1. Bowen and Dinaburg gave a alternative definition of topological entropy by calculating the exponential growth rate of the -covers. This definition resembles the box dimension of Euclidean subset , and gives the same value while using the definition given by Adler, Konheim, and McAndrew. In particular, the entropy is time-reversal invariant: .

2. Later Bowen introduced another definition of topological entropy for noncompact subset in 1973, which resembles the Hausdorff dimension.

Let be a homeomorphism on a compact metric space, and be Bowen’s topological entropy of (may not be compact).

Bowen proved that, for any ergodic measure , , where is the set of -generic points. This identity has been generalized to general invariant measures of transitive Anosov systems:

**Theorem 1.** (Pfister–Sullivan link) Let be a transitive Anosov diffeomorphism. Then for any invariant measure .

Note that whenever is invariant but non-ergodic.

3. An interesting fact is that may **not** be time-reversal invariant.

**Example 2.** Let be a transitive Anosov diffeomorphism, be a periodic point, . Then , but .

Now let be two different invariant measures of , be the set of -generic points with respect to , and be the set of -generic points with respect to . Let (resemble the heteroclinic intersection of different saddles). Then it is proved (Proposition D in here) that

**Proposition 3.** Let be a transitive Anosov diffeomorphism. Then and .

A well known fact is that, for any , there exists some invariant measure with . So a direct corollary of Proposition 3 is:

**Corollary.** Let be a transitive Anosov diffeomorphism. Then for any , there exists an invariant subset such that and .

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