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