DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 10,Number3,April2004 pp. 827–833
ON BOWEN’S DEFINITION OF TOPOLOGICAL ENTROPY
Micha l 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 MICHA L 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
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. Nevertheless, in order to prove that if they are Lipschitz continuous 830 MICHA L MISIUREWICZ
(like the tent maps, given by the formula f(z)=1+cz2/|z| for |c| > 1) then the Hausdorff dimension of their Julia sets is positive, we do not have to analyze their structure – we just apply Corollary 2.2. The inequality of Corollary 2.2 can be also used in a reverse direction. Since the Hausdorff dimension is not larger than the topological dimension, we get im- mediately that the topological entropy of a smooth map in dimension d is bounded from above by the maximal norm of its derivative to power d. Thisisawellknown theorem of Bowen [6], but we get its new simple proof.
3. Another definition of topological entropy. While we are considering the Bowen’s definition of topological entropy, we will use the opportunity to propose (later in this section) a simpler definition in case when Y = X. It is kind of a hybrid between the Bowen’s definition and the original definition (of Adler, Konheim and McAndrew [1]). Let us recall the original definition of the topological entropy. Let X be a nonempty compact Hausdorff space, f : X → X a continuous map, and A an open cover of X (not necessarily finite). Set −1 −(n−1) An = A∨f (A) ∨···∨f (A).
That is, An consists of the sets of the form −1 −(n−1) A0 ∩ f (A1) ∩···∩f (An−1), where Ai ∈Afor i =0, 1,...,n − 1. Let Nn be the minimal cardinality of a subcover of X chosen from An. The sequence (Nn) is submultiplicative, and we set 1 1 h(f,A) = lim log Nn =inf log Nn. n→∞ n n n Then h(f) is the supremum of h(f,A) over all open covers A of X. FixanopencoverA. Consider a finite cover B and nonnegative integers n(B) such that B ∈An(B) for each B ∈B. We will call such cover A-admissible and will always understand that it comes with the function n(·). The function ϕ = ϕB, given by ϕ(t)= e−tn(B) B∈B is decreasing (note that often we have n(B)=nf,A(B) for all B ∈Band then ϕ(t)=DA(B,t)). Since ϕ(0) ≥ 1 and limt→∞ ϕ(t) = 0, there is a unique λ(B) ≥ 0 such that ϕ(λ(B)) = 1. With the above notation, we have the following lemma. Its proof is a part of the proof of Proposition 1 of [7]. However, since formally the assumptions in [7] are stronger, and some important details are omitted there, we supply a proof. Lemma 3.1. If ϕ(t) < 1 then h(f,A) ≤ t. k Proof: For B =(B1,...,Bk) ∈B ,set i−1 ni(B)= n(Bj) j=1 if 1 ≤ i ≤ k + 1; then n(B)=nk+1(B)and k −(ni(B)) C(B)= f (Bi). i=1 ON BOWEN’S DEFINITION OF TOPOLOGICAL ENTROPY 831
Since
−(ni(B)) −(ni(B)) −(ni(B)+1) −(ni+1(B)−1) f (Bi) ∈ f (A) ∨ f (A) ∨···∨f (A), we have C(B) ∈An(B). If x ∈ X then we can construct by induction a sequence (Bi) such that x ∈ C(B1,...,Bk) for every k.IfM = max{n(B):B ∈B}, then for every n we can choose k such that that n ≤ n(B1,...,Bk) Cn = {C(B):n ≤ n(B) Now Lemma 3.1 suggests the following theorem. Theorem 3.2. For a given open cover A of X, the infimum h of the numbers λ(B) over all A-admissible covers B is equal to h(f,A). Proof: By Lemma 3.1 and since ϕB is decreasing, if t>λ(B) then h(f,A) ≤ t. Thus h(f,A) ≤ λ(B) for every A-admissible cover B,soh ≥ h(f,A). On the other hand, for every n the subcover Bn of An of minimal cardinality n B n B ∈B A ϕ t N e−tn (with ( )= for every n)is -admissible. We have Bn ( )= n and 1 thus λ(Bn)= n log Nn. Therefore infn λ(Bn)=h(f,A), so h ≤ h(f,A). By this theorem, we can define the topological entropy as follows. For a given open cover A of X we define h(f,A) as the infimum of the numbers λ(B)over all A-admissible covers B and then h(f) is the supremum of h(f,A) over all open covers A of X. Compared to the original definition, after fixing the cover A, we define the mea- sure of complexity of certain other covers. Then we define their minimal possible complexity in one step, while in the original definition this is done in two steps (although each one is simpler). To see possible applications of the new definition, consider a simple example, when X consists of all one-sided 0-1 sequences without two consecutive 1’s and f is the shift (the “golden mean” shift). This is one of the simplest subshifts of finite type. The usual way to compute its entropy is to consider the generating cover A by cylinders of length 1 and count the number of cylinders of length n (that is, the number of elements of An. This leads to looking at the powers of the transition matrix, and the entropy is the logarithm od the spectral root of this matrix. This method is not very complicated, but uses strong algebraic machinery. The cover A is a partition into two sets, corresponding to blocks 0 and 1, so cylinders of length m correspond to initial blocks of length m. Observe that if 832 MICHA L MISIUREWICZ the last symbol in the block is 1 then by adding 0 at the end we get a block that corresponds to a cylinder of length m + 1, which nevertheless is equal to our initial cylinder of length m. We can recode in a unique way every infinite 0-1 sequence from X, beginning with 0, as an a-b sequence, where a replaces the block 0 (if it is followed by another 0) and b replaces the block 01. In such a way we get all possible a-b sequences. Now every a-b cylinder is a 0-1 cylinder, whose length is the number of a’s plus twice the number of b’s plus 1 (we add 1, since the next 0-1 symbol has to be 0). For instance, the a-b cylinder corresponding to the block abaaba is the 0-1 cylinder corresponding to the block 001000100; note an additional 0 at the end. Let Bm be the partition of X corresponding to a-b blocks of length m and 1 followed by such blocks. This partition is an A-admissible cover. As we saw, for B ∈Bm we can take as n(B) the number of a’s plus twice the number of b’s plus 1 if there is no 1 in front of the block, and a number by 1 larger if there is such 1. Taking the sum of the expressions e−tn(B) over all a-b blocks of length m not starting with 1, we get m m m m e−t(m+k+1) e−t e−t m−k e−2t k e−t e−t e−2t m . k = k = + k=0 k=0 Therefore the sum over all a-b blocks of length m will be ϕ t e−t e−t e−t e−2t m e−t e−2t m+1 , Bm ( )= 1+ + = + so λ(Bm) is the solution to the equation e−t + e−2t =1. Let us call this number simply λ. As we saw in the proof of Theorem 3.2, we have h(f,A) ≤ λ. On the other hand, every 0-1 cylinder is an element B of some Bm. It can be divided into two elements B1,B2 of Bm+1 and we get e−λn(B1) + e−λn(B2) = e−λ + e−2λ e−λn(B) = e−λn(B). Thus, for any partition C of X into cylinders we can arrive by a finite number of divisions like above to some Bm andwegetϕC(λ) = 1. Too good to be true? Indeed, we assumed here that the function n(·)forC is the same as for the partitions Bm. If a 0-1 cylinder B corresponds to a block ending at 1 or 10, we have two choices of n(B). The choice we made above is the larger n(B). If we make n(B) smaller, the value of ϕC(λ) will become larger, so the solution to the equation ϕC(λ)=1 will increase. This proves that h(f,A) ≤ λ. Since A is a topological generator, we have h(f,A)=h(f), so the topological entropy of f is λ. This method of computing entropy of this system seems to be more natural than the standard one. It exploits the structure of the corresponding Markov graph rather than the algebraic properties of the transition matrix and is closely related to the rome method (see [4], [2]). 4. The rest of the proof. In order to make this paper self-contained, and to show to the reader that the full proof of Corollary 2.2 is simple, we complete here the proof of Bowen’s Proposition 1 of [7] (which we call here Proposition 4.1). Proposition 4.1. If X is a compact metric space and f : X → X is continuous then h(f,X)=h(f). ON BOWEN’S DEFINITION OF TOPOLOGICAL ENTROPY 833 Proof: Take a finite open cover A of X. Assume that λ is such that µA,λ(X)=0. This implies that there is a cover B of X with DA(B,λ) < 1. We can modify B so that it becomes A-admissible and still ϕ(λ) < 1. At the beginning we set n(B)=nf,A(B) for every B ∈B.Now,ifB ∈Band n(B) is finite, we may replace this B by an element of An(B) containing it. If n(B) is infinite, we can replace B by an element of An containing it and set n(B)=n, with n so large that ϕ(λ) < 1 still holds. If we perform infinitely many replacements of this type, we should keep ϕ(λ) less than 1 − ε for some ε>0 fixed at the beginning of the construction, so that after all replacements ϕ(λ) ≤ 1 − ε<1. After we are finished with replacements, B is an open cover of X, so we can replace it by its finite subcover. Now we apply Lemma 3.1, and get that h(f,A) ≤ λ. This proves that h(f,A) ≤ h(f,A,X). To get the opposite inequality, we observe that if Bn is a subcover of An with −nλ Nn elements, then DA(Bn,λ) ≤ Nne . Therefore 1 µA,λ(X) ≤ lim exp(n(−λ + log Nn)). n→∞ n Thus, if λ>h(f,A) then µA,λ(X) = 0. This proves that h(f,A) ≥ h(f,A,X). In such a way we get h(f,A)=h(f,A,X), and by taking the supremum of both sides over open finite covers of X,wegeth(f)=h(f,X) (in the classical definition of topological entropy it is enough to consider finite covers). 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