Topological entropy on subsets for fixed-point free flows

By considering all possible reparametrizations of the flows instead of the time- \begin{document} $1$ \end{document} maps, we introduce Bowen topological entropy and local entropy on subsets for flows. Through handling techniques for reparametrization balls, we prove a covering lemma for fixed-point free flows and then prove a variational principle.

In the literature of entropy for flows via reparametrizations, (t, ε, φ)−balls are used in place of the usual Bowen balls. Topological entropy for one parameter flows on compact metric spaces is defined by Bowen [1,2]. To investigate the topological entropies of mutually conjugate expansive flows, Thomas [9] first defined the entropy for flows arised from allowing reparametrizations of orbits. Later on, he developed this study in [10] and showed that his definition of entropy is equivalent to Bowen's definition for any flow without fixed points on compact metric spaces. Sun and Vargas studied measure-theoretic aspect of this manner in [7,8].
In 1973, Bowen [3] introduced the topological entropy for any subset in a way resembling the Hausdorff dimension for discrete dynamical systems, which is called Bowen topological entropy. In particular, Bowen topological entropy for the whole space coincides with the original topological entropy for compact discrete dynamical systems. This definition plays a key role in topological dynamics and dimension theory [6]. Since the variational principles are fundamental results in ergodic theory and dynamical systems, it is nature to find a variational principle for Bowen topological entropy. Inspired by a classical result in dimension theory, Feng and Huang [4] proved that for any non-empty compact subset K, Bowen topological entropy on K is the supremum of the measure theoretic local entropies, where the supremum is taken over all the Borel probability measures that concentrate on K. The proof is along the following steps: 1. define the weighted entropy; 2. give the relation between Bowen entropy and weighted entropy (actually they coincide); 3. prove a dynamical Frostman's Lemma via weighted entropy; 4. prove the result for compact subsets. In this paper, we will introduce Bowen entropy on subsets for compact metric flows through reparametrization balls and then apply Feng and Huang's steps to prove a variational principle for compact metric flows without fixed points. We should emphasize here that the technical difficulties arising from allowing reparametrizations of orbits need to be overcome. The paper is organized as follows. In section 2, we introduce Bowen topological entropy and local measure theoretic entropy for flows. Some basic properties are also listed therein. In section 3, we give some lemmas related to the reparametrization balls and then prove a covering lemma. These lemmas will play a key role for proving the main theorem. Finally, in section 4, we follow Feng and Huang's technical line to prove the theorem.
2. Bowen topological entropy and local measure theoretic entropy. Let (X, φ) be a flow and Z a subset of X. For s ≥ 0, N ∈ N, and ε > 0, define where the infimum is taken over all finite or countable families of reparametrization balls The quantity M s N,ε dose not decrease as N increases and ε decreases, hence the following limits exist: Proposition 1. Let (X, φ) be a flow. 1. For any s ≥ 0, N ∈ N and ε > 0, M s N,ε (φ, ·) is an outer measure on X.
2. For any s ≥ 0, M s (φ, ·) is a metric outer measure on X. Proof.
(1) is a direct result from the definition of M s N,ε (φ, ·) and we only need to prove (2).
Suppose d = d(E, F ) > 0 and let 0 < ε < d/2, N ∈ N. For any δ > 0, we choose a family of reparametrization balls Then for some For x ∈ X, ε > 0, t ≥ 0 and n ∈ N, we can define the following two classes of usual Bowen balls: Replacing the reparametrization balls by the usual Bowen balls B t (x, ε, φ), we can have the definition of the usual Bowen topological entropy on a subset Z for the flow (X, φ), denote it byh B top (Z, φ). If we replace the reparametrization balls by the Bowen balls B n (x, ε, φ 1 ), we can have the definition of the usual Bowen topological entropy on a subset Z for the time-1 map, denote it by h B top (Z, φ 1 ). Remark 1. For any ε > 0, since X is compact and φ is continuous, there exists δ > 0 such that for any 0 ≤ s ≤ 1 and x, y ∈ X, we have d(φ s x, φ s y) < ε whenever d(x, y) < δ. Then it is easy to see that where t is the largest integer which is not smaller than t. Hence from the definitions of the above Bowen topological entropies, . But it is not clear whether the equality holds for every Z ⊂ X.
Let µ ∈ M(X). The measure-theoretical lower and upper local entropies of µ are defined respectively by and

Remark 2. Similar to Remark 1, it holds that
It is also not clear whether the equalities hold for every µ ∈ M(X).

Now we state the main theorem.
Theorem 2.1. Let (X, φ) be a compact metric flow without fixed points. If K is a non-empty compact subset of X, then We suggest here that there are some further results related to Theorem 2.1 for flows without fixed points. Due to Feng and Huang [4], the compact subsets K's can be improved to analytic sets under the finite entropy or even zero mean dimension assumption. And one can also consider another kind of concept, named as packing entropy, then there will be a variational principle via the measure-theoretical upper local entropy. The proofs may involve more results in ergodic theory to flows and more techniques in geometric measure theory.
3. Properties about reparametrization balls and a covering lemma. In this section, we first will give some properties about reparametrization balls for flows without fixed points. Then we will apply these results to prove a related covering lemma(Theorem 3.5). This lemma is crucial in the proof of Theorem 2.1.
Proof. It is obvious true for t 1 ≥ t 2 . Now we assume t 1 < t 2 . For Then for 0 ≤ s ≤ t 1 , d(φ α2(s) x, φ s y) = d(φ α1(s) x, φ s y) < ε. Since X is a compact space and φ is continuous, there exists δ > 0 such that d(φ t x, x) < ε 2 for any x ∈ X whenever 0 < t < δ. And hence for t 1 < s ≤ t 2 , Now we give our covering lemma which is a variation of the classical 5r-coving Lemma in fractal geometry(see, for example, Theorem 2.1 of [5]). Choose an arbitrary x 1 ∈ A 1 and then inductively choose For each x i , we only choose one such t i , noticing that there may exist different t i 's with (x i , t i ) ∈ I. Firstly we show that B(x i , t i , ε, φ)'s are mutually disjoint. Suppose we have chosen x i and x j , i > j and there exists y ∈ B(x i , t i , ε, φ) ∩ B(x j , t j , ε, φ). As |t i − t j | < δ, by Lemma 3.4,

By (1) of Lemma 3.3, it holds that
And thus by Lemma 3.2, x i ∈ B(x j ,t j , 3ε, φ). This contradicts the choice of x i .
Secondly we claim that there exists a finite k 1 such that To see this, we note that there exists r > 0 such that d(φ s x, φ s y) < ε for any s ∈ [0, M + δ], whenever d(x, y) < r. So each reparametrization ball B(x i , t i , ε, φ) contains an ordinary ball B(x i , r). Since a compact metric space cannot contain infinite many mutually disjoint balls with the same radius r, we can conclude that k 1 is finite.
For any x ∈ A 1 , we can choose an x i from {x 1 , · · · , x k1 } such that x ∈ B(x i ,t i , 3ε, φ). Then for (x, t) ∈ I with M ≤ t < M + δ, by Lemma 3.2 and 3.4, we have that

This yields that
Choose x k1+1 ∈ A 2 arbitrarily and then inductively choose As above there is a finite k 2 such that the reparametrization balls B(x i , t i , ε, φ), i = 1, 2, · · · , k 2 , are pairwise disjoint and Combining with (5), using the same argument as above, we get x∈A2,(x,t)∈I Repeating the above process, we finish the proof.
4. Proof of Theorem 2.1. With the preparation in Section 3, we can now proceed Feng-Huang's steps to prove Theorem 2.1.
Step 1. Defining a weighted entropy for flows. Let (X, φ) be a compact metric flow. For any bounded function f : X → R, N ∈ N and ε > 0, define where the infimum is taken over all finite or countable families where B i := B(x i , t i , ε, φ) and χ A denotes the characteristic function of set A.
For Z ⊆ X, we set W s N,ε (φ, Z) = W s N,ε (φ, χ Z ). The quantity W s N,ε (φ, Z) does not decrease as N increases and ε decreases, hence the following limits exist: Clearly, there exists a critical value of the parameter s, which will be denoted as We call h W B top (φ, Z) the weighted Bowen topological entropy (or just weighted entropy for short) of the flow φ on Z.
Proof. Taking f = χ Z and c i ≡ 1 in (6), it is clear that the second inequality holds for each N ∈ N. In the following, we prove the first inequality when N is large enough.
where as in Step 1, we denote B i := B(x i , t i , ε, φ). Then we will show that and hence M s+δ . For simplicity, in the rest of the proof, we denoteB i := B(x i ,t i , ε, φ) and 5B i := B(x i ,t i , 5ε, φ), for i ∈ I. Now we decompose I into subsets I n := {i ∈ I : t i ∈ (n − 1, n]} and decompose each I n into finite subsets I n,k := {i ∈ I n : i ≤ k} for n ≥ N and k ∈ N. For τ > 0, set For each n ≥ N, k ∈ N and τ > 0, let us consider the set Z n,k,τ . We may assume that each c i is a positive integer. This could be done as follows. Since I n,k is finite and by approximating the c i 's from above, we may first assume c i 's are positive rational numbers. Also notice that Z n,k,dτ for dc i 's is equal to Z n,k,τ for c i 's, so by multiplying with a common denominator d, we may then assume that each c i is a positive integer. Let m = τ , the smallest integer no less than τ . Denote B = {B i : i ∈ I n,k } and define u : B → Z by u(B i ) = c i . We now inductively define integer-valued functions v 1 , v 2 , · · · , v m on B and subfamilies B 1 , B 2 , · · · , B m of B starting with v 0 = u. Using Theorem 3.5, we find a pairwise disjoint subfamily and hence Z n,k,τ ⊆ B∈B1 5B. Then by repeatedly using Theorem 3.5, for j = 1, . . . , m, we can define inductively disjoint subfamilies B j of B such that This is possible since for j < m, Z n,k,τ ⊆ {x : whence every x ∈ Z n,k,τ belongs to some reparametrization ball Moreover, due to the construction of B j0 , Z n,k,τ ⊆ i∈J n,k,τ 5B i . We next show that for each n ≥ N and τ > 0, we have Assume Z n,τ = ∅. Since Z n,k,τ ↑ Z n,τ , we have that Z n,k,τ = ∅ when k is large enough. Let J n,k,τ be the sets constructed in the previous discussion. Denote E n,k,τ = {x i : i ∈ J n,k,τ }. Note that the family of all non-empty compact subsets of X is compact under the Hausdorff metric. So there exists a subsequence {k j } of natural numbers and a non-empty compact set E n,τ ⊂ X such that E n,kj ,τ converges to E n,τ in the Hausdorff metric as j goes to infinity.
Since any two points in E n,k,τ can not be contained in the same B i , any two points in E n,τ also can not. Note that each B i for i ∈ J n,k,τ contains a ball with radius r > 0 (for the choice of r, one may refer to the proof of Theorem 3.5). Thus E n,τ is a finite set, moreover, #(E n,kj ,τ ) = #(E n,τ ) when j is sufficiently large. By choosing subsequence of {k j }(still denoted by {k j }), when x ij ∈ E n,kj ,τ tends to x ∈ E n,τ , we can make the corresponding parameters t ij converges to a number denoted by t x . We note that each t x (x ∈ E n,τ ) lies in the interval [n − 1, n].