## 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 $h_B(f^{-1},E)=b$.