UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
HAUSDORFF DIMENSION OF SOME INVARIANT SETS BY XIAOPING 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  TABLE OF CONTENTS 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 XiaoPing 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 noninteger 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 twodimensional 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 ndimensional 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, maxlTzfk 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 n1 A = limsup1 n log(b+ 21 + < 0. noo 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 nonnegative 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 ets (K) = 0 Hs(K) = 0. This also implies that if t < s and V8(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 noninteger 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 wellknown 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 [p2m, (p+ 1)2m], 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 2ms 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 2s(m1), or we multiply pim by a constant less than 1 in such a way that the resulting measure has mass 2s(m1); that way we get a measure Pm1. For each k > 1, the measure 1L'k is obtained from m_ in the same way, that is mk1 = (D) mk on each dyadic cube D of order m k 1, with A(D) = min{1,2s(mk1)tmk(D)1}. Clearly p'k is independent of k when k is large enough. Then we can write mmk Pmk = Pm Notice the measure pm is carried by the dyadic cubes of order m which intersect K and we have pm(D) < 2s(mk) 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 711DI 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 for each dyadic cube D. Now, given an interval of length b, find k such that 2k1 < b < 2k; then the interval is covered by two dyadic intervals of length 2k. Given a ball of diameter b, it is covered by 21 dyadic cubes of order k. Therefore, pi(B) 5 21711B1 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 e0 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) e0 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). E0  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 , e0 log e noo 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. e0O 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 ITlfk 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 lTzfk 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 JTzfk 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 ndimensional 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 = 71(2A), or D = n if #1f +2+""'+Pn > 2A. If AB2 < 1, where A = limAk, B = limBk, and Ak = max{max ITf Imax IITrfk }, 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 f1a is also an open cover, with N(fla) < N(a). Thus we can introduce the open cover V1 fia 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 nIf a) < H(V1fia) + H(fk vn1 fia) < ak + an. The subadditivity implies that lim lan= lim 1H(f Vn1 fa) noo n noo 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). n4oo 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 60 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 ndimensional 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 ndimensional 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 Txf1 : 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). That gives the subadditivity oflog f,(fk). By the subadditivity the limit lim log Cf( koo 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) = supzKwi(Tzf1'I E). The definition of wi implies that ou(fp+q) < fyu(fp) (Zu((fq). That gives the subadditivity oflog '(fk). By the subadditivity the limit 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 f1. 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 IIv 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 < 411Ji 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 welldefined 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 f1 to get fX1(v) = Txfl(v) + x(v), where IIzx(v)II < AdvI2. 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 f1, 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 fl(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, j1i()1f ()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(Ofm(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 A1. 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(Ofm+(I), a)) n Tfm(B(O f +1 (),a))}, and in particular we can pick some W1 E B(Ofm+1(W), a)  27  with (1 + e)mTfm(wl) = u. Moreover, IPr(u)I I5 (1 + e)mBgma, and 1ru(u)11 5 (1 + e)mBma (because IITfm IE, II < Bfm and IITfm IEu II < Blm). It follows that lull (1 + e)mBma < a < 6, and thus ff)(u) = Tf(x)fl(u) + gf(x)(u) is welldefined, with II1f() (u) I < Alu112 < A (1 + e)2mB2ma2 < e(1 + e)". aAm1. Since maxzEK IITzf Jll A1, we have IlTxf1(v)ll > Al1 vlj for all x E K and v Ex. Thus Tzf1(B(O, a)) Al B(Of1(xa), and follows. Hence Yf(z)(u) e(1 + e)mAlml(B(Oz, a)) C e(1 + e)mTfml(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 = Tf1((1 + e)mTfm(wi)) + e(1 + e)mTf(m+l)(w2) = (1 + )m+Tfml(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 fl(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 (aail(L),...,a.an(L)) in F. PROOF: See Temam (1989). Lemma 3.4.3 shows that rsTfmB(O,,a) and ruTfmB(O, a) are ellip soids with certain axes in E8f and Em 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(Ofm(x), a)) n Tfm(B(Ofm((),a))}. Using Lemma 3.4.3, we know that rxTfm(B(Ofm(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(Ofm(z), a)) 3 ruB(Ox, a). Similarly, we know that ruTfm(B(Ofm(), a)) is an ellipsoid in EU of axes {aPnn+l,m(x),  apn,m(x)} where Pnl+i,m = ai(Tfm I Ex), with Pnl+l,m(x) D *pm+i,m(x) (p P' )m and rsTfm(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(Ofm (), 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 fm(), a) n Tf m(B(Ofm(x), a)) is covered by no more than J Pl,m(x) ... Pjl,m()/l(Pj,m(x))j1 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) ... Pjl,m(x)l(pj,m(x)Yj1 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))j1 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 fi[B(fi(x)a/C)], i=m by taking fm, 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 premeasure 'H7 is given by inf rf III and has been proved to be a nonincreasing function of e's. Therefore, sCa(K) = inf E rf < p(C(1 + e)mapT, s) III < Pm(x (pj,m(x))" [C(1 + e)ma]s zEM [p,m()i < exp[m(2A + e)] expm(p1 + ... + pj1 + (s j + 1)pj + se) Sexp[m log(1 + e)]. C'Ja = exp m[2A + pl + ... + lj1 + (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. a0 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(TzftlE). 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  So by subadditivity both 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 ndimensional 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 = r1(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 IITzft I 11, zEK xEK B = max{ma.lftx IT IE IImax ITft 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, IITf1 IE 1} < 1, zEK where the norm is obtained from an appropriate Riemannian metric on X. We call r = maxxK({IITf IE II, ITaf1 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, IITZf1E 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, IITxf111} 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(Txf1^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 nonnegative 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 realvalued 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) = En1 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 fia}, 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). noo 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). EO C0 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, ), noo n Now the topological pressure is given by P(f, A) = lim Q(f,A,e) = lim P(f,A,e). *0 e0 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 welldefined 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 (fix), 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 {OzfmBm(Ofym(z),2a)x E K1} is an open cover for K. Since 0f = fO, it follows that fmOfm(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 moo 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(fl,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(Ofm(Z), r) n Tfm(z)fmB(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)fmruB(Ofm(z), r). Reshuffle a1(Tfm()fmlE*),... ,an(Tfm(x)fm IE' ); al(Tfm(I)fm Eu)," *, In2(Tfm(x)fmIE ) 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)tj+1 zEK2 = C(1 + e)m E Wt2()(Tfm(X)fm IE )Wt (x) Tfm() fm lE) xEK2 m 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(f1 IK, ) + P(f IKAu) < O. Thus we can require 1 log[Pm(f1IK 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, (fix))} tl+t2=t zEK2 i=1 xEK2 i=1 < Cexp(me) : Pm(f1 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) = Ol0(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(0fm(x), r) n T ( (z)fmB(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(fmx, 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(fmx, 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)tj+l = W(TfmfmlEs) < wt (Tfm f IES)W(Tym+lJEf .E). T (Tff IE8) = exp As,t(fm.x).. .exp As,t(fx) m = exp As,t(fix) 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( Asfix) 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 nonzero aij we give a contraction map Pyij : R  R1 with lpij(x) pij(y)l < cx yl, 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 kfold 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 nonempty 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... zzn+l(Exn+l). It is clear that r is a H6lder continuous, surjective map. We will denote the composition Voxl " XniXn 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 jth Lyapunov number of a linear map L, denoted by aj(L), is the square root of the jth 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 Kpinching 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) in the discussions concerning topological pressure, see also Chapter 4. Theorem 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 nonempty 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 Kpinching 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 Iw < c, we have Irzozx(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 + )nlTz ,...xZB(0,a)]. Using (5.2.2): Pxoxi k~z...Xn(x) + (1 + e)n1Tvx,...X(w)] = pzo...n (x) + (1 + E)nlTvzo...x,(w) + rxox,((1 + e)nlTvxl...x,,n(w), IP...,,( ()). 58  Because of (5.2.3), 11(1 + e)n1T l...x,(w) < 11wI < a, where w E B(O,a). Using (5.2.4), we have Ilrzxoz((1 + e)n1TwzZ...Z,,(w), wx...x,(x))II < cl(1 + e)n1Tlz....,,(w) 1l+'c < C(1 + e)(n1)(l+) [alXz12(z2...n(z,()) ... acl,znx,,(x)IIwII]1+ < C(1 + e)n1al,,,, z))...a,2,n 1 (x)Illwll1+ < (1 + E)nla"CCal,T(,,1( ..., (x))... al,,n_X(X)IIwII < e(1 + e)nlao,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.nlX,, (x)B(O, a). Hence rx0zo((1 + e)n1lTW ... ,, (w), cp...:,,(x)) E e(1 + e)n1Tzx zo...z,,B(O, a). Therefore, Vxzoz [kx...xx) + (1 + E)n1T Z,...X(w)] E cpx...x,(x) + (1 + e)"1Txzo ...,B(O, a) + e(1 + e)n1Tz~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 Kpinching 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 2m1 < a' < 2m. 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 {anB(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{7anB(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" "xWZoo.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 tmeasure of E, using the smaller balls of radius a(1 + e)n aj(Tiranxxo 0...X) < a to cover the 61  open set poz...,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 Cwi1(j"rX o.X )a (T xxzoxo... x,)[a(1 + )naj(Trax Pzox...z,)]t xEK = (1 + )ntat( : "t(Tr0onx0...,X) xEK < (1 + e)ntC W(TTX zozi)wt(Tra,2xrzXz)2) wt(Tra"xVP._zz,) _EK = (1 + e)ntC exp[At(x) + At(ax) + + At(anx)] XEK (1 + e)ntCPn(a, At,a') < (1 + e)ntCPn(a, At, a) < C exp(nte) exp(2nte) + 0, as n + oo. Thus 7t = 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(,ranlx)R, which implies that al,zozi(7rox) al,,nlx,,(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 31clptK 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 < 31clp. 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 [dl, d] exp(P(a, At)n + SnAt(x)), for each cylinder Cn(x) in E. Thus P(Cn(x)) E [dl,d] exp(SnAt(x)),  64  since P(7, At) = 0. So, p(Cn(s)) < dexp SnAt(x) < d[al,,or1 (rTx) al,znlzn (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 [31cldrt/(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,te/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 _,ti/4) > P(a, Ap,te/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 te= 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 nonconformal 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 127128. 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 xaxis, one knows that HD(E, ) > 1 for A > 0. But Eo is a Cantor set contained in the yaxis 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 nonexpansive, 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 n1 2 xi 1 A = limsup log(b + 21i+ l noo 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 H1, 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, yv < 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 n1 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+n1 H (m2b +2 an, ++ 1) r balls of radius . But mn i+n1 II (m2b+2lni +r+ 1)< j=i ni+n1 S(m2b+ 21jl + r + 1) j=0 N+n1 < lI (m2b+2lxjl+r+l). j=0 Therefore the number of balls of radius needed to cover A is at most mn m" N+n1 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 noo log k inlj (m2b+ 2xI + r + 1) log(r/mn) log k + ENn1 log(m2b + 21xj + r + 1) n log m + log r ENn1 2b+log(m2xl+r+1) j=0 ( 21 + r 1) n log m 1 1 = lim sup 1 log m n+oo n N+n1 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 n0oo n N+n1 j= j=0 1 1 = 2 + lim sup  log m n+oo n log(m2b + 21xl + 1) N+n1 j=0 2mljl + 1 log(b + ) m2 1 n1 2mlxl + 1 =2+ 1 lim sup log(b + 2 ) log m noo = 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 nonexpansive, 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), 89100. Bedford, T., Hausdorff dimension and box dimension in self similar sets, Proc. Conf. Topology and Measure V (1988), Greifswald, 1726. Benedicks, M., and Carleson, L., The dynamics of the Henon map, Ann. Math., 133 (1991), 73169. Besicovitch, A., and Ursell, H., Sets of fractional dimensions, V: On dimensional numbers of some continuous curves, J. Lond. Math. Soc., 12 (1937), 1825. 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) 406426. Dekking, F. M., Recurrent sets, Adv. in Math. 44 (1982) 78104. Devaney, R. L., An Introduction to Chaotic Dynamical Systems, The Benjamin Cummings Publishing Company Inc., Menlo Park, California, 1986. 77 78  Devaney, R., and Nitecki, Z., Shift automorphisms in the Henon mapping, Com mun. Math. Phys. 67 (1979) 137146. Douady, A., and Oesterl6, J., Dimension de Hausdorff des attracteurs, C. R. Acad. Sci. Paris S&r. I Math., 290 (1980) 11351138. Falconer, K., Fractal Geometry, Mathematical Foundations and Applications, John Wiley and Sons, Chichester, England, 1990. Fathi, A., Some compact invariant subsets for hyperbolic linear automorphisms of torii, Ergodic Theory and Dynamical Systems, 8 (1988), 191204. Fathi, A., Expansiveness, hyperbocility and Hausdorff dimension, Communications in Mathematical Physics, 126 (1989), 249262. Frederickson, P., Kaplan, J., Yorke, E., and Yorke, J., The Lyapunov dimension of strange attractors, Journal of Diff. Eq. 49 (1983), 185207. Hausdorff, F., Dimension und iusseres Mass, Math. Annalen, 79(1919), 157179. Henon, M., A twodimensional mapping with a strange attractor, Commun. Math. Phys. 50 (1976) 6977. Henon, M., and Pomeau, Y., Two strange attractors with a simple structure, Lec ture Notes in Mathematics, 565 (1975) 2968. Hirsch, M., and Pugh, C., Stable manifolds and hyperbolic sets, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, vol. XIV, (1970) 133164. 79  Hurewicz, W., and Wallman, H., Dimension Theory, Princeton University Press, Princeton, 1948. Hutchinson, J., Fractals and self similarity, Indiana U. Math. J. 30(1981) 713747. Jiang, Y., On nonconformal semigroups, Preprint Article, State University of New York at Stony Brook (1991). Kahane, J., Some Random Series of Functions, Second Edition, Cambridge Univer sity Press, Cambridge, England, 1985. Ledrappier, F., Some relations between dimension and Lyapunov exponents, Comm. Math. Phys. 81 (1981), pp 229238. Ledrappier, F., and Young, L. S., The metric entropy of diffeomorphisms part II: Relations between entropy, exponents and dimension, Ann. Math. 122 (1985), 540574. Mandelbrot, B., Fractals: Form, Chance and Dimension, W. H. Freeman, San Francisco, 1977. Mather, J., Characterization of Anosov diffeomorphisms, Indag. Math. 30 (1968), No. 5. Mauldin, R., and Williams, S., Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988) 811829. McCluskey, H., and Manning, A., Hausdorff dimension for horseshoes, Erg. Theo. & Dyn. Sys., 3(1983), 251260. 80  Milnor, J., Nonexpansive Henon maps, Adv. in Math., 69 (1988) 109114. Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747817. Temam, R., The Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988. Walters, P., An Introduction to Ergodic Theory, SpringerVerlag, New York, Hei 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 Dean, Graduate School UNIVERSITY OF FLORIDA 1 22III IIIIII II l5I9i 688111111111111111 3 1262 08556 9688 