DE-JUN FENG AND WEN HUANG

Transcription

1 VARIATIONAL PRINCIPLES FOR TOPOLOGICAL ENTROPIES OF SUBSETS DE-JUN FENG AND WEN HUANG Abstract. Let (X, T ) be a topological dynamical system. We define the measuretheoretical lower and upper entropies h µ (T ), h µ (T ) for any µ M(X), where M(X) denotes the collection of all Borel probability measures on X. For any non-empty compact subset K of X, we show that h B top(t, K) = sup{h µ (T ) : µ M(X), µ(k) = 1}, h P top(t, K) = sup{h µ (T ) : µ M(X), µ(k) = 1}, where h B top(t, K) denotes Bowen s topological entropy of K, and h P top(t, K) the packing topological entropy of K. Furthermore, when h top (T ) <, the first equality remains valid when K is replaced by any analytic subset of X. The second equality always extends to any analytic subset of X. 1. Introduction Throughout this paper, by a topological dynamical system (TDS) (X, T ) we mean a compact metric space X together with a continuous self-map T : X X. Let M(X), M(X, T ), and E(X, T ) denote respectively the sets of all Borel probability measures, T -invariant Borel probability measures, and T -invariant ergodic Borel probability measures on X. By a measure theoretical dynamical system (m.t.d.s.) we mean (Y, C, ν, T ), where Y is a set, C is a σ-algebra over Y, ν is a probability measure on C and T is a measure preserving transformation. A probability measure µ M(X, T ) induces a m.t.d.s. (X, B X, µ, T ) or just (X, µ, T ), where B X is the σ-algebra of Borel subsets of X. In 1958 Kolmogorov [17] associated to any m.t.d.s. (Y, C, ν, T ) an isomorphic invariant, namely the measure-theoretical entropy h ν (T ). Later on in 1965, Adler, Konheim and McAndrew [1] introduced for any TDS (X, T ) an analogous notion of topological entropy h top (T ), as an invariant of topological conjugacy. There is a basic relation between topological entropy and measure-theoretic entropy: if (X, T ) is a TDS, then h top (T ) = sup{h µ (T ) : µ M(X, T )}. This variational principle 2000 Mathematics Subject Classification. Primary: 37B40, 37A35, 37B10, 37A05. Key words and phrases. Topological entropies, Measure-theoretical entropies, Variational principles. 1

2 2 DE-JUN FENG AND WEN HUANG was proved by Goodman [12], and plays a fundamental role in ergodic theory and dynamical systems (cf. [24, 26]). In 1973, Bowen [4] introduced the topological entropy h B top(t, Z) for any set Z in a TDS (X, T ) in a way resembling Hausdorff dimension, which we call Bowen s topological entropy (see Sect. 2 for the definition). In particular, h B top(t, X) = h top (T ). Bowen s topological entropy plays a key role in topological dynamics and dimension theory [24]. A question arises naturally whether there is certain variational relation between Bowen s topological entropy and measure-theoretic entropy for arbitrary non-invariant compact set, or Borel set in general. However, when K X is T -invariant but not compact, or K is compact but not T -invariant, it may happen that h B top(t, K) > 0 but µ(k) = 0 for any µ M(X, T ) (see Example 1.5). Hence we don t expect to have such variational principle on the class M(X, T ). For our purpose, we need to define the measure-theoretic entropy for elements in M(X). Fix a compatible metric d on X. For any n N, the n-th Bowen metric d n on X is defined by (1.1) d n (x, y) = max { d ( T k (x), T k (y) ) : k = 0,..., n 1 }. For every ɛ > 0 we denote by B n (x, ɛ), B n (x, ɛ) the open (resp. closed) ball of radius ɛ in the metric d n around x, i.e., (1.2) B n (x, ɛ) = {y X : d n (x, y) < ɛ}, B n (x, ɛ) = {y X : d n (x, y) ɛ}. Following the idea of Brin and Katok [6], we give the following. Definition 1.1. Let µ M(X). The measure-theoretical lower and upper entropies of µ are defined respectively by h µ (T ) = h µ (T, x) dµ(x), h µ (T ) = h µ (T, x) dµ(x), where h µ (T, x) = lim ɛ 0 lim inf n + 1 n log µ(b n(x, ɛ)), h µ (T, x) = lim lim sup 1 ɛ 0 n log µ(b n(x, ɛ)). n + Brin and Katok [6] proved that for any µ M(X, T ), h µ (T, x) = h µ (T, x) for µ-a.e x X, and h µ (T, x) dµ(x) = h µ (T ). Hence for µ M(X, T ), h µ (T ) = h µ (T ) = h µ (T ). To formulate our results, we need to introduce an additional notion. A set in a metric space is said to be analytic if it is a continuous image of the set N of

3 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 3 infinite sequences of natural numbers (with its product topology). It is known that in a Polish space, the analytic subsets are closed under countable unions and intersections, and any Borel set is analytic (cf. Federer [11, ]). The main results of this paper are the following two theorems. Theorem 1.2. Let (X, T ) be a TDS. (i) If K X is non-empty and compact, then h B top(t, K) = sup{h µ (T ) : µ M(X), µ(k) = 1}. (ii) Assume that h top (T ) <. If Z X is analytic, then (1.3) h B top(t, Z) = sup{h B top(t, K) : K Z is compact }. Theorem 1.3. Let (X, T ) be a TDS. (i) If K X is non-empty and compact, then h P top(t, K) = sup{h µ (T ) : µ M(X), µ(k) = 1}, where h P top(t, K) denotes the packing topological entropy of K (see Sect. 2 for the definition). (ii) If Z X is analytic, then (1.4) h P top(t, Z) = sup{h P top(t, K) : K Z is compact }. The above two theorems establish the variational principles for Bowen and packing topological entropies of arbitrary Borel sets in a dual manner. They provide as a kind of extension of the classical variational principle for topological entropy of compact invariant sets. In the reminder of this section, we give two examples which motivated this paper. Example 1.4. Let (X, T ) denote the one-sided full shift over a finite alphabet {1, 2,..., l}, where l is an integer 2. Endow X with the metric d(x, y) = e n for x = (x j ) j=1 and y = (y j ) j=1, where n is the largest integer such that x j = y j (1 j n). It is easy to check by definition that for any E X, h B top(t, E) = dim H E, h P top(t, E) = dim P E, where dim H E, dim P E denote respectively the Hausdorff dimension and the packing dimension of E in the ultra-metric space (X, d) (cf. [23]). It is a well known fact in geometric measure theory (cf. [23]) that, for any analytic set Z X with dim H Z > 0, and any 0 s < dim H Z, 0 t < dim P Z, there exist compact sets K 1, K 2 Z such that 0 < H s (K 1 ) <, 0 < P t (K 2 ) <,

4 4 DE-JUN FENG AND WEN HUANG where H s, P s denote respectively the s-dimensional Hausdorff measure and packing measure, and hence dim H K 1 = s, dim P K 2 = t. Furthermore, for H s -a.e x K 1, and P t -a.e y K 2, lim inf r 0 log H s (K 1 B r (x)) log r = s, lim sup r 0 log P t (K 2 B r (x)) log r where B r (x) denotes the open ball centered at x of radius r. This can derive Theorems in the full shift case with some additional density arguments as in [23, p.99, Exercises 6-7]. Example 1.5. Again let (X, T ) denote the one-sided full shift over a finite alphabet {1, 2,..., l}. Define ϕ : X R as { 1 if x1 = 1 ϕ(x) = 0 otherwise for x = (x i ) i=1 X. Let E denote the set of non-typical points associated with the Birkhoff average of ϕ, i.e., { } 1 n 1 E = x X : lim inf ϕ(t i 1 n 1 x) lim sup ϕ(t i x). n n n n i=0 It is easy to see that E is T -invariant and Borel. By the Birkhoff ergodic theorem, µ(e) = 0 for any µ M(X, T ). However h B top(t, E) = h top (T ) = log l (cf. [2]). Furthermore, as we mention in Example 1.4 that for any 0 s < log l, there exists a compact set K E such that h B top(t, K) = dim H K = s. In our proofs of Theorems , we use and extend some ideas and techniques in geometric measure theory and topological dynamical systems. We remark that the assumption h B top(t ) < in Theorem 1.2(ii) can be weaken somehow (see Remark 3.15). However it remains open whether this assumption can be removed. The paper is organized as follows. In Sect. 2 we give the definitions and some basic properties of several topological entropies of subsets in a TDS: upper capacity topological entropy, Bowen s topological entropy, the packing topological entropy. In Sect. 3.3, we prove Theorem 1.2. In Sect. 4, we prove Theorem 1.3. i=0 2. Topological entropies of subsets In this section, we give the definitions and some basic properties of several topological entropies of subsets in a TDS: upper capacity topological entropy, Bowen s topological entropy and packing topological entropy. Let (X, d) be a compact metric space and T : X X a continuous transformation. Let d n and B n (x, ɛ) be defined as in (1.1)-(1.2). = t,

5 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES Upper capacity topological entropy. Let Z X be a non-empty set. For ɛ > 0, a set E Z is called a (n, ɛ)-separated set of Z if x, y E, x y implies d n (x, y) > ɛ; E X is called (n, ɛ)-spanning set of Z, if for any x Z, there exists y E with d n (x, y) ɛ. Let r n (Z, ɛ) denote the largest cardinality of (n, ɛ)- separated sets for Z, and r n (Z, ɛ) the smallest cardinality of (n, ɛ)-separated sets of Z. The upper capacity topological entropy of T restricted on Z, or simply, the upper capacity topological entropy of Z is defined as h UC top(t, Z) = lim ɛ 0 lim sup n 1 n log r n(z, ɛ) = lim ɛ 0 lim sup n 1 n log r n(z, ɛ). We remark that the second equality holds for each Z X (cf. [26, P. 169]). The quantity h UC top(t, Z) is the straightforward generalization of the Adler-Konheim- McAndrew definition [1] of the topological entropy to arbitrary subsets Bowen s topological entropy. Suppose that U is a finite open cover of X. Denote diam(u) := max{diam(u) : U U}. For n 1 we denote by W n (U) the collection of strings U = U 1... U n with U i U. For U W n (U) we call the integer m(u) = n the length of U and define X(U) = U 1 T 1 U 2... T (n 1) U n = { x X : T j 1 x U j for j = 1,..., n }. Let Z X. We say that Λ n 1 W n(u) covers Z if U Λ X(U) Z. For s R, define M s N(U, Z) = inf e sm(u), Λ where the infimum is taken over all Λ j N W j(u) that cover Z. Clearly M s N (U, ) is a finite outer measure on X, and U Λ (2.1) M s N(U, Z) = inf{m s N(U, G) : G Z, G is open}. Note that M s N (U, Z) increases as N increases. Define Ms (U, Z) = lim N M s N (U, Z) and Set h B top(t, U, Z) = inf{s : M s (U, Z) = 0} = sup{s : M s (U, Z) = + }. (2.2) h B top(t, Z) = sup h B top(t, U, Z), U where U runs over finite open covers of Z. We call h B top(t, Z) the Bowen s topological entropy of T restricted to Z or, simply, the topological entropy of Z. This quantity was first introduced by Bowen in [4]. It is known (see, i.e. [24, Theorem 11.1]) that (2.3) sup h B top(t, U, Z) = lim h B top(t, U, Z). U diam(u) 0

6 6 DE-JUN FENG AND WEN HUANG Bowen s topological entropy of subsets can be defined in an alternative way. For Z X, s 0, N N and ɛ > 0, define M s N,ɛ(Z) = inf exp( sn i ), i where the infimum is taken over all finite or countable families {B ni (x i, ɛ)} such that x i X, n i N and i B n i (x i, ɛ) Z. The quantity M s N,ɛ (Z) does not decrease as N increases and ɛ decreases, hence the following limits exist: M s ɛ(z) = lim N Ms N,ɛ(Z), M s (Z) = lim ɛ 0 M s ɛ(z). Bowen s topological entropy h B top(t, Z) can be equivalently defined as a critical value of the parameter s, where M s (Z) jumps from to 0, i.e. 0, s > h B top(t, Z), M s (Z) =, s < h B top(t, Z). For details, see [24, Page 74] Packing topological entropy. Let Z X. For s 0, N N and ɛ > 0, define PN,ɛ(Z) s = sup exp( sn i ), i where the supermum is taken over all finite or countable pairwise disjoint families {B ni (x i, ɛ)} such that x i Z, n i N for all i, where B n (x, ɛ) := {y X : d n (x, y) ɛ}. The quantity PN,ɛ s (Z) does not decrease as N, ɛ decrease, hence the following limits exist: Pɛ s (Z) = lim P N,ɛ(Z). s N Define { } Pɛ s (Z) = inf Pɛ s (Z i ) : Z i Z. i=1 Clearly, Pɛ s satisfies the following property: if Z i=1 Z i, then Pɛ s (Z) i=1 Ps ɛ (Z i ). There exists a critical value of the parameter s, which we will denote by h P top(t, Z, ɛ), where Pɛ s (Z) jumps from to 0, i.e. 0, s > h P top(t, Z, ɛ), Pɛ s (Z) =, s < h P top(t, Z, ɛ). Note that h P top(t, Z, ɛ) increases when ɛ decreases. We call i=1 h P top(t, Z) := lim ɛ 0 h P top(t, Z, ɛ)

7 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 7 the packing topological entropy of T restricted to Z or, simply, the packing topological entropy of Z, when there is no confusion about T. This quantity is defined in way which resembles the packing dimension. We remark that an equivalent definition of packing topological entropy was given earlier in [15] Some basic properties. Proposition 2.1. (i) For Z Z, h UC top(t, Z) h UC top(t, Z ), h B top(t, Z) h B top(t, Z ), h P top(t, Z) h P top(t, Z ). (ii) For Z i=1 Z i, s 0 and ɛ > 0, we have M s ɛ(z) i=1 M s ɛ(z i ), h B top(t, Z) sup i 1 (iii) For any Z X, h B top(t, Z) h P top(t, Z) h UC top(t, Z). (iv) Furthermore, if Z is T -invariant and compact, then h B top(t, Z i ), h P top(t, Z) sup h P top(t, Z i ). i 1 h B top(t, Z) = h P top(t, Z) = h UC top(t, Z). Proof. (i) and (ii) follow directly from the definitions of topological entropies. To see (iii), let Z X and assume 0 < s < h B top(t, Z). For any n N and ɛ > 0, let R = R n (Z, ɛ) be the largest number so that there is a disjoint family {B n (x i, ɛ)} R i=1 with x i Z. Then it is easy to see that for any δ > 0, R B n (x i, 2ɛ + δ) Z, i=1 which implies that M s n,2ɛ+δ (Z) Re ns Pn,ɛ(Z) s for any s 0, and hence M s 2ɛ+δ (Z) P ɛ s (Z). By (ii), M s 2ɛ+δ (Z) Ps ɛ (Z). Since 0 < s < h B top(t, Z), we have M s (Z) = and thus M s 2ɛ+δ (Z) 1 when ɛ and δ are small enough. Hence Pɛ s (Z) 1 and h P top(t, Z, ɛ) s when ɛ is small. Therefore h P top(t, Z) = lim ɛ 0 h P top(t, Z, ɛ) s. This implies that h B top(t, Z) h P top(t, Z). Next we show that h P top(t, Z) h UC top(t, Z). Our argument is modified slightly from the proof of [10, Lemma 3.7]. Assume that h P top(t, Z) > 0; otherwise there is nothing left to prove. Choose 0 < t < s < h P top(t, Z). Then there exists δ > 0 such that for 0 < ɛ < δ, h P top(t, Z, ɛ) > s and thus Pɛ s (Z) Pɛ s (Z) =. Thus for any N, there exists a countable pairwise disjoint families {B ni (x i, ɛ)} such that x i Z, n i N for all i, and 1 < i e nis. For each k, let m k be the number of i so that n i = k. Then we have 1 < m k e ks. k=n

8 8 DE-JUN FENG AND WEN HUANG There must be some k N with m k > e kt (1 e t s ), otherwise the above sum is at most k=1 ekt ks (1 e t s ) < 1. Let r k (Z, ɛ) denote the largest cardinality of (k, ɛ)-separated sets for Z. Then r k (Z, ɛ) m k > e kt (1 e t s ). Hence 1 lim sup log r n n(z, ɛ) t. Letting ɛ 0, we obtain h UC top(t, Z) t. This is true for n any 0 < t < h P top(t, Z) so h UC top(t, Z) h P top(t, Z). When Z X is T -invariant and compact, Bowen [4] proved that h B top(t, Z) = h UC top(t, Z); this together with (iii) yields (iv). 3. Variational principle for Bowen s topological entropy of subsets 3.1. Weighted topological entropy. For any function f : X [0, ), N N and ɛ > 0, define (3.1) WN,ɛ(f) s = inf c i exp( sn i ), i where the infimum is taken over all finite or countable families {(B ni (x i, ɛ), c i )} such that 0 < c i <, x i X, n i N and c i χ Bi f, i where B i := B ni (x i, ɛ), and χ A denotes the characteristic function of A, i.e, χ A (x) = 1 if x A and 0 if x X\A. For Z X and f = χ Z we set WN,ɛ s (Z) = Ws N,ɛ (χ Z). The quantity WN,ɛ s (Z) does not decrease as N increases and ɛ decreases, hence the following limits exist: Wɛ s (Z) = lim N Ws N,ɛ(Z), W s (Z) = lim Wɛ s (Z). ɛ 0 We remark that W s is defined in a way which resembles the weighted Hausdorff measure in geometric measure theory (cf. [11, 23]). Clearly, there exists a critical value of the parameter s, which we will denote by h W top B (T, Z), where W s (Z) jumps from to 0, i.e. { 0, s > h W s W B top (T, Z), (Z) =, s < h W top B (T, Z). We call h W top B (T, Z) the weighted Bowen s topological entropy of T restricted to Z or, simply, the weighted Bowen s topological entropy of Z Equivalence of h B top and h W B top. The following properties about M s (cf. Sect. 2.2) and W s can be verifies directly from the definitions. Proposition 3.1. (i) For any s 0, N N and ɛ > 0, both M s N,ɛ and Ws N,ɛ are outer measures on X. (ii) For any s 0, both M s and W s are metric outer measures on X.

9 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 9 We remark that M s and W s depend not only s but also the TDS (X, T ). However, M s and W s are purely topological and independent of the special choice of the metric d. The main result of this subsection is the following. Proposition 3.2. Let Z X. Then for any s 0 and ɛ, δ > 0, we have M s+δ N,6ɛ (Z) Ws N,ɛ(Z) M s N,ɛ(Z), when N is large enough. As a result, M s+δ (Z) W s (Z) M s (Z) and h B top(t, Z) = h W B top (T, Z). To prove Proposition 3.2, we need the following lemma. Lemma 3.3 ([23], Theorem 2.1). Let (X, d) be a compact metric space and B = {B(x i, r i )} i I be a family of closed (or open) balls in X. Then there exists a finite or countable subfamily B = {B(x i, r i )} i I of pairwise disjoint balls in B such that B B(x i, 5r i ). B B i I Proof of Proposition 3.2. Let Z X, s 0, ɛ, δ > 0. Taking f = χ Z and c i 1 in the definition (3.1), we see that WN,ɛ s (Z) Ms N,ɛ (Z) for each N N. In the following, we prove that M s+δ N,6ɛ (Z) Ws N,ɛ (Z) when N is large enough. Assume that N 2 such that n 2 e nδ 1 for n N. Let {(B ni (x i, ɛ), c i )} i I be a family so that I N, x i X, 0 < c i <, n i N and (3.2) c i χ Bi χ Z, where B i := B ni (x i, ɛ). We show below that (3.3) M s+δ N,6ɛ (Z) c i e nis, i I which implies M s+δ N,6ɛ (Z) Ws N,ɛ (Z). i Denote I n := {i I : n i = n} and I n,k = {i I n : i k} for n N and k N. Write for brevity B i := B ni (x i, ɛ) and 5B i := B ni (x i, 5ɛ) for i I. Obviously we may assume B i B j for i j. For t > 0, set { } Z n,t = x Z : and Z n,k,t = { x Z : c i χ Bi (x) > t i I n } c i χ Bi (x) > t. i I n,k We divide the proof of (3.3) into the following three steps.

10 10 DE-JUN FENG AND WEN HUANG Step 1. For each n N, k N and t > 0, there exists a finite set J n,k,t I n,k such that the balls B i (i J n,k,t ) are pairwise disjoint, Z n,k,t i J n,k,t 5B i and #(J n,k,t )e ns 1 c i e ns. t i I n,k To prove the above result, we adopt the method of Federer [11, ] used in the study of weighted Hausdorff measures (see also Mattila [23, Lemma 8.16]). Since I n,k is finite, by approximating the c i s from above, we may assume that each c i is a positive rational, and then multiplying with a common denominator we may assume that each c i is a positive integer. Let m be the least integer with m t. Denote B = {B i, i I n,k } and define u : B Z by u(b i ) = c i. We define by induction integer-valued functions v 0, v 1,..., v m on B and sub-families B 1,..., B m of B starting with v 0 = u. Using Lemma 3.3 (in which we take the metric d n instead of d) we find a pairwise disjoint subfamily B 1 of B such that B B B B B 1 5B, and hence Z n,k,t B B 1 5B. Then by repeatedly using Lemma 3.3, we can define inductively for j = 1,..., m, disjoint subfamilies B j of B such that B j {B B : v j 1 (B) 1}, Z n,k,t B B j 5B and the functions v j such that { vj 1 (B) 1 for B B v j (B) = j, v j 1 (B) for B B\B j. This is possible since for j < m, Z n,k,t { x : B B: B x v j(b) m j }, whence every x Z n,k,t belongs to some ball B B with v j (B) 1. Thus m m #(B j )e ns = (v j 1 (B) v j (B))e ns j=1 j=1 B B j m j 1 (B) v j (B))e B B j=1(v ns u(b)e ns = c i e ns. B B i I n,k Choose j 0 {1,..., m} so that #(B j0 ) is the smallest. Then #(B j0 )e ns 1 c i e ns 1 c i e ns. m t i I n,k i I n,k Hence J n,k,t = {i I : B i B j0 } is desired. Step 2. For each n N and t > 0, we have (3.4) M s+δ N,6ɛ (Z n,t) 1 c n 2 i e ns. t i I n To see this, assume Z n,t ; otherwise this is nothing to prove. Since Z n,k,t Z n,t, Z n,k,t when k is large enough. Let J n,k,t be the sets constructed in Step 1. Then

11 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 11 J n,k,t when k is large enough. Define E n,k,t = {x i : i J n,k,t }. Note that the family of all non-empty compact subsets of X is compact with respect to the Hausdorff distance (cf. Federer [11, ]). It follows that there is a subsequence (k j ) of natural numbers and a non-empty compact set E n,t X such that E n,kj,t converges to E n,t in the Hausdorff distance as j. Since any two points in E n,k,t have a distance (with respect to d n ) not less than ɛ, so do the points in E n,t. Thus E n,t is a finite set, moreover, #(E n,kj,t) = #(E n,t ) when j is large enough. Hence B n (x, 5.5ɛ) B n (x, 5ɛ) = 5B i Z n,kj,t x E n,t x E n,kj,t i J n,kj,t when j is large enough, and thus x E n,t B n (x, 6ɛ) Z n,t. By the way, since #(E n,kj,t) = #(E n,t ) when j is large enough, we have #(E n,t )e ns 1 t i I n c i e ns. This forces M s+δ N,6ɛ (Z n,t) #(E n,t )e n(s+δ) 1 c e nδ i e ns 1 c t n 2 i e ns. t i I n i I n Step 3. For any t (0, 1), we have M s+δ N,6ɛ (Z) 1 t i I c ie n is. As a result, (3.3) holds. To see this, fix t (0, 1). Note that n=n n 2 < 1. It follows that Z n=n Z n,n 2 t from (3.2). Hence by Proposition 3.1(i) and (3.4), we have M s+δ N,6ɛ (Z) n=n M s+δ N,6ɛ (Z n,n 2 t) which finishes the proof of the proposition. n=n 1 t c i e ns = 1 c i e nis, t i I n i I 3.3. A dynamical Frostman s lemma and the proof of Theorem 1.2 (i). To prove Theorem 1.2(i), we need the following dynamical Frostman s lemma, which is an analogue of the classical Frostman s lemma in in compact metric space. Our proof is adapt from Howroyd s elegant argument (cf. [13, Theorem 2], [23, Theorem 8.17]). Lemma 3.4. Let K be a non-empty compact subset of X. Let s 0, N N and ɛ > 0. Suppose that c := WN,ɛ s (K) > 0. Then there is a Borel probability measure µ on X such that µ(k) = 1 and µ(b n (x, ɛ)) 1 c e ns, x X, n N. Proof. Clearly c <. We define a function p on the space C(X) of continuous real-valued functions on X by where WN,ɛ s is defined as in (3.1). p(f) = (1/c)W s N,ɛ(χ K f),

12 12 DE-JUN FENG AND WEN HUANG Let 1 C(X) denote the constant function 1(x) 1. It is easy to verify that (1) p(f + g) p(f) + p(g) for any f, g C(X). (2) p(tf) = tp(f) for any t 0 and f C(X). (3) p(1) = 1, 0 p(f) f for any f C(X), and p(g) = 0 for g C(X) with g 0. By the Hahn-Banach theorem, we can extend the linear functional t tp(1), t R, from the subspace of the constant functions to a linear functional L : C(X) R satisfying L(1) = p(1) = 1 and p( f) L(f) p(f) for any f C(X). If f C(X) with f 0, then p( f) = 0 and so L(f) 0. Hence combining the fact L(1) = 1, we can use the Riesz representation theorem to find a Borel probability measure µ on X such that L(f) = fdµ for f C(X). Now we show that µ(k) = 1. To see this, for any compact set E X\K, by the Uryson lemma there is f C(X) such that 0 f 1, f(x) = 1 for x E and f(x) = 0 for x K. Then f χ K 0 and thus p(f) = 0. Hence µ(e) L(f) p(f) = 0. This shows µ(x\k) = 0, i.e. µ(k) = 1. In the end, we show that µ(b n (x, ɛ)) (1/c)e ns for any x X and n N. To see this, for any compact set E B n (x, ɛ), by the Uryson lemma, there exists f C(X) such that 0 f 1, f(y) = 1 for y E and f(y) = 0 for y X\B n (x, ɛ). Then µ(e) L(f) p(f). Since f χ K χ Bn(x,ɛ) and n N, we have W s N,ɛ (χ K f) e ns and thus p(f) 1 c e sn. Therefore µ(e) 1 c e ns. It follows that µ(b n (x, ɛ)) = sup{µ(e) : E is a compact subset of B n (x, ɛ)} 1 c e sn. Remark 3.5. There is a related known result (see, e.g. [22, 25]) that, for any Borel set E X and any Borel probability measure µ on E, if h µ (T, x) s for all x E, then h B top(t, E) s; conversely if h µ (T, x) s for all x E, then h B top(t, E) s, where h µ (T, x) is defined as in Sect. 1. Now we are ready to prove Theorem 1.2(i). Proof of Theorem 1.2(i). We first show that h B top(t, K) h µ (T ) for any µ M(X) with µ(k) = 1. Let µ be a given such measure. Write h µ (T, x, ɛ) = lim inf n 1 n log µ(b n(x, ɛ))

13 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 13 for x X, n N and ɛ > 0. Clearly h µ (T, x, ɛ) is nonnegative and increases as ɛ decreases. Hence by the monotone convergence theorem, lim h µ (T, x, ɛ)dµ = h µ (T, x)dµ = h µ (T ). ɛ 0 Thus to show h B top(t, K) h µ (T ), it is sufficient to show h B top(t, K) h µ (T, x, ɛ)dµ for each ɛ > 0. Fix ɛ > 0 and l N. Denote u l = min{l, h µ (T, x, ɛ)dµ(x) 1 }. Then there l exist a Borel set A l X with µ(a l ) > 0 and N N such that (3.5) µ(b n (x, ɛ)) e nu l, x A l, n N. Now let {B ni (x i, ɛ/2)} be a countable or finite family so that x i X, n i N and i B n i (x i, ɛ/2) K A l. We may assume that for each i, B ni (x i, ɛ/2) (K A l ), and choose y i B ni (x i, ɛ/2) (K A l ). Then by (3.5), e n iu l µ(b ni (y i, ɛ)) µ(b ni (x i, ɛ/2)) i i i µ(k A l ) = µ(a l ) > 0. It follows that M u l (K) M u l N,ɛ/2 (K) Mu l N,ɛ/2 (K A l) µ(a l ). Therefore h B top(t, K) u l. Letting l, we have the desired inequality h B top(t, K) hµ (T, x, ɛ)dµ. Hence h B top(t, K) h µ (T ). We next show that h B top(t, K) sup{h µ (T ) : µ M(X), µ(k) = 1}. We can assume that h B top(t, K) > 0, otherwise we have nothing to prove. By Proposition 3.2, h BW top (T, K) = h B top(t, K). Let 0 < s < h B top(t, K). Then there exist ɛ > 0 and N N such that c := WN,ɛ s (K) > 0. By Proposition 3.4, there exists µ M(X) with µ(k) = 1 such that µ(b n (x, ɛ)) 1 c e sn for any x X and n N. Clearly h µ (T, x) h µ (T, x, ɛ) s for each x X and hence h µ (T ) h µ (T, x)dµ(x) s. This finishes the proof of Theorem 1.2(i) The proof of Theorem 1.2(ii). To prove Theorem 1.2(ii), we first prove the following. Theorem 3.6. Let (X, T ) be a TDS. Assume that X is zero-dimensional, i.e., for any δ > 0, X has a closed-open partition with diameter less than δ. Then for any analytic set Z X, h top (T, Z) = sup{h top (T, K) : K Z, K is compact}. The following proposition is needed for the proof of Theorem 3.6. Proposition 3.7. Assume U is a closed-open partition of X. Let N N. Then

14 14 DE-JUN FENG AND WEN HUANG (i) If E i E, i.e., E i+1 E i and i E i = E, then M s N(U, E) = lim i M s N(U, E i ). (ii) Assume Z X is analytic. Then M s N(U, Z) = sup{m s N(U, K) : K Z, K is compact}. Proof. We first show that (i) implies (ii). Assume that (i) holds. Let Z be analytic, i.e., there exists a continuous surjective map φ : N Z. Let Γ n1,n 2,...,n p be the set of (m 1, m 2,...) N such that m 1 n 1, m 2 n 2,..., m p n p and let Z n1,...,n p be the image of Γ n1,...,n p under φ. Let (ɛ p ) be a sequence of positive numbers. Due to (i), we can pick a sequence (n p ) of positive integers recursively so that M s N (U, Z n 1 ) M s N (U, Z) ɛ 1 and M s N(U, Z n1,...,n p ) M s N(U, Z n1,...,n p 1 ) ɛ p, p = 2, 3,... Hence M s N (U, Z n 1,...,n p ) M s N (U, Z) i=1 ɛ i for any p N. Let K = Z n1,...,n p. p=1 Since φ is continuous, we can show that p=1 Z n 1,...,n p = p=1 Z n 1,...,n p by applying Cantor s diagonal argument. Hence K is a compact subset of Z. If Λ j N W j(u) is a cover of K (of course it is an open cover), then it is a cover of Z n1,...,n p when p is large enough, which implies U Λ e sm(u) lim p M s N(U, Z n1,...,n p ) M s N(U, Z) ɛ i. Hence M s N (U, K) Ms N (U, Z) i=1 ɛ i. Since i=1 ɛ i can be chosen arbitrarily small, we prove (ii). Now we turn to prove (i). Our argument is modified from the classical proof of the increasing sets lemma for Hausdorff outer measures (cf. [7, Sect. II] and [9, Lemma 5.3]). Note that any two non-empty elements in W n (U) are disjoint, and each element in W n+1 (U) is a subset of some element in W n (U). We call this the net property of (W n (U)). Let E i E be given. Let (δ i ) be a sequence of positive numbers to be specified later and for each i, choose a covering Λ i j N W j(u) of E i such that (3.6) e sm(u) MN(U, s E i ) + δ i. U Λ i By the net property of (W n (U)), we may assume that for each i, the elements in Λ i are disjoint. i=1

15 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 15 For any x E, choose U x i=1 Λ i containing x such that m(u x ) is the smallest. By the net property of (W n (U)), the collection {U x : x E} consists of countable many disjoint elements. Relabel these elements by U i s. Clearly E i U i. We now choose an integer k. Use A 1 to denote the collection of those U i s that are taken from Λ 1. They cover a certain subset Q 1 of E k. The same subset is covered by a certain sub-collection of Λ k, denoted as Λ k,1. Since Λ k,1 also covers the smaller set Q 1 E 1, by (3.6), (3.7) e sm(u) e sm(u) + δ 1. U A 1 U Λ k,1 To see this, assume that (3.7) is false. Then by (3.6), U (Λ 1 \A 1 ) Λ k,1 e sm(u) < M s N(U, E 1 ), which contradicts the fact that (Λ 1 \A 1 ) Λ k,1 j N W j(u) is an open cover of E 1. Next we use A 2 to denote the collection of those U i s that are taken from Λ 2 but not from Λ 1. Define Λ k,2 similarly. As above, we find (3.8) e sm(u) e sm(u) + δ 2. U A 2 U Λ k,2 We repeat the argument until all coverings Λ n, n k, have been considered. Note that U Λ k,i U U A i U for i k. For different i, i k, the elements in Λ k,i are disjoint from those in Λ k,i. The k inequalities (3.7), (3.8),..., are added which yields k k e sm(u) + δ n M s N(U, E k ) + δ n + δ k. e sm(u) U k n=1 An Letting k, we have i U k n=1 Λ k,n n=1 e sm(u i) lim k M s N(U, E k ) + Since n=1 δ n can be chosen arbitrarily small we have M s N(U, E) lim k M s N(U, E k ). Since the opposite inequality is trivial we have proved (i). δ n. Proof of Theorem 3.6. Let Z be an analytic subset of X with h B top(t, Z) > 0. Let 0 < s < h B top(t, Z). By (2.2), there exists a closed-open partition U so that h B top(t, U, Z) > s and thus M s (U, Z) =. Hence M s N (U, Z) > 0 for some N N. By Proposition 3.7, we can find a compact set K Z such that M s N (U, K) > 0. It implies h B top(t, K) h B top(t, U, K) s. n=1 n=1

16 16 DE-JUN FENG AND WEN HUANG Before we prove Theorem 1.2(ii), we still need some notation and additional results in topological dynamical systems. Let us define the natural extension ( X, T ) of a TDS (X, T ) with a metric d and a surjective map T where X = {(x 1, x 2, ) : T (x i+1 ) = x i, x i X, i N} is a subspace of the product space X N = Π i=1x endowed with the compatible metric d T as d(x i, y i ) d T ((x 1, x 2, ), (y 1, y 2, )) =, 2 i T : X X is the shift homeomorphism with T (x1, x 2, ) = (T (x 1 ), x 1, x 2, ), and π i : X X is the projection to the i-th coordinate. Clearly, πi : ( X, T ) (X, T ) is a factor map. Lemma 3.8. Let (X, T ) be a TDS with a metric d and a surjective map T, ( X, T ) be the natural extension of (X, T ) and π 1 : X X be the projection to the first coordinate. Then sup x X h UC top( T, π1 1 (x)) = 0. Proof. Fix x X. For any ɛ > 0, take N N large enough such that i=n ɛ. i=1 diam(x) 2 i < Let E N π1 1 (x) be a finite (N, ɛ)-spanning set of π1 1 (x). Next we are to show that E N is also a (n, ɛ)-spanning set of π1 1 (x) for n > N. Fix n N with n > N. For any ỹ π1 1 (x), since E N is a (N, ɛ)-spanning set of π1 1 (x) there exist x E N such that d T ( T i x, T i ỹ) < ɛ for i = 0, 1,, N 1. Now for k {N, N + 1,, n 1}, we have π j ( T k x) = π j ( T k ỹ)) = T k j+1 (x) for j = 1,, k, k + 1. Thus d T ( T k x, T k d(π j ( ỹ) = T k x), π j ( T k x)) d(π j ( = T k x), π j ( T k x)) 2 j 2 j j=1 j=k+2 diam(x) 2 j j=n j=k+2 diam(x) 2 j < ɛ. This implies (d T ) n ( x, ỹ) < ɛ. Hence E N is also a (n, ɛ)-spanning set of π1 1 (x) for n > N. Let r n (π 1 (x), ɛ) denote the smallest cardinality of (n, ɛ)-spanning sets of π 1 (x). Then r n (π 1 (x), ɛ) #(E N ). Hence h UC top( T, π1 1 1 (x)) = lim lim sup ɛ 0 n n log r n(π 1 (x), ɛ) lim This ends the proof of the lemma. ɛ 0 lim sup n 1 n log #(E N) = 0. In the following part we will lift general TDSs having finite topological entropy to zero dimensional TDSs by the so called principal extensions.

17 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 17 Definition 3.9. [18] An extension π : (Z, R) (X, T ) between two TDSs is a principal extension if h ν (R) = h ν π 1(T ) for every ν M(Z, R). The following general result is needed in our proof of Theorem 1.2(ii). Proposition 3.10 (Proposition 7.8 in [5]). Every invertible TDS (X, T ) with h top (T ) < has a zero dimensional principal extension (Z, R) with R being invertible. Let π : (Y, S) (X, T ) be a factor map between two TDSs. Bowen proved that h top (S) h top (T ) + sup x X h UC top(s, π 1 (x)) (cf. [3, Theorem 17]). In fact, Bowen s proof is also valid for the following result (see, i.e. Theorem 7.3 in [14] for a detailed proof). Theorem Let π : (X, T ) (Y, S) be a factor map between two TDSs. Then for any E X one has (3.9) h B top(s, π(e)) h B top(t, E) h B top(s, π(e)) + sup h UC top(t, π 1 (y)). y Y We also need the following variational principle of conditional entropies. Proposition Let π : (X, T ) (Y, S) be a factor map between two TDSs. Then we have (3.10) sup h UC top(t, π 1 (y)) = sup (h µ (T ) h µ π 1(S)). y Y µ M(X,T ) Proof. It is the direct combination of [8, Theorem 3] and [19, Theorem 2.1]. Lemma Let (X, T ) be a TDS with h top (T ) <. Then there exists a factor map π : (H, Γ) (X, T ) such that (H, Γ) is zero dimensional and sup h UC top(γ, π 1 (x)) = 0. x X Proof. First, we take D = { 1 } n n N {0} and let Z = X D. Define R : Z Z 1 satisfying R(x, ) = (x, 1 ), n N; R(x, 1) = (T x, 1) and R(x, 0) = (x, 0) for n+1 n x X. Then (Z, R) is a TDS and R is surjective. If we identity (x, 1) with x for each x X, then X can be viewed as a closed subset of Z and R X = T. It is also clear that h top (R) = h top (T ) <. Let ( Z, R) be the natural extension of (Z, R) and π 1 : Z Z be the projection to the first coordinate. Then (3.11) sup h UC top( R, π1 1 (z)) = 0 z Z

18 18 DE-JUN FENG AND WEN HUANG by Lemma 3.8, and, so h top ( R) = h top (R) <. Since R is homeomorphism on Z, by Lemma 3.10, there exists a factor map ψ : (W, G) ( Z, R) such that (W, G) is a zero-dimensional TDS and ψ is principal extension. Since h top ( R) < and ψ is principal extension, we have the following variational principle of condition entropy (3.12) sup h UC top(g, ψ 1 ( z)) = sup (h θ (G) h θ ψ 1( R)) = 0. z Z θ M(W,G) The first equality in (3.12) follows from (3.10). Let H = ψ 1 (π 1 1 X), Γ = G H and π = π 1 ψ H. Then (H, Γ) be a zerodimensional TDS and π : (H, Γ) (X, T ) be a factor map. Applying Proposition 3.12 to the factor map π : (H, Γ) (X, T ), we obtain sup h UC top(γ, π 1 (x)) = sup (h µ (Γ) h µ π 1(T )) x X µ M(H,Γ) sup (h µ (Γ) h µ π 1(T )) µ M(W,G) = sup (h µ (Γ) h µ ψ 1(T ) + h µ ψ 1(T ) h µ π 1(T )) µ M(W,G) sup µ M(W,G) This shows sup x X h UC top(γ, π 1 (x)) = 0. (h µ (Γ) h µ ψ 1(T )) + sup ν ( Z, R) = sup h UC top(g, ψ 1 ( z)) + sup h UC top( R, π z Z 1 1 (z)) z Z = 0 ( by (3.12), (3.11)). (h ν ( R) h ν π 1 (R)) 1 Proof of Theorem 1.2(ii). By Lemma 3.13, there exists a factor map π : (Y, S) (X, T ) such that (Y, S) is zero dimensional and sup x X h UC top(s, π 1 (x)) = 0. By Theorem 3.11, we have that for any F Y, (3.13) h B top(s, F ) = h B top(t, π(f )). Let Z be an analytic subset of X. Then π 1 (Z) is also an analytic set of Y (cf. Federer [11, ]). By (3.13) and Theorem 3.6, h B top(t, Z) = h B top(s, π 1 (Z)) = sup{h B top(s, E) : E π 1 (Z), E is compact} = sup{h B top(t, π(e)) : E π 1 (Z), E is compact} sup{h B top(t, K) : K Z, K is compact}. The reverse inequality is trivial, so This finishes the proof. h B top(t, Z) = sup{h B top(t, K) : K Z, K is compact}.

19 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 19 Remark For an invertible TDS (X, T ), Lindenstrauss and Weiss [21] introduced the mean dimension mdim(x, T ) (an idea suggested by Gromov). It is well known that for an invertible TDS (X, T ), if h top (T ) < or the topological dimension of X is finite, then mdim(x, T ) = 0 (see [21, Definition 2.6 and Theorem 4.2]). In general, one can show that for an invertible TDS (X, T ), if mdim(x, T ) = 0 then (X, T ) has a zero dimensional principal extension (Z, R) with R being invertible. Indeed, let (Y, S) be an irrational rotation on the circle. Then (X Y, T S) admits a nonperiodic minimal factor (Y, S) and mdim(x Y, T S) = 0. Hence (X Y, T S) has the so called small boundary property [20, Theorem 6.2], which implies the existence of a basis of the topology consisting of sets whose boundaries have measure zero for every invariant measure. With these results it is easy to construct a refining sequence of small-boundary partitions for (X Y, T S), where the partitions have small boundaries if their boundaries have measure zero for all µ M(X Y, T S). Then by a standard construction (see p in [5]), which associates to this sequence a zero dimensional principal extension (Z, R) of (X Y, T S) with R being invertible. Finally note that (X Y, T S) is a principal extension of (X, T ), we know that (Z, R) is also a zero dimensional principal extension of (X, T ) since the composition of two principal extensions is still a principal extension. Remark By Remark 3.14, we may strengthen Theorem 1.2(ii) as follows: Let (X, T ) be a TDS with mdim(x, T ) = 0. Then for any analytic set Z X, h B top(t, Z) = sup{h B top(t, K) : K Z, K is compact}. 4. Variational principle for the packing topological entropy In this section we prove Theorem 1.3. We first give a lemma. Lemma 4.1. Let Z X and s, ɛ > 0. Assume P s ɛ (Z) =. Then for any given finite interval (a, b) R with a 0 and any N N, there exists a finite disjoint collection {B ni (x i, ɛ)} such that x i Z, n i N and i e n is (a, b). Proof. Take N 1 > N large enough such that e N1s < b a. Since Pɛ s (Z) =, we have PN s 1,ɛ (Z) =. Thus there is a finite disjoint collection {B n i (x i, ɛ)} such that x i Z, n i N 1 and i e nis > b. Since e nis < b a, by discarding elements in this collection one by one until we can have i e nis (a, b). Proof of Theorem 1.3. We divide the proof into two parts:

20 20 DE-JUN FENG AND WEN HUANG Part 1. Z X. h P top(t, Z) sup{h µ (T ) : µ M(X), µ(z) = 1} for any Borel set To see this, let µ M(X) with µ(z) = 1 for some Borel set Z X. We need to show that h P top(t, Z) h µ (T ). For this purpose we may assume h µ (T ) > 0; otherwise we have nothing to prove. Let 0 < s < h µ (T ). Then there exist ɛ, δ > 0, and a Borel set A Z with µ(a) > 0 such that h µ (T, x, ɛ) > s + δ, x A, where h µ (T, x, ɛ) := lim sup n 1 n log µ(b n(x, ɛ)). Next we show that Pɛ/5 s (Z) =, which implies that hp top(t, Z) h P top(t, Z, ɛ/5) s. To achieve this, it suffices to show that Pɛ/5 s (E) = for any Borel E A with µ(e) > 0. Fix such a set E. Define E n = {x E : µ(b n (x, ɛ)) < e n(s+δ) }, n N. Since E A, we have n=n E n = E for each N N. µ( n=n E n) = µ(e), and hence there exists n N such that Fix N N. Then 1 µ(e n ) n(n + 1) µ(e). Fix such n and consider the family {B n (x, ɛ/5) : x E n }. By Lemma 3.3 (in which we use d n instead of d), there exists a finite pairwise disjoint family {B n (x i, ɛ/5)} with x i E n such that B n (x i, ɛ) B n (x, ɛ/5) E n. i x E n Hence P s N,ɛ/5(E) P s N,ɛ/5(E n ) i e ns e nδ i e n(s+δ) e nδ i µ(b n (x i, ɛ)) e nδ µ(e n ) e nδ n(n + 1) µ(e). Since e nδ as n, letting N we obtain that P s n(n+1) ɛ/5 (E) =. Part 2. Let Z X be analytic with h P top(t, Z) > 0. For any 0 < s < h P top(t, Z), there exists a compact set K Z and µ M(K) such that h µ (T ) s. Since Z is analytic, there exists a continuous surjective map φ : N Z. Let Γ n1,n 2,...,n p be the set of (m 1, m 2,...) N such that m 1 n 1, m 2 n 2,..., m p n p and let Z n1,...,n p be the image of Γ n1,...,n p under φ. Take ɛ > 0 small enough so that 0 < s < h P top(t, Z, ɛ). Take t (s, h P top(t, Z, ɛ)). We are going to construct inductively a sequence of finite sets (K i ) i=1 and a sequence of finite measures (µ i ) i=1 so that K i Z and µ i is supported on K i for each i.

21 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 21 Together with these two sequences, we construct also a sequence of integers (n i ), a sequence of positive numbers (γ i ) and a sequence of integer-valued functions (m i : K i N). The method of our construction is inspired somehow by the work of Joyce and Preiss [16] on packing measures. The construction is divided into several small steps: Step 1. Construct K 1 and µ 1, as well as m 1 ( ), n 1 and γ 1. Note that P t ɛ(z) =. Let H = {G X : G is open, P t ɛ(z G) = 0}. Then P t ɛ(z H) = 0 by the separability of X. Let Z = Z\H = Z (X\H). For any open set G X, either Z G =, or P t ɛ(z G) > 0. To see this, assume P t ɛ(z G) = 0 for an open set G; then P t ɛ(z G) P t ɛ(g Z ) + P t ɛ(z H) = 0, implying G H and hence Z G =. Note that Pɛ(Z t ) = Pɛ(Z) t = (because Pɛ(Z) t Pɛ(Z t ) + Pɛ(Z t H) = Pɛ(Z t )). It follows Pɛ s (Z ) =. By Lemma 4.1, we can find a finite set K 1 Z, an integervalued function m 1 (x) on K 1 such that the collection {B m1 (x)(x, ɛ)} x K1 is disjoint and e m1(x)s (1, 2). x K 1 Define µ 1 = x K 1 e m1(x)s δ x, where δ x denotes the Dirac measure at x. Take a small γ 1 > 0 such that for any function z : K 1 X with d(x, z(x)) γ 1, we have for each x K 1, ( ) ( ) (4.1) B(z(x), γ 1 ) B m1 (x)(z(x), ɛ) B(z(y), γ 1 ) B m1 (y)(z(y), ɛ) =. y K 1 \{x} Here and afterwards, B(x, ɛ) denotes the closed ball {y X : d(x, y) ɛ)}. Since K 1 Z, P t ɛ(z B(x, γ 1 /4)) P t ɛ(z B(x, γ 1 /4)) > 0 for each x K 1. Therefore we can pick a large n 1 N so that Z n1 K 1 and P t ɛ(z n1 B(x, γ 1 /4)) > 0 for each x K 1. Step 2. Construct K 2 and µ 2, as well as m 2 ( ), n 2 and γ 2. By (4.1), the family of balls {B(x, γ 1 )} x K1, are pairwise disjoint. For each x K 1, since P t ɛ(z n1 B(x, γ 1 /4)) > 0, we can construct as Step 1, a finite set and an integer-valued function such that E 2 (x) Z n1 B(x, γ 1 /4) m 2 : E 2 (x) N [max{m 1 (y) : y K 1 }, )

22 22 DE-JUN FENG AND WEN HUANG (2-a) Pɛ(Z t n1 G) > 0 for each open set G with G E 2 (x) ; (2-b) The elements in {B m2 (y)(y, ɛ)} y E2 (x) are disjoint, and µ 1 ({x}) < e m2(y)s < ( )µ 1 ({x}). y E 2 (x) To see it, we fix x K 1. Denote F = Z n1 B(x, γ 1 /4). Let H x := {G X : G is open P t ɛ(f G) = 0}. Set F = F \H x. Then as in Step 1, we can show that P t ɛ(f ) = P t ɛ(f ) > 0 and furthermore, P t ɛ(f G) > 0 for any open set G with G F. Note that P s ɛ (F ) = (since s < t), by Lemma 4.1, we can find a finite set E 2 (x) F and a map m 2 : E 2 (x) N [max{m 1 (y) : y K 1 }, ) so that (2-b) holds. Observe that if a open set G satisfies G E 2 (x), then G F, and hence P t ɛ(z n1 G) P t ɛ(f G) > 0. Thus (2-a) holds. Since the family {B(x, γ 1 )} x K1 is disjoint, E 2 (x) E 2 (x ) = for different x, x K 1. Define K 2 = x K 1 E 2 (x) and µ 2 = y K 2 e m 2(y)s δ y. By (4.1) and (2-b), the elements in {B m2 (y)(y, ɛ)} y K2 are pairwise disjoint. Hence we can take 0 < γ 2 < γ 1 /4 such that for any function z : K 2 X with d(x, z(x)) < γ 2 for x K 2, we have (4.2) ( B(z(x), γ 2 ) B m2 (x)(z(x), ɛ) ) ( y K 2 \{x} ) B(z(y), γ 2 ) B m2 (y)(z(y), ɛ) = for each x K 2. Choose a large n 2 N such that Z n1,n 2 K 2 and Pɛ(Z t n1,n 2 B(x, γ 2 /4)) > 0 for each x K 2. Step 3. Assume that K i, µ i, m i ( ), n i and γ i have been constructed for i = 1,..., p. In particular, assume that for any function z : K p X with d(x, z(x)) < γ p for x K p, we have (4.3) ( B(z(x), γ p ) B mp(x)(z(x), ɛ) ) ( y K p\{x} ) B(z(y), γ p ) B mp(y)(z(y), ɛ) = for each x K p ; and Z n1,...,n p K p and P t ɛ(z n1,...,n p B(x, γ p /4)) > 0 for each x K p. We construct below each term of them for i = p + 1 in a way similar to Step 2. Note that the elements in {B(x, γ p )} x Kp are pairwise disjoint. For each x K p, since P t ɛ(z n1,...,n p B(x, γ p /4)) > 0, we can construct as Step 2, a finite set E p+1 (x) Z n1,...,n p B(x, γ p /4)

23 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 23 and an integer-valued function such that m p+1 : E p+1 (x) N [max{m p (y) : y K p }, ) (3-a) P t ɛ(z n1,...,n p G) > 0 for each open set G with G E p+1 (x) ; and (3-b) {B mp+1 (y)(y, ɛ)} y Ep+1 (x) are disjoint and satisfy µ p ({x}) < e mp+1(y)s < (1 + 2 p 1 )µ p ({x}). y E p+1 (x) Clearly E p+1 (x) E p+1 (x ) = for different x, x K p. Define K p+1 = x K p E p+1 (x) and µ p+1 = y K p+1 e mp+1(y)s δ y. By (4.3) and (3-b), {B mp+1 (y)(y, ɛ)} y Kp+1 are disjoint. Hence we can take 0 < γ p+1 < γ p /4 such that for any function z : K p+1 X with d(x, z(x)) < γ p+1, we have for each x K p+1, (4.4) ( ) ( B(z(x), γ p+1 ) B mp+1 (x)(z(x), ɛ) y K p+1 \{x} Choose a large n p+1 N such that Z n1,...,n p+1 K p+1 and for each x K p+1. P t ɛ(z n1,...,n p+1 B(x, γ p+1 /4)) > 0 ) B(z(y), γ p+1 ) B mp+1 (y)(z(y), ɛ) =. As in the above steps, we can construct by induction the sequences (K i ), (µ i ), (m i ( )), (n i ) and (γ i ). We summarize some of their basic properties as follows: (a) For each i, the family F i := {B(x, γ i ) : x K i } is disjoint. Each element in F i+1 is a subset of B(x, γ i /2) for some x K i. (b) For each x K i and z B(x, γ i ), B mi (x)(z, ɛ) B(y, γ i ) = and µ i (B(x, γ i )) = e m i(x)s y K i \{x} y E i+1 (x) where E i+1 (x) = B(x, γ i ) K i+1. e m i+1(y)s (1 + 2 i 1 )µ i (B(x, γ i )), The second part in (b) implies, µ i (F i ) µ i+1 (F i ) = µ i+1 (F ) (1 + 2 i 1 )µ i (F i ), F F i+1 : F F i F i F i

24 24 DE-JUN FENG AND WEN HUANG Using the above inequalities repeatedly, we have for any j > i, j (4.5) µ i (F i ) µ j (F i ) (1 + 2 n )µ i (F i ) Cµ i (F i ), F i F i, n=i+1 where C := n=1 (1 + 2 n ) <. Let µ be a limit point of (µ i ) in the weak-star topology. Let K = K i. n=1 i n Then µ is supported on K. Furthermore K = K i n=1 i n Z n1,...,n p. However by the continuity of φ, we can show that p=1 Z n 1,...,n p = p=1 Z n 1,...,n p by applying Cantor s diagonal argument. Hence K is a compact subset of Z. On the other hand, by (4.5), e m i(x)s = µ i (B(x, γ i )) µ(b(x, γ i )) Cµ i (B(x, γ i )) = Ce m i(x)s, x K i. In particular, 1 x K 1 µ 1 (B(x, γ 1 )) µ(k) x K 1 Cµ 1 (B(x, γ 1 )) 2C. Note that K x K i B(x, γ i /2). By the first part of (b), for each x K i and z B(x, γ i ), p=1 µ(b mi (x)(z, ɛ)) µ(b(x, γ i /2)) Ce m i(x)s. For each z K and i N, z B(x, γ i /2) for some x K i. Hence µ(b mi (x)(z, ɛ)) Ce m i(x)s. Define µ = µ/ µ(k). Then µ M(K), and for each z K, there exists a sequence k i such that µ(b ki (z, ɛ)) Ce kis / µ(k). It follows that h µ (T ) s. 5. Main notation and conventions For the reader s convenience, we summarize in Table 1 the main notation and typographical conventions used in this paper. Acknowledgements The first author was partially supported by the RGC grant and the Focused Investments Scheme B in CUHK. The second author was partially supported by NSFC ( , ), Fok Ying Tung Education Foundation, FANEDD (Grant ), and the Fundamental Research Funds for the Central Universities (WK ,WK ). The authors thank Hanfeng Li for helpful comments.

25 VARIATIONAL PRINCIPLES OF TOPOLOGICAL ENTROPIES 25 Table 1. Main notation and conventions (X, T ) A topological dynamical system (Sect. 1) M(X) Set of all Borel probability measures on X M(X, T ), E(X, T ) Set of T -invariant (resp. ergodic) Borel probability measures on X d n n-th Bowen s metric (cf. (1.1)) B(x, ɛ), B(x, ɛ) Open (resp. closed) ball in (X, d) centered at x of radius ɛ B n (X, ɛ), B n (x, ɛ) Open (resp. closed) ball in (X, d n ) centered at x of radius ɛ h µ (T ), h µ (T ) Measure-theoretic upper (resp. lower) entropy of T with respect to µ M(X) (Sect. 1) h UC top (T, Z) Upper capacity topological entropy of Z (Sect. 2) h B top(t, Z) Bowen s topological entropy of of Z (Sect. 2) h P top(t, Z) Packing topological entropy of Z (Sect. 2) h top (T ) Topological entropy of T (Sect. 2) M s N,ɛ (Z), Ms ɛ(z), M s (Z) (Sect. 2) WN,ɛ s (Z), Ws ɛ (Z), W s (Z) (Sect. 2) PN,ɛ s (Z), P ɛ s (Z), Pɛ s (Z) (Sect. 2) M s N (U, Z), Ms (U, Z) (Sect. 2) h B top(t, U, Z) (Sect. 2) N the set of infinite sequences of natural numbers endowed with product topology. References [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), [2] L. Barreira and J. Schmeling, Sets of non-typical points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), [3] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), [4] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), [5] M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), [6] M. Brin and A. Katok, On local entropy. Geometric dynamics (Rio de Janeiro, 1981), 30 38, Lecture Notes in Math., 1007, Springer, Berlin, [7] L. Carleson, Selected problems on exceptional sets. Van Nostrand Co., Inc., Princeton, [8] T. Downarowicz and J. Serafin, Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math., 172 (2002), [9] K. J. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge, [10] K. J. Falconer, Fractal geometry. Mathematical foundations and applications. Second edition. John Wiley & Sons, Inc., Hoboken, NJ, [11] H. Federer, Geometric measure theory. Springer-Verlag, New York Inc., [12] T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), [13] J. D. Howroyd, On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc. (3) 70 (1995), [14] W. Huang, X. D. Ye and G. H. Zhang, Lowering topological entropy over subsets, Ergodic Theory Dynam. Systems, 30 (2010), [15] W. Huang, X. D. Ye and P. F. Zhang, A new topological entropy for non-compact set and its application, draft.

26 26 DE-JUN FENG AND WEN HUANG [16] H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure. Mathematika 42 (1995), [17] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Sci. SSSR., 119 (1958), [18] F. Ledrappier, A variational principle for the topological conditional entropy, Lecture Notes in Math., 729 (1979), [19] F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), [20] E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Publ. Math. I.H.E.S., 89 (1999), [21] E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), [22] J. H. Ma and Z. Y. Wen, A Billingsley type theorem for Bowen entropy. C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, [23] P. Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge University Press, [24] Ya. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, IL, [25] L. Shu, Poincare recurrence, measure theoretic and topological entropy. Ph. D Thesis, The Chinese University of Hong Kong, [26] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York-Berlin, Department of Mathematics, The Chinese University of Hong Kong Shatin, Hong Kong. address: djfeng@math.cuhk.edu.hk Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, , P.R. China address: wenh@mail.ustc.edu.cn