DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 10,Number3,April2004 pp. 827–833 ON BOWEN’S DEFINITION OF TOPOLOGICAL ENTROPY Michal Misiurewicz Department of Mathematical Sciences IUPUI, 402 N. Blackford Street Indianapolis, IN 46202-3216, USA (Communicated by Lluis Alseda) Abstract. About 5 years ago, Dai, Zhou and Geng proved the following result. If X is a metric compact space and f : X → X a Lipschitz continuous map, then the Hausdorff dimension of X is bounded from below by the topological entropy of f divided by the logarithm of its Lipschitz constant. We show that this is a simple consequence of a 30 years old Bowen’s definition of topological entropy for noncompact sets. Moreover, a modification of this definition provides a new insight into the entropy of subshifts of finite type. 1. Introduction. Although the Bowen’s paper [7] has been published almost 30 years ago, some of the ideas that are hidden there remain unexploited. The first general idea is that the connection between the topological entropy and the Hausdorff dimension can be studied using simple topological tools. In particu- lar, one can use Bowen’s methods to give a simple proof that if the map is Lipschitz continuous then the Hausdorff dimension of the space is bounded from below by the topological entropy of the map divided by the logarithm of the Lipschitz con- stant (Corollary 2.2). This result has been proved only several years ago by Dai, Zhou and Geng [8] and remains relatively unknown. The original proof follows a more complicated path, via metric entropies and the Variational Principle. Earlier results in this direction (see e.g. the book [10] and the bibliography there, or [9]) were weaker and required additional assumptions. This theorem can be applied in several interesting situations. A natural way is to apply it to the restriction of a map to an invariant closed set. In particular, it implies that if the topological entropy of the map is positive and the map is Lipschitz continuous then the set of nonwandering points has positive Hausdorff dimension. For the smooth maps results of this type exist in the literature, for instance for rational complex maps. However, we do not need any smoothness, so we can apply Corollary 2.2 for instance to various branched coverings of the sphere (cf. [3]). In the classical definition of the topological entropy via open covers [1], as well as in another Bowen’s definition via separated or spanning sets [6], one looks at the pieces of trajectories of length n, and only later takes some limit as n →∞. The second general idea of [7] is that we can look at pieces of orbits of different length at the same time. This leads to a new definition of the topological entropy, which is in the spirit of the classical definition, but takes into account this idea. It allows us to look at the entropy of certain subshifts of finite type from a different 1991 Mathematics Subject Classification. 37B40, 37F35, 54F45. Key words and phrases. Topological entropy, Hausdorff dimension. The author was supported by NSF grant DMS 0139916. 827 828 MICHAL MISIUREWICZ perspective and gives another insight into the “rome” method for computing the topological entropy, introduced in [4]. The paper is organized as follows. Section 2 is devoted to the first of the ideas described above, and Section 3 to the second one. Then in Section 4 we complete the proof of the proposition from [7], in order to make this paper more self-contained. Moreover, it shows that once we know the definitions of the topological entropy, the full detailed proof of Corollary 2.2 takes less than 2 pages and does not involve any other strong theorems, like the Variational Principle or an equivalence of the Brin-Katok definition of the metric entropy ([5]) with the usual one. 2. Entropy and Hausdorff dimension. Let X be a nonempty compact metric space and f : X → X a Lipschitz continuous map with Lipschitz constant L. Let us recall Bowen’s definition ([7]) of the topological entropy h(f,Y )off on an arbitrary subset Y ⊂ X. Let A be a finite open cover of X.ForasetB ⊂ X we write B ≺Aif B is contained in some element of A.Letnf,A(B) be the largest nonnegative integer n k such that f (B) ≺Afor k =0, 1,...,n− 1. If B ≺Athen nf,A(B)=0andif k f (B) ≺Afor all k then nf,A(B)=∞. Now we set diamA(B) = exp(−nf,A(B)), diamA(B)=supdiamA(B) B∈B and λ DA(B,λ)= (diamA(B)) B∈B for a family B of subsets of X and a real number λ. Then µA,λ(Y ) = lim inf{DA(B,λ):B is a cover of Y and diamA(B) <ε} ε→0 has similar properties as the classical Hausdorff measure λ µλ(Y ) = lim inf (diam(B)) : B is a cover of Y and diam(B) <ε ε→0 B∈B (where diam(B)=supB∈B diam(B)), that is, there exists h(f,Y,A) such that µA,λ(Y )=∞ for λ<h(f,Y,A)andµA,λ(Y )=0forλ>h(f,Y,A). Finally, we set h(f,Y )=sup{h(f,Y,A):A is a finite open cover of Y }. This number h(f,Y ) is the topological entropy of f on the set Y .IfY = X then by Proposition 1 of [7] h(f,X)=h(f) is equal to the topological entropy of f. The main technical result of this section is the following theorem. Theorem 2.1. For any Y ⊂ X the Hausdorff dimension of Y is larger than or h(f,Y ) equal to log L . Proof: We fix a finite open cover A of X. It has a Lebesgue number δ, that is a positive number such that if B ⊂ X and diam(B) <δthen B ≺A.If n−1 k diam(B) <δ/L then diam(f (B)) <δfor k =0, 1,...,n− 1, so nf,A(B) ≥ n. n n< log δ−log diam(B) In other words, for every nonnegative integer ,if log L + 1 then n ≤ nf,A(B). Hence, log δ − log diam(B) ≤ nf,A(B). (2.1) log L ON BOWEN’S DEFINITION OF TOPOLOGICAL ENTROPY 829 c log δ f A The number = log L is a constant, as long as and are fixed. We can rewrite (2.1) as −c 1/ log L diamA(B) ≤ e (diam(B)) . (2.2) Therefore if B is a cover of Y then for any λ we have −cλ λ/ log L DA(B,λ) ≤ e (diam(B)) . (2.3) B∈B Fix λ>HD(Y ) log L, where HD(Y ) is the Hausdorff dimension of Y . Then µλ(Y )= 0, so for every ε>0 there is a cover B of X such that e−c(diam(B))1/ log L <ε and e−cλ (diam(B))λ/ log L <ε. B∈B For this B we get diamA(B) <εby (2.2) and DA(B,λ) <εby (2.3). Hence, µA,λ(Y ) = 0, and thus h(f,A) ≤ λ. Taking supremum over all finite covers A of X,wegeth(f,Y ) ≤ λ whenever λ>HD(Y ) log L,soh(f,Y ) ≤ HD(Y ) log L. Setting Y = X we get the following corollary (see [8]). X h(f) Corollary 2.2. The Hausdorff dimension of is larger than or equal to log L . Let µ be an f-invariant probability measure on X. Its Hausdorff dimension HD(µ)is defined as the infimum of the Hausdorff dimensions of the sets of full measure (see e.g. [10]). The support of µ is the smallest compact set of full measure, and hµ(f) denotes the metric entropy of f with respect to µ. The following theorem also appears in [8]. Theorem 2.3. If µ is an f-invariant probability measure on X and f restricted to µ L µ ≥ hµ(f) the support of is Lipschitz continuous with constant then HD( ) log L . Proof: Let Y ⊂ Z be a set of full measure, where Z is the support Z of µ. Since µ is invariant, Z is an invariant set. Applying Theorem 2.1 with X replaced by Z Y ≥ h(f,Y ) h f ≤ h f,Y we get HD( ) log L . By Theorem 1 of [7], µ( ) ( ), so we get Y ≥ hµ(f) HD( ) log L . In the definition of the Hausdorff dimension of a measure it is enough to take the infimum over the sets Y that are contained in the support of µ, µ ≥ hµ(f) and thus HD( ) log L . Remark 2.4. Often one can improve Theorems 2.1 and 2.3 and Corollary 2.2 by 1 replacing log L by inf{ n log Ln : n =1, 2,...}, where Ln is the Lipschitz constant for f n. This follows from the fact that the entropy of f n is equal to n times the entropy of f. Moreover, since we used the Lipschitz property only on sets of arbitrarily small diameter, we can replace L by the local Lipschitz constant (as in [8]), which is the infimum of Lipschitz constants of the map restricted to sets of a given diameter. Since there is no assumption of smoothness in Corollary 2.2, it can be used for expanding polymodials (see [3]). They are like complex polynomials, but the nature of their critical (singular) points is different. In particular, they are not differen- tiable there.