Asymmetry of Bowen’s dimensional entropy

1. Bowen and Dinaburg gave a alternative definition of topological entropy h_{\text{top}}(f) by calculating the exponential growth rate of the (n,\epsilon)-covers. This definition resembles the box dimension of Euclidean subset E\subset\mathbb{R}^k, and gives the same value while using the definition given by Adler, Konheim, and McAndrew. In particular, the entropy is time-reversal invariant: h_{\text{top}}(f^{-1})=h_{\text{top}}(f).

2. Later Bowen introduced another definition of topological entropy for noncompact subset in 1973, which resembles the Hausdorff dimension.
Let f:X\to X be a homeomorphism on a compact metric space, E\subset X and h_B(f,E) be Bowen’s topological entropy of E (may not be compact).

Bowen proved that, for any ergodic measure \mu, h_B(f,G_{\mu})=h(f,\mu), where G_{\mu} is the set of \mu-generic points. This identity has been generalized to general invariant measures of transitive Anosov systems:

Theorem 1. (Pfister–Sullivan link) Let f:M\to M be a transitive Anosov diffeomorphism. Then h_B(f,G_{\mu})=h(f,\mu) for any invariant measure \mu.

Note that \mu(G_\mu)=0 whenever \mu is invariant but non-ergodic.

3. An interesting fact is that h_B(f,E) may not be time-reversal invariant.

Example 2. Let f:M\to M be a transitive Anosov diffeomorphism, p be a periodic point, D=W^u(x,\epsilon). Then h_B(f,D) > 0, but h_B(f^{-1},D)=0.

Now let \mu,\nu be two different invariant measures of f, W^s(\mu,f)=G_\mu be the set of \mu-generic points with respect to f, and W^u(\nu,f)=W^s(\nu,f^{-1}) be the set of \mu-generic points with respect to f^{-1}. Let H_f(\mu,\nu)=B^s(\mu,f)\cap B^u(\nu,f) (resemble the heteroclinic intersection of different saddles). Then it is proved (Proposition D in here) that

Proposition 3. Let f:M\to M be a transitive Anosov diffeomorphism. Then h_B(f,H_f(\mu,\nu))=h_\mu(f) and h_B(f^{-1},H_f(\mu,\nu))=h_\nu(f).

A well known fact is that, for any 0\le t\le h_{\text{top}}(f), there exists some invariant measure \mu with h_\mu(f)=t. So a direct corollary of Proposition 3 is:

Corollary. Let f:M\to M be a transitive Anosov diffeomorphism. Then for any a, b\in [0, h_{\text{top}}(f)], there exists an invariant subset E such that h_B(f,E)=a and h_B(f^{-1},E)=b.

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