manifold X. We say that K C U is hyperbolic for a diffeomorphism f if f(K) =

K and if there is a Tf invariant splitting of the tangent bundle E = TKX =

37 -

- 38 -

Es e Eu such that

maxf{||llf IE, II, IITf-1 IE 1} < 1,
zEK

where the norm is obtained from an appropriate Riemannian metric on X.

We call r = maxxK({IITf IE II, ITaf-1 IEu II} the skewness of K. This
definition of hyperbolicity is slightly different from the ones given in the previous
chapter and Fathi (1989). However, if K is hyperbolic in the sense of Chapter
3, we can adapt a metric to K as Mather (1968) did. Then under this Mather

metric, K is hyperbolic under our definition here. Refer to Walters (1982) or
the next section of this chapter for the definition of the topological pressure for

a function A and the map f K : K -+ K, and the variational principle. In this
chapter, we let the real valued negative function A : U -- R be defined by

A(x) = log maxi{ ITf lE II, IITZf-1E II},

Clearly the function A is continuous if f is C1. For the Hausdorff dimension of a
hyperbolic set K, first we have

THEOREM 4.1. If f is a C1 diffeomorphism with a hyperbolic set K, and at
some real positive value s we have the topological pressure P(flK, s) = 0, then
HD(K) < s.

We say f satisfies a pinching condition if AB2 < 1 where

A = max{ lTf |I, IITxf-111}
zEK

and B = maxZEK{A(x)}. We also define two characteristic functions AX and A"
as follows.

39 -
For L : E -- F a linear map between n dimensional normed spaces E and
F, we write

ai(L) = sup inf ILp IIF,
SCE,dim S=i IIlsIE=l,PES
and wi(L) = al(L) .. ai(L) where i = 1, *. ,n. Here L induces A'L : A'E -
A'F; and in fact wi(L) = | A' LIIF. If t is not an integer, we write

wt(L) = al(L)... a[t](L)a[tl+(L)t-[t.

It is noticed the linear map L sends any ball of radius r in E to an ellipsoid in F

with axes given by (ral(L), ran(L)) .
Now consider the splitting of TKX = Es8 Eu. Suppose that dimE8 =

ni, Eu = n2, where n1 + n2 = n. When x E K define

A'(t,x) = logwt(TxflE ) for t E [0,ni],

and

A"(t, z) = log wt(Txf-1^E) for t E [O, n].

Write A)(x) = AU(t, ) and Af(x) = A'(t, z). Under the pinching condition and
using functions As and Au, we have

THEOREM 4.2. If f is a C2, pinched diffeomorphism with a hyperbolic set K,
then there is a unique real non-negative value t such that

max{P(f lK, A) + P(flK, A))Itl + t2 = t} =0.

Moreover the Hausdorff dimension of the hyperbolic set K satisfies HD(K) < t.

We also give the upper bounds for the Hausdorff dimension of the transver-
sals of the hyperbolic set to the stable and unstable manifolds.

40 -

THEOREM 4.3. If f is a C2, pinched diffeomorphism with a hyperbolic set K,
and at some real positive value t we have the topological pressure P(f K, A') = 0,

then the Hausdorff dimension in the direction that transverses to WU(x,f) is
given by

HD(K n (x, f)) < t.

THEOREM 4.4. If f is a C2, pinched diffeomorphism with a hyperbolic set K,
and at some real positive value t we have the topological pressure P(fIK, N) = 0,

then the Hausdorff dimension in the direction that transverses to W(x, f) is
given by

HD(K n W"(x,f)) < t.

REMARK: The inequalities in Theorems 4.3 and 4.4 are qualities when K is the
basic set of a two dimensional horseshoe. This fact can be found in McCluskey

and Manning (1983).

4.2. Topological Pressure

We first give the definition of topological pressure using the open coverings.

Then we also give the definition using either spanning sets or separated sets.

Both definitions are equivalent. See Walters (1982) for more details and the

variational principle. The concept of topological pressure in this type of setting

was introduced by Ruelle and studied in the general case by Walters.

Let f : X --+ X be a continuous transformation of a compact metric space

(X, d). Let C(X, R) denote the Banach algebra of real-valued continuous func-
tions on X equipped with the supremum norm. The topological pressure of f will

be a map P(f,.) : C(X, R) -- R U {oo} which will have good properties relative

41 -

to the structures on C(X, R). It contains topological entropy in the sense that

P(f, 0) = ent(f) where 0 denote the member of C(X, R) which is identically zero.

Let A E C(X,R). For x E X denote Sn~(x) = En-1 A(fx). Let a be an

open cover of X and denote

qn(f, A, a) = inf{ I inf exp((SnA)(x)): / is a finite subcover of VnO1 f-ia},
BE# zEB
pn(f, A, a) = inf{ sup exp((SnA)(x)): /3 is a finite subcover of Vno1 -i a}.
BE#zEB

Notice that qn(f, A, a) < Pn(f, A, a) and the subadditivity of log pn(f, A, a) is due

to the relation pn(f, A, a) Pm(f, A, a) > pn+m(f, A, a). Let

q(f, a) = lim sup log qn(f, A, a),
n--+oo n
p(f, A,a) = lim -logpn(f,A, a).
n-oo n

Recall lal denotes the largest diameter of sets in a. Write

q(f,A, e) = sup{q(f, A, a): a is an open cover of X, laI < e},

p(f, A, e) = sup{p(f, A, a) : a is an open cover of X, Ic a e}.
a

Now both sides of the following identity gives the quantity P(f, A) which is called

the topological pressure:

lim q(f, ,e) = lim p(f, e).
E-O C--0

We have defined the (n, e) spanning set and the (n, e) separated set in the

previous chapter. Now we can use them to give alternative definitions of the

- 42 -

topological pressure. Let

Qn(f, A,e) = inf{ exp((SnA)(z)) : F is (n,e) spanning for X},
zEF
Pn(f A, e) = sup{ exp((SnA)(x)) : F is a (n,e) separated subset of X}.
zEF

Then define

Q(f, A,e) = limsup -log Qn(f, A,e),
n-+oo nf
P(f, A, e) = lim sup log Pn(f, A, ),
n--oo n

Now the topological pressure is given by

P(f, A) = lim Q(f,A,e) = lim P(f,A,e).
--*0 e-0

The variational principle shows the topological pressure can be computed

using f invariant measures.

THEOREM 4.2.1. (The Variational Principle) Let f : X -- X be a continuous

self mapping of a compact metric space and A a real valued function defined on

X. Then the topological pressure can be given by

P(f, A) = sup{hp(f) + Ad : p E M(X, f)}

where hp(f) is the entropy of f as a map that preserves the measure p and

M(X, f) is the set of all measures invariant under f.

See Walters (1982) for an introduction to the properties of the topological

pressure, as well as the proof of the variational principle and equilibrium states,

and how pressure determines the invariant measures.

43 -
4.3. A Rough Estimate on Hausdorff Dimension

Lemmas 4.3.1 and 4.3.2 are obvious.

LEMMA 4.3.1. Iff : K -- K with K compact and A1 < A2 are two real functions
on K, then P(f, l) < P(f,A2).

LEMMA 4.3.2. IfA is a negative real valued function then P(fIK, sA) is a strictly
decreasing, continuous function of s.

For x E K let TzX = E, E E, be the tangent space and 7r : TzX -
E r, : TxX -+ E, be the projections. If v E TxX, and if 1I'(v)|jIE, IIru(v) IIE
are norms induced from the adapted metric, then the norm

|Ilvl = max{llJ7(v)liE,, II7r(v)lE }

is equivalent to the one induced from the adapted metric. With the new norm K
is still hyperbolic under our definition. Let Ox : B(Oz, a) X be the exponential
map. Since K compact and since TOz = I the identity, we can find 6 > 0 such
that Ox is well-defined for all x E K and a < 6, with

1
lIv wil < d(Oz(v),0x(w)) < 211v wll.

Define fz = Of1 fOx to be the lifting of f. Define the set

Bm(O,,a) = {v E EIIIffi(,)... fx(v)II < a,

1ff1-()---- (v)11 < a, i = 0,...,m- 1}

as in Chapter 3.

44 -
PROPOSITION 4.3.3. Let f, K, be as in Theorem 3.1. IfP(f K,sA) < 0, then
the Hausdorff dimension HD(K) < s.

PROOF: Write fj(v) = Tzf(v) + g(v). Let e be a small number such that

e < -P(f,sA) s.

There is 6 > 0 with fxB(Oz,a) C (1 + e)TzfB(Oz,a) when a < 6. Thus
Bm(0, a) C (1 + e)mB(Ox, rm(x)) where

rm(x) = a max{exp(SmA(x)),exp(SmA(x))},

with SmA(z) = ,1 (f-ix), and SmA(x) = E 1 A(fix). Notice that

rm(X) < a[exp mA(x) + exp SmA(x)].

Let K1 C K be a maximal (2m + 1, -a) separated subset. Then for any
x E K In K, there exists y E K1 such that

max{d(x,y),... ,d(f2m+l(x),f2m+l(y))} < a.
4

So {Ozf-mBm(Ofym(z),2a)x E K1} is an open cover for K. Since 0f = fO,
it follows that f-mOfm(z)Bm(Ofm(x), 2a)lx E K1} is an open cover for K. Let
K2 = fmK1 which is clearly (m, ,a) separated. {OzBm(Oz, 2a)|x E K2} is also
an open cover for K. Let the number M be large enough and 6 small enough
that

1 1 1
log Pm(f, sA -a) < P(f, sA -a) + se/2 < P(f, sA) + se < -2se.
m 4 4

- 45 -

Here as in section 4.2,

Pm(f, sA, a) = sup{ expSm(sA)(x)IE is an
zEE
1
(m, 4a) separated subset of K},
4

and

1 1 1
P(f,sA, A ) = limsup- logPm(f, s, -a)
4 m-oo m 4

P(f, sA)= lim P(f, sA, a).
a--+O 4

Since P(f-,sA) = P(f,sA) < 0, we can require

1 1
-log Pm(f-,sA, a) < -2se.
m 4

Thus we have

1 1
Pm(f, sA, a) < exp(-2mse) and Pm(f-l,sA, -a) < exp(-2mse).
4 4

In particular, since Pm(f, sA, a) is the supremum,

exp(SmsA(x)) < exp(-2mse), and E exp(msA(x)) < exp(-2mse).
xEK2 xEK2

Now consider the Hausdorff s measure corresponding to the open cover

{OxBm(Ox,2a)Ix E K2}.

46 -
The radius of each open set OxBm(Oz, 2a) is less than 4a. Let a < 6/2. Thus

H4ha(K) E \ OzxBm(Ox,2a) 8
ZEK2
< 4s8 Bm(OX,2a)\8
xEK2
< 48s (2a)s(1 + e)sm(exp SmA(x) + exp SmA(x))
xEK2
< (8a)8 exp(sme)2exp(-2mse)

= 2(8a)8 exp(-sme) -+ 0,

as m -+ oo. So i a(K) = 0; and HD(K) < s. I
Lemma 4.3.1 and Proposition 4.3.3 give a proof of Theorem 4.1.

PROOF OF THEOREM 4.1: Let t = inf{sj P(fIK,sA) < 0}. Then HD(K) < t.
It is clear that P(f K,tA) = 0 and t is unique in view of Lemma 4.3.1. I

4.4. Under the Pinching Condition

The following results will prove Theorem 4.2.

PROPOSITION 4.4.1. Let f be C2, pinched, and K its hyperbolic set. Ift is such
that
max {P(f K, A ) + P(flK, A)} < 0,
tl+t2=t
then the Hausdorff dimension HD(K) < t.

PROOF: By Lemma 3.4.2, for any e > 0 there is b > 0, when r < 6,

Bm(Oz, r) C (1 + e)m{Tf-.m(z)fmB(Of-m(Z), r) n Tfm(z)f-mB(Ofm(z), r)}
Thus, considering their projections we have

rsBm(Ox, r) C (1 + e)mTfm(z)fmrsB(Of-,m(),r),

- 47 -

and

ruBm(Oz, r) C (1 + e)mTf,(z)f-mruB(Ofm(z), r).

Reshuffle

a1(Tf-m()fmlE*),... ,an(Tf-m(x)fm IE' );

al(Tfm(I)f-m Eu)," *, In2(Tfm(x)f-mIE )

to get al(m,x) < ..- < an(m,x). It follows that Bm(Ox,r) is contained in an
"ellipsoid" of axes {ral(m, x), ,rann(m, x)}.

Let j = [t] + 1. Use balls of radius raj(m, x) to cover the above ellipsoid of
axes {rca(m,x ), ,ran(m, x)}. At most

Qail(m, x) ... aj(m, x)/aj(m, x)3

balls are needed where Q depends only on the dimension n, the metric we choose
and the compact hyperbolic set K. Notice that the number aj(m, x) < 1.

Let K1, K2 be two subsets of K as in the proof of Proposition 4.3.3, where
the collection {0xBm(Oz, r)jx E K2} is an open cover for K. The balls of radius
raj(m,x) needed to cover Bm(Oz, r) is at most

Qal(m, x) aj(m, x)/aj(m, x)s.

The image in K under {Oz : x E K2} is an open cover for K, with radius less

- 48

than 4r, and the corresponding Hausdorff t measure is estimated as the following:

'Ht4r(K) < (1 + e)m Z rtQ4t al(m, x) .. aj(m, x)/aj(m, )j aY(m, (x)
zEK2
= C(1 + )m E al(m,x). ..a-_l(m,x)aj(m,x)t-j+1
zEK2
= C(1 + e)m E Wt2()(Tfm(X)f-m IE )Wt (x) Tf-m() fm lE)
xEK2
m
zEK2 i=1

Here C = rtQ4t. At least one of t1, t2 is an integer. Also t1 + t2 = t. There are
at most n(n 1) pairs of such (tl(x), t2(x)) for each x. For each pair of available

(t1, t2), we have

P(f-1 IK, ) + P(f IKAu) < O.

Thus we can require

1 log[Pm(f-1IK s,tl)Pm(flK, Au,t2)] < -2e
m

for all available pairs (tl, t2) and all large m. So,

m m
r(K) 5 Cexp(mexp() exp( Au,t2(fix)). exp( As, (f-ix))}
tl+t2=t zEK2 i=1 xEK2 i=1
< Cexp(me) : Pm(f-1 K,~s,tj)Pm(fIK, u,t2)
tl+t2=t
< C exp(me) exp(-me)

= Cexp(-me) -> 0.

Therefore -t4,(K) = 0; and HD(K) < t.
The proof of Theorem 4.2 follows immediately from the above Proposition

and Lemma 4.3.2.

49 -
4.5. The Transversals

By the Stable Manifold Theorem, the stable manifold WS(z, f) is locally
embedded in X, with TzWs(x, f) = E.. Similarly TxWU(x, f) = Eu. Now let

X0 : TzW(x, f) = E W(x, f) C X

be the exponential map for W8(x, f) at x and

0 : TzW"(x,f) = EU WU(x,f) C X

be the exponential map for WU(x, f) at x. Define : TzX = Es Eu --E T X
by

(C(w ) v) = Ol-0(w) + OOleu(v)

for w E E' and v EE. Now the exponential map O can be modified to obtain
a map

z : TzX = Es D Eu -+ X

by O ,x = Cx, with the property that TCxz(Ox) = identity. By the stable
manifold theorem, TzWS(x, f) = Ex and

ox(E n B(O, r)) C W(x, f).

It is noticed that zx is defined locally and since K compact there is r > 0 such
that Ox is well defined on B(0x, r) C TxX for all x E K. Then similarly to
Chapter 3, we let f(v) = f( )fz(v) and define the set Bm(Oz,r) through f.

A distortion lemma similar to Lemma 3.4.2 can be obtained.

-50 -
PROPOSITION 4.5.1. Let f and K be as in Theorem 4.3. For any x E K, let
W8(x,f) be the stable manifold at x. If the topological pressure P(f, A) < 0,
then the Hausdorff dimension HD(K n Ws(x, f)) < t.

PROOF: Similar to Lemma 3.4.2,

Bm(Oz, r) C (1 + e)m{Tf -m(Z)fmB(0f-m(x), r) n T ( (z)f-mB(0fm(,), r)},

we have
7rsBm(Ox, r) C (1 + e)mTf_m(x)fmrsB(-m(z), r).

The right side is an ellipsoid of axes (1 + e)mr{al(m,x), an,(m, x)}. Let
j = [t] + 1. The number of balls of radius raj(m, x) needed to cover rsBm(Oz, r)
is at most
Caq(m, x) aj(m,x)/aj(m, x)j.

Let M1 C K n Ws(f-mx, f) be a maximal (2m + 1, rr) separated subset,
such that M1 U {y} is not (2m + 1, 1r) separated if y E K n WS(f-mx, f) and

y M1. Then the set M2 given by

M2 = fmMI C K n W(x, f)

is maximal (m, r) separated; and {txzrsBm(z0,r)jx E M2} is an open cover
for K n WS(x,f). Under the collection of maps {xzlx E M2}, the image of
small balls of radius rarj(m, x) used to cover Bm(0x, f) have radius less than 4r.
The corresponding Hausdorff t measure for K n W(x, f), the quantity 7H4r(K n
Ws(x, f)) is less than

E 4Cr(1 + e)maj(m, x)tal(m, x) ... aj(m, z)/aj(m, x)
xEM2

51-

Since

aj(m, z)tal(m, x) .. aj(m, z)/aj(m, x)

= ; a(m, z) ... j_ (m, x)aj(m, x)t-j+l

= W(Tf-mfmlEs)

< wt (Tf-m f IES)W(Ty-m+lJEf .E). T (Tf-f IE8)

= exp As,t(f-m.x).. .exp As,t(f-x)
m
= exp As,t(f-ix)
i=1

and r(1 + e)maj(m, x) < r, we have

m
M",r(K n Ws(x, f)) < 4Cr(1 + e)m E exp(- Asf-ix)
zEM2 i=1
< 4Cr exp(me)Pm(f, As,t, -r)

< 4Cr exp(me)exp(-2me) -+ 0,

as m -+ oo. So t4rr(K n W(x, f)) = 0. It follows that Vit(K n W8(x,f)) = 0

and the Hausdorff dimension HD(K n W8(x, f)) < t. I

The proof of of Theorem 4.3 follows immediately from the above Proposition.
The proof of Theorem 4.4 works similarly.

CHAPTER 5

THE SELF SIMILAR SETS

5.1. The Self Similar Sets

In this chapter we shall construct the self similar sets. Then we use the
topological pressure defined in Chapter 4 to bound the Hausdorff dimension of

self similar sets. Finally we discuss the continuity of Hausdorff dimension at
conformal constructions.

The construction of a self similar set starts with a k x k matrix A = (aij)
which has entries zeroes and ones, with all entries of AN positive for some N > 0,

see Bedford (1988). For each non-zero aij we give a contraction map Pyij : R -
R1 with ||lpij(x) pij(y)l| < c||x y|l, where c < 1 is a constant and we are using

the Euclidean norm on R1. Define the Hausdorff metric by

d(E, F) = inf{6d(x, F) < 6 for all x E E, and d(y, E) < 6 for all y E F}

in the space C of all nonempty compact subsets of RI. See, for example, Hutchin-

son (1981) or Falconer (1990). The map ( on the k-fold product space Ck given
by

(FI,.. ,F) = (UiP(F,),.. ,U=1Pkj(F))

is a contraction map. By the Banach Fixed Point Theorem the contraction map

- has a unique fixed point in Ck, i.e., a vector of compact non-empty subsets of
52 -

53 -
R1, denoted by (El,..., Ek) E Ck, with

Uaii=1Pij(Ej) = Ei.

The union E = U=l Ei is called a self similar set.
Let

E = = {(xO, l,..., n,...)I1 < zi < k and a.,iz+l = 1 for all i > 0}

be the shift space with the following metric: for x = (xo, xl,... ), y = (yo, l- *)
in E, d(x,y) = 2-" if and only if n = min{mlxm Z ym}. Let a be the shift map
of E and let r : E E be given by

7r(xo, X1,..., n,...) = the only point in nn> l (xozz1 X 22... zz-n+l(Exn+l).

It is clear that r is a H6lder continuous, surjective map. We will denote the
composition Voxl "- Xn-iXn by Vzo--.,. Also, we assume all pij be C1 diffeo-
morphisms and denote the derivative of Cpij at a point x by Tzxpij or Tyij(x).

DEFINITION 5.1.1. The j-th Lyapunov number of a linear map L, denoted by
aj(L), is the square root of the j-th largest eigenvalue of LL*, where L* is the
conjugate of L. Write

wt(L) = al(L)... a[t](L)a[t]+l(L)t-[t]

For a set of construction diffeomorphisms {Qpij}, the function At : E -- R for
each t E [0, 1] and x = (x0xl ...) E E is defined by

At(z) = log al(Tzozl(iroz)) + -+ log a[t](Tvczo-(7roa))

+ (t [t]) loga[t]+l(T0Zxo1(7rza))

= logWt(Tzoxz1(7ra)).

54 -
The constructions and dimensions of self similar sets have been studied by
several authors under various restrictions. In this chapter we relax the restrictions
on construction diffeomorphisms to a K pinching condition, which is defined as
follows.

DEFINITION 5.1.2. We say that a C1 smooth homeomorphism Pij satisfies the
K-pinching condition if the derivatives satisfy the following

IITz Piill+ IIT,,io(z)Pj'I < 1

for all x E E.

REMARK: If TzijTx*- has eigenvalues

al,ij(x)2 > ... > al,ij(x)2

where TZxcp denotes the conjugate of Txrij, the numbers a, ij(x),...,a ,ij(x)

are Lyapunov numbers with

1 > al,ij(x) > ... > al,ij(x) > 0.

Then the pinching condition is equivalent to al,ij(x)1+' < alij(x).

For the definition and properties of Hausdorff dimension, refer to Chapter
2 or Kahane (1985). Also, we use the definitions and notions of Walters (1982)
III given in the introduction is proved in this chapter through Theorem 5.1 and
Theorem 5.2.

55 -

THEOREM 5.1. Let {(yij} be the C1 construction diffeomorphisms for the self
similar set E, satisfying the K pinching condition for some positive number K < 1.

Suppose the derivatives of all {yij} are Holder continuous of order K. Ift is the
unique positive number such that the topological pressure P(a, At) = 0, then the
Hausdorff dimension HD(E) < t.

Let us recall the disjoint open set condition on the construction of self similar
sets, see Hutchinson (1981). It states that for each integer i from 1 to k there is
a non-empty open set Ui such that

U pij (Uj)CUi, and Pij(Uj) n ik(Uk) = if j k.
aij=1

For n > 0, denote Un(x) = WX2ozz X2 ... xn,_-z(Uz,). It follows that Ei C Ui;
and that the collection {Un(x) : x E E} is pairwisely disjoint for each fixed n.

THEOREM 5.2. Let {1oij} be the C1 construction diffeomorphisms for the self
similar set E in RI, satisfying both the K pinching condition for some positive
number K < 1, and the disjoint open set condition. Suppose the derivatives of all

{fcij} are Hilder continuous of order K. If t is the unique positive number such
that the topological pressure P(a, At) = 0, then the Hausdorff dimension

HD(E) > lK.
1+K

REMARK: We call a C1 diffeomorphism C1+4 if its derivative is H6lder continu-
ous of order K. If we fix the construction to be C1+) for some # > 0 but let -- 0
for the i pinching condition, then our upper and lower bounds will coincide with

the estimate for conformal cases as in Bedford (1988).

56 -
Theorem 5.1 is proved in Section 5.2, and Theorem 5.2 is proved in Section

5.3. As a corollary of Theorems 5.1 and 5.2, in Section 5.4 we will also discuss

some continuity in the C1 topology of the Hausdorff dimension at conformal
Cl+' constructions under the disjoint open set condition. For discussions of

the constructions of self similar sets using similitudes and their dimensions, see
Hutchinson (1981), Mauldin and Williams (1988). For the constructions using
conformall" contraction maps, see Bedford (1988). Other related works can be

found in Dekking (1982), Bedford (1986) and Falconer (1990).

5.2. The Upper Bound

LEMMA 5.2.1. If all construction diffeomorphisms Wij are C1+' and satisfy the

K-pinching condition, then for any e > 0, there exists 6 > 0 depending only on e,

such that for all x E E, all a with 0 < a < 6, and all x = (0,xI, .) in E, all
integers n > 0, we have

(5.2.1) po...-zB(x,a) C o-...zX(x) + (1 + e)nT zo...zB(O,a).

Here B(x, a) denotes a ball of radius a centered at x in RI.

PROOF: Using Taylor's formula, for any y, w E R1,

(5.2.2) 'poxl(y + w) = zxoxl(y) + Tyxzozx(w) + rzozx(w,y).

Since E is compact, we can find some constants C > 0 and c > 0, such that for

all y E E and w E R1 with I|w|| < c, we have I|rzozx(w,y)ll < CIIwl1+K. We will

set also b = minzEE,i,j{al,ij(x)}.

-57 -
Fix any small e > 0. Since E is compact and all construction diffeomor-
phisms satisfy the r. pinching condition, without loss of generality we can assume
e to be so small that for all pairs (i,j),

(5.2.3) 11(1 + e)Tzcijll < 1 for all x E E;

(5.2.4) (1 + e)cal,ij(x)l+' < aij(x) for all x E E.

Pick 6 > 0, with 6 < min{c, (be/C)1/"}. Thus b < eatij(x)/C, for all x E E
and all pairs of (i,j). Let a < 6, and pick any w E R1 with IIwll < a. For any
x in E, lIrxoz,(w,x)jl < CIIwll+I < Cal+' < aCb t < aeal,zox,(x), and thus
rzozx(w,x) E eai,xox1(z)B(O,a). Since ealj,o(a(x)B(O, a) C eTzoxl B(O, a), it
follows from (5.2.2) that

xzoz (x + w) = xox, (x) + TxxPoxl (w) + rZol (w, x)

E pzozx (x) + Tzcpzox B(0, a) + eTroxozl B(0, a)

= zozlX(X) + (1 + e)TxzPxoxB(0,a).

That gives (5.2.1) for n = 1. Now the induction hypothesis gives

PxoXZ...x B(x, a) = Pzxox s ...zx B(x, a)

C zoz~l x[-x...z(x) + (1 + )n-lTz ,...xZB(0,a)].

Using (5.2.2):

Pxoxi k~z...Xn(x) + (1 + e)n-1Tvx,...X(w)]

= pzo...n (x) + (1 + E)n-lTvzo...x,(w)
+ rxox,((1 + e)n-lTvxl...x,,n(w), IP...,,( ()).

58 -
Because of (5.2.3), 11(1 + e)n-1T l...x,(w) < |11wI < a, where w E B(O,a).
Using (5.2.4), we have

Ilrzxoz((1 + e)n-1TwzZ...Z,,(w), wx...x,(x))II
< cl|(1 + e)n-1Tlz....,,(w) 1l+'c

< C(1 + e)(n-1)(l+) [alXz12(z2...n(z,()) ... acl,znx,,(x)IIwII]1+
< C(1 + e)n-1al,,,, z))...a,2,n 1 (x)Illwll1+

< (1 + E)n-la"CCal,T(,,1( ..., (x))... al,,n-_X(X)IIwII
< e(1 + e)n-lao,xozx(I -x ... (x)) ai,zxli2(zx...x,(x))...al,zn-_X,,(x)a.

On the other hand Tz' ijB(O, a) D al,ij(x)B(O, a), and it follows that

Tzxpxo...x,,B(O, a) D aq,zoz (Czlr...z,, (X)) **' al.n-lX,, (x)B(O, a).

Hence rx0zo((1 + e)n-1lTW ... ,, (w), cp...:,,(x)) E e(1 + e)n-1Tzx zo...z,,B(O, a).
Therefore,

Vxzoz [kx...xx) + (1 + E)n-1T Z,...X(w)]
E cpx...x,(x) + (1 + e)"-1Txzo ...,B(O, a) + e(1 + e)n-1Tz~zo...,B(O, a)

C Vo...zxn (X) + (1 + E)nTz zo.... B(0,a).

Thus (5.2.1) is true for n. That completes the induction process. I
I have learned that Jiang (1991) has a distortion lemma for a regular non-
conformal semigroup, which is a semigroup of pinched contracting diffeomor-
phisms. His version is stronger than our version here. However, for our purpose
of estimating Hausdorff dimensions here, our version is strong enough.

59 -

PROPOSITION 5.2.2. If all construction diffeomorphisms pij are Cl+ and sat-

isfy the K-pinching condition where 0 < i < 1, and if the topological pressure

P(o, At) < 0 where a is the shift map in E, then the Hausdorff dimension

HD(E) < t.

PROOF: Choose small e > 0 satisfying both (5.2.3) and (5.2.4), with

P(a, At) < -2te.

By Lemma 5.2.1, there exists 6 > 0 such that (5.2.1) holds for all integers n > 0

and each x E E, when 0 < a < 6.

We fix a < 6 small enough, and a positive integer n big enough, such that
(See Walters (1982) or Section 4.2 for notations)

log Pn(a, At, a) < -2nte.

Recall that 7 is Holder continuous. Suppose that 7 is the exponent such that

there exists a constant D with

r(x) ir(y)I < D d(x,y)7

for all x, y in E. Fix a' < min{D-/lalh/7,a}. Pick m with 2-m-1 < a' < 2-m.

Let

K' = {(x, -- m+n)I there exists x E E with x = (x0,- -. Xm+n, **)}

Choose for each word (0, xm+n) in K' a point x in E with the initial of

o,..., Xm+n, to form a subset K of E. The subset K is (n, a') separated, and is

60 -
maximal in the sense that one cannot add another point to K such that it is still

(n,a') separated. Thus, the collection {a-nB(anx, a') x E K} is an open cover
for E. Notice that 7rx = Wpx0,roax. Since irB(x,a') C B(7r(x),a) and

7{7a-nB(anx, a')lx E K} C {WXozx ...B(rnx, a)lx = (xo, x1,......n...) K}

it follows that

{XzozX...XnB(ranx,a)x != (x0,xl,...,Xn... ) E K}

is an open cover for E = U=1,Ei. Also, using (5.2.1) of Lemma 5.2.1 we have

(5.2.5) po...znB(7ranx,a) C xro- r x ( ) + (1 + e)nT" "xWZo-o.n B(0, a).

The right side of (5.2.5) is an ellipsoid with axes

{a(1 + e)n Qj(TXr x o )...)I1 < j < l}.

Pick j with j 1 < t < j. Then that ellipsoid can be covered by

C "( rjTraxroXZ...x ) aj(TTr"x.Oo... )/aj(TvanxCWxo...n)

= C wj-_1 (TrnxPo...xn)aj +l(Tlun"Xo0-.-xn)

balls of radius a(1 + e)naj(T r"xO.nXoX...z), where the constant C > 0 depends
only on the dimension of Rt. Now we calculate the Hausdorff t-measure of E,
using the smaller balls of radius a(1 + e)n aj(Tiranxxo 0...X) < a to cover the

-61 -
open set po-z...,nB(7ra"x, a). If {P: i E I} is an open cover for E where Pi is a

ball of radius ri, then we define I = maxiE ri and

= inf r t
III iEI

We have

74e< :E Cwi-1(j"rX o.X )a (T xxzoxo... x,)[a(1 + )naj(Trax Pzox...z,)]t
xEK
= (1 + )ntat( : "t(T-r0onx0...,X)
xEK
< (1 + e)ntC W(TTX zozi)wt(T-ra,2xrzXz)2) wt(Tra"xVP._z-z,)
_EK
= (1 + e)ntC exp[At(x) + At(ax) + + At(an-x)]
XEK
(1 + e)ntCPn(a, At,a')

< (1 + e)ntCPn(a, At, a)

< C exp(nte) exp(-2nte)

-+ 0,

as n -+ oo. Thus 7-t = 0. Since a can be arbitrarily small, p(t) = 0. It follows

that HD(E) < t. I

PRooF OF THEOREM 5.1: P(a, At) is a decreasing function of t's since E is

compact and At is strictly decreasing with respect to t. So there is only one real
number t such that P(a, At) = 0. Also, the unique t with P(a, At) = 0 is equal to

inf{t : P(a, At) < 0}.

Consequently, we have HD(E) < t where P(a, At) = 0. I

-62 -
5.3. The Lower Bound

PROOF OF THEOREM 5.2: Notice that for each t, the map A is H6lder contin-
uous on E. So there exists an equilibrium state p for At, in the sense that

P(, At) = hp(a) + f tdu.

Fix any p > 0, let's estimate the p measure of a ball B(z, p) centered at z with
radius p. For each x E E choose the unique n = n(x) > 0 such that the diameters
satisfy
diam(Un(x)) < p < diam(U,,_n(x)).

LEMMA 5.3.1. There exists a constant c > 0 such that for all x E E, the open
set Un(x)(x) is contained in a ball of radius p and contains a ball of radius cpl+1.

PROOF: It is clear that Un(x) is contained in a ball of radius p. Since the radius
of Un(x) decreases to 0 as n grows to infinity, without loss of generality we can
assume the maximum diameter R of all Ui is less than the number 6 given in
Lemma 5.2.1. Also pick r small enough that each Ui contains a ball of radius r.
Then Un(x) contains a ball of radius

r* ai,xox,(i}) .. a/,_ (7rcT "x)> ra x *x ..o. a n.* "i "x.

But on the other hand

p 5 diam(Un-_(x)) < al,,ozx(sax_) alX,_z,_l(,ran-lx)R,

which implies that

al,zozi(7rox) al,,n-lx,,(Or n) aOlp/R

63 -
where the constant ia = minyEE,i,j{al,ij(Y)} > 0 does not depend on either n
or x. Therefore Un(x) contains a ball of radius > rpl+ac +K/Rl+K. Writing
c = cal*+r/Rl+' a constant, then Un(x) contains a ball of radius cpl+ as
desired. I
For two points x, y E E, since the construction maps satisfy the open set
condition, Un(x)(x) and Un(y)(y) are either equal or disjoint. Let F C E be
a subset such that {Un(_x)(x)lx E F} is a disjoint collection which contains all

Un(x)(x) for x E E. Notice that {Utn(x)(x)x E r} covers E.

LEMMA 5.3.2. (Similar to Hutchinson (1981) 5.3 (a)) At most 31c-lp-tK of

{Un()(Z) E F} can meet B(z,p).

PROOF: Suppose that VI, ,Vm in {Un(x)(x) x E F} meet B(z,p). Then each
of them is a subset of B(z, 3p). By the definition of F the sets in the collection
{Un(x)(x)lx E r} are disjoint. Comparing the volumes we have

mJc1pl(l1+) < J31pl

where J is the volume of a unit ball in RI. Hence m < 31c-lp-. I
Let
Cn(X) = {y = (Yo, Y1, -) E6 IY = x0, ,yn = Xn}

be an n cylinder. Recall that p is a Gibbs measure (see Bowen (1975) for a
discussion or Bedford (1988) for a summary). There exists a constant d > 0,
with
p(Cn(x)) E [d-l, d] exp(-P(a, At)n + SnAt(x)),

for each cylinder Cn(x) in E. Thus

P(Cn(x)) E [d-l,d] exp(SnAt(x)),

- 64 -

since P(7, At) = 0. So,

p(Cn(s)) < dexp SnAt(x)

< d[al,,or1 (rTx) al,z-n-lzn (roanx)]t
< d[al,zozi (7Trx) al ,z,,- ((ran)lt/(1+x)]

< d [ diam (Un(x))/r]t/(l+)

Hence if n = n(x) we obtain

p(Cn(x)(_x)) < dpt/(l+)/rt/(l+c).

Noticing IrCn(x) D Un(x) n E, by Lemma 5.3.2,

lrp( B(z, p)) 5 [31c-ldr-t/(l+)]pt/(1+K)-IK.

By the Frostman lemma (see Chapter 2 or Kahane (1985) for a proof),

HD(E) > l.

That completes the estimate for the lower bound. |

5.4. Some Continuity in the C1 Topology

The construction of the self similar set Ep depends on the contracting dif-
feomorphisms { ij }. Now let us fix 0 < / < 1, and consider a C1 perturbation to
a C1+O conformal construction with diffeomorphisms {oij}, and obtain another

65 -
matrix of contracting diffeomorphisms {f4ij}, which is not necessarily conformal.
Denote the new self similar set for 4 by E,. Define

dcl(, ') = max dc{ ((ij, ij)},

where the later dci is the C1 metric. Note that for any K < f/, when 4 is
sufficiently C1 close to p, 4 must be C1+' and also K pinched. The following
theorem is a corollary of Theorems 5.1 and 5.2. It tells that at a Cl+4f conformal
construction satisfying the open set condition for self similar sets, the Hausdorff
dimension HD(EO) depends continuously on {f4ij} in C1 topology.

THEOREM 5.4.1. Let {pyij} be a matrix of Cl+ conformal construction diffeo-
morphisms for the self similar set Ep, satisfying the open set condition. For any
e > 0, there exists 6 > 0, such that for any C1+# construction 4 satisfying the
open set condition, with dci (V, 4) < 6, we have

IHD(Ep) HD(E)I
PROOF: Let

A,s(x>) = log w,(Tvoxx(7rrxs)),

A,,s(,x) = log ws(rfxoxIoB(7rax)),

be two real functions on E as defined in Definition 5.1.1. Let t be such that
P(a, Aq,t) = 0. Because pij's are conformal, the Hausdorff dimension of E'
equals t. Also, remark that P(a, AX,t+e) < 0 for any e > 0.
Now fix any e > 0. Let K = min{fl,e/41} and let

(5.4.1) e, = min{-P(a, A,t+c), P(a, A,t-e/4)} > 0.

66 -
Since p is C1+# and conformal, there is 6 > 0 such that the C'1+ diffeomor-
phism 4 is Cl+N and K pinched with JA,,(x) A~,,(x)l < e' for all s E [0,1], if

dci (c, 0) < S. Then

P(a, Ak,t+c) < P(7, Ap,t+E + e') < P(a, Ap,t+e) + e' < 0.

So

HD(E0) < t + e= HD(Ep) + e,

by Proposition 5.2.2.
On the other hand, by (5.4.1), when dci(<, 4) < 6, we have

P(a, A _,t-i/4) > P(a, Ap,t-e/4 e') > P(a, A _,,t-/4) e' > 0.

So we have some s > t e/4 with P(o, A,s,) = 0 since P(o, Ap,s) is strictly
decreasing with respect to s. By Theorem 5.2,

HD(E,) > s/(1 + I) In > s/(1 + e/41) l(e/41)

> s(l e/41) /4 s e/2 > t -e.

It then follows that

HD(E) 2 t-e= HD(EV) e,

as desired. |
We say a construction W with diffeomorphisms {pij} satisfies the strong
open set condition if there are open sets U1, U1 in R1 with Vij(Uj) C Ui for
all i, j. If the construction p satisfies the strong open set condition, then 4 must
also satisfy the strong open set condition if it is C1 close enough to p. Thus we
have obtained an immediate corollary of the above Theorem 5.4.1:

67 -
COROLLARY 5.4.2. Let {ij} be a matrix of C1+P conformal construction dif-
feomorphisms for the self similar set Eo, satisfying the strong open set condition.
For any e > 0, there exists 6 > 0, such that for any C1+ construction 4 with

dC (p, 0) < 6, we have

IHD(E,) HD(E,)I

Finally we have a remark on the continuity of the Hausdorff dimension in
the C1 topology at non-conformal constructions.

REMARK 5.4.3: The following example shows that if the conformall" condition
for the construction diffeomorphisms {pjij} fails, then the results in Theorem
5.4.1 and Corollary 5.4.2 can be false. The example is derived from Example
9.10 of Falconer (1990), pp 127-128.
Let S, T : R2 -+ R2 be given by

S(x,y) = (x/2,y/3 + 2/3), T,(x,y) = (x/2 + A, y/3)

where A E [0, 1/2) and (x, y) E R2. Let cpll = 321 = S, P12 = VP22 = To. Take

OA = {'ij,A} where b11,, = 021,A = S, and 012,A = 022,A = TA. The strong open
set condition is met for {f ij}. In fact if we let U1 = U2 = (-1/8,9/8)2 C R2
then pij(Uj) C Ui.
Let E(, EP, be the self similar sets for p and OA. Considering the projection
of Ep, to the x-axis, one knows that HD(E, ) > 1 for A > 0. But Eo is a Cantor
set contained in the y-axis with the Hausdorff dimension

HD(E() = log 2/log 3 < 1.

-68 -
Since dci(t/4, o) = A, so letting A -+ 0 we know the Hausdorff dimension is not

continuous at p. We notice that {Pij} are not conformal although { ij} and

{tij,A,} are all 3/4 pinched.

CHAPTER 6

THE HINON ATTRACTOR

In the 1970's H6non et al. conducted numerical experiments on the following

quadratic automorphisms of the plane, H : R2 -t R2 defined by

H(X, Y) = (1 + Y aX2, bX), where a, b are parameters ,

which "show" a "strange" attractor. We now call this map a Henon map. See

Henon (1976), H6non and Pomeau (1975) for numerical results. Say A is an

attractor of H if it is invariant under H and there is a neighborhood U of A

such that every point z E U has limn~ood(Hn(z),A) = 0. A Henon map is

regarded as the simplest smooth example that computer simulations indicate the

existence of a strange attractor. Since Henon's numerical experiments a number

of authors have studied the dynamical behavior of the Henon mappings. Devaney

(1986) gives an introduction to this subject. Milnor (1988) noticed many H6non

mappings are non-expansive, and the topological entropies of the H6non mappings

fall within [0, log 2]. In this chapter, we give an upper bound for the capacity and

the Hausdorff dimension of a H6non attractor.

Under the coordinate change suggested Devaney and Nitecki (1979), X =

x/a, Y = by/a we have the following convenient form of the Henon mapping:

H(x,y) = (a + by x2, ), where a, b are parameters.
69 -

70-
Benedicks and Carleson (1991) shows the following: Let W" = WU(z, H)

be the unstable manifold of H at the fixed point z of H in the first quadrant.

Then there is a b0 > 0 such that for all b E (0, bo) there is a set Eb of positive

one dimensional Lebesgue measure such that for all a E Eb:

(1)There is an open set U = U(a, b) such that for all z E U,

d(Hn(z), W") -4 0, as n -+ oo.

(2)There is a point z0 = zo(a, b) E WU such that {H"(zo)}n=0 is dense in

W".
1
We consider the case when b0 < -. Also, it is clear that WU is C" and one

dimensional. Thus

C(A) > C(A) > HD(A) > 1.

Now let R be the bound for A such that for every (x, y) E A we have Ixl, Iyl R,

and denote
2R+ 1
Zn = (xn,Yn) = Hn(zo), m =- 2
b
and
n-1 2 xi 1
A = limsup log(b + 21i+ -l
n--oo n m m2
i=0
Since

2Ix _I 1
b+ + 1 2b< 1,
m 2 -
we have A < 0. For the capacity of an attractor, we shall prove the following:

THEOREM IV. If A is a compact invariant attractor under the Henon map H,

as proved existing by Benedicks and Carleson (1991) then the upper capacity

A
C(A) <2+ < 2.
log m

71 -

Douady and Oesterle have an upper bound for the Hausdorff dimension of

attractors. Since HD(A) < C(A), our theorem IV also gives an upper bound for

the Hausdorff dimension. In the case of a Henon attractor our estimate is better

than Douady and Oesterl6's result applied to a Henon attractor.

Let R1 be the larger root of the equation x2 (1 + Ibj)x a = 0. Let

S = [RI,R11 x [R1, R1]. Then any point outside S on the plane will flee to

infinity under either the Henon map H or H-1, see Devaney (1986). In other

words, any invariant set A is contained in S, and thus bounded.

PROOF OF THEOREM IV: First take a look at the image of a ball under the

Henon map. Let |I(x,y)ll = max{x,y} and d be the induced metric. After a

coordinate change x = x/m, 9 = y the Henon map has the following form:

H(, y) = (ma + mby -1 2-).
m m

Let us denote the new coordinate system also by (x, y) from now on.

LEMMA 6.1. Let

B= {(x,y): x- uI < r, y-v < r}

be a ball centered at (u, v) of radius r in our metric. Then H(B) is contained in

the following rectangle:

r r
[ul 1, u1 + I] x [vl -, r + -]
m m

where

(ul, v1) = H(u, v), and I = mbr + 2 lu + -
m m

72 -

PROOF OF LEMMA 6.1: Since B is bounded by x = u r, y = v r, the image

of B, H(B) will be bounded by

u r 2
y = -, and x = ma + mbv my mbr.
m m

Pick any point (x, y) from the parabola

2 u r
x = ma + mbv my -mbr with y < -.
m m

Then
r ul r2
Ix u11 < mbr + 2 + -,
m m
which completes the proof for the lemma.

Therefore, H(B) can be covered by at most

rjuj r2 r
1 + (mbr + 2u + -)/- = m2b+ 2u + r +1
m mm

balls of radius r/m. Now let {Bi : i = 1, k} be a finite cover of A where

every ball Bi has a radius less than r/2. Since {zn : n = 0,1, } is dense in

A, one can pick some ni such that zn, E Bi. Then {B(zn,, r) : i = 1, k} is a

cover of A. Let us write in order: ni < n2 < .. < nk. Using Lemma, if a ball B

centered at (u,v) has radius r and we write H'(u,v) = (ui, vi) then Hn(B) can

be covered by at most
n-1
l (m2b+ 2ui + r + 1)
i=0
r
balls of radius --. Since {B(zn,,r) : i = 1, ,k} is a cover of A, the image

under the diffeomorphism Hn,

{HnB(zn,,r) : i= 1,... ,k}

- 73 -

is also a cover of A. Suppose N = max{ni : i = 1,.-- ,k}.

Notice that

HnB(zni, r) is covered by at most

i+n-1
H (m2b +2 an, ++ 1)

r
mn

i+n-1
II (m2b+2lni +r+ 1)<
j=i

ni+n-1
S(m2b+ 21jl + r + 1)
j=0

N+n-1
< lI (m2b+2lxjl+r+l).
j=0

Therefore the number of balls of radius needed to cover A is at most
mn
m"

N+n-1
k- I (m2b+21xjl+r+l).
j=0

By Lemma 1.3.1, we have

C(A) < lim sup
n-*oo

= lim sup
n-+oo

= lim sup
n--oo

log k in-lj (m2b+ 2xI + r + 1)
log(r/mn)
log k + ENn-1 log(m2b + 21xj + r + 1)
n log m + log r
ENn-1 2b+log(m2xl+r+1)
j=0 ( 21 + r 1)
n log m

1 1
= lim sup 1
log m n-+oo n

N+n-1
E log(m2b+2xjl +r + 1).
j=0

- 74 -

Since x1;i < R for all i, and the function log is differentiable, there is a constant

C such that for all r with 0 < r < 1 and for all j > 0,

log(m2b + 21xj + 1) < log(m2b + 21xjl + r + 1)

5 log(m2b + 2xjl + 1) + Cr.

This implies by letting r -- 0 that

1 1
C(A) < lim sup-
log m n-0oo n

N+n-1
j=
j=0

1 1
= 2 + lim sup -
log m n-+oo n

log(m2b + 21xl + 1)

N+n-1

j=0

2mljl + 1
log(b + )
m2

1 n-1 2mlxl + 1
=2+ 1 lim sup log(b + 2-- )
log m n-oo = m
j=O
A
=2+
log m

That completes the proof. I

CHAPTER 7

SUMMARY AND CONCLUSIONS

In this paper we are mainly concerned with the Hausdorff dimension of

hyperbolic sets, the self similar sets and the H6non attractors. These are all

invariant sets under some smooth diffeomorphisms.

For a hyperbolic set of a diffeomorphism or a flow of diffeomorphisms, the

uniform stable and unstable Lyapunov exponents have been defined. Using the

uniform Lyapunov exponents we define a characteristic function. At some real

positive number t the characteristic function equals the value of the topological

entropy of the diffeomorphism on the hyperbolic set. We prove that t is an upper

bound for the Hausdorff dimension of this hyperbolic set. Technically, we use a

pinching condition to cope with the nonlinearity of the diffeomorphism, to prove

that the iterated image of a ball is somehow contained in the image of the ball

under the derivative of the iterated diffeomorphism. Some earlier results are thus

improved.

In Chapter 4 we have also given upper bounds for the Hausdorff dimension

of the transverse of a hyperbolic set with stable and unstable manifolds, using

the topological pressure on the hyperbolic set.

Several authors have studied the self similar sets of iterated mapping sys-

tems in the cases where construction diffeomorphisms are conformal. In chapter

5 we study the case where the construction diffeomorphisms are not necessarily

conformal. We give a new distortion lemma. Using topological pressure on a

75 -

76 -

shift space we give an upper estimate of the Hausdorff dimension, when the con-

struction diffeomorphisms are C1+" and satisfy a K pinching condition for some

K < 1. Moreover, if the construction diffeomorphisms also satisfy the disjoint

open set condition we then give a lower bound for the Hausdorff dimension. We

believe that the characteristic function A can be revised to achieve an even bet-

ter estimate for the Hausdorff dimension. We also obtain the continuity of the

Hausdorff dimension in the C1 topology at conformal constructions.

In chapter 6 we study the image of a disc under the iteration of a Henon map,

which enables us to find an upper bound for the Hausdorff dimension and capacity

of the Henon attractor. That improves some earlier estimates when applied to the

case of a Henon attractor. Milnor (1988) noticed that many Henon mappings are

non-expansive, and the topological entropies of the Henon mappings fall within

[0, log 2]. Since we cannot establish a distortion lemma similar to Lemma 3.4.2

for Henon maps, it is unknown to us whether the Hausdorff dimension of a Henon

attractor can be related to its topological entropy. It is still an open problem.

REFERENCES

Bedford, T., Dimension and dynamics for fractal recurrent sets, J. London Math.

Soc., (2) 33 (1986), 89-100.

Bedford, T., Hausdorff dimension and box dimension in self similar sets, Proc. Conf.

Topology and Measure V (1988), Greifswald, 17-26.

Benedicks, M., and Carleson, L., The dynamics of the Henon map, Ann. Math.,

133 (1991), 73-169.

Besicovitch, A., and Ursell, H., Sets of fractional dimensions, V: On dimensional

numbers of some continuous curves, J. Lond. Math. Soc., 12 (1937), 18-25.

Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms,

Lecture notes in Mathematics 470 (1975), Springer, Berlin.

Caratheodory, C., Uber das lineare Mass von Punktmengen, eine Verallgemeinerung

das Lingenbegriffs, Nach. Ges. Wiss. G6ttingen, (1914) 406-426.

Dekking, F. M., Recurrent sets, Adv. in Math. 44 (1982) 78-104.

Devaney, R. L., An Introduction to Chaotic Dynamical Systems, The Benjamin

Cummings Publishing Company Inc., Menlo Park, California, 1986.

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78 -

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delberg, Berlin, 1982.

BIOGRAPHICAL SKETCH

I was born October 31, 1963, in Jiangsu Province, eastern China. I attended

the University of Science and Technology of China in Hefei, Anhui Province,

China, from 1979 to 1983. I was a graduate student at the Institute of Mathe-

matics, Chinese Academy of Sciences, from 1983 to 1987, and at the University

of Florida since 1987.

-81 -

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Albert Fathi, Chairman
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Louis Block
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully ad quate, in scope and
quality, as a dissertation for the degree of Doctor of Phi phy.

a s Keesligg
ssor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Stephen umr rs
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

David Groisser
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Robert Buchler
Professor of Physics

This dissertation was submitted to the Graduate Faculty of the Department
of Mathematics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.

May, 1992

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