Pointwise dimension, entropy and Lyapunov exponents for $C^1$ maps

Abstract:

Let $f$ be a $C^1$ self-map on a smooth Riemannian manifold $M$, $\mu$ be an $f$-invariant ergodic Borel probability measure with a compact support $\Lambda$ and $\chi ^1_\mu >\cdots >\chi ^s_\mu$ be the Lyapunov exponents of $\mu$ with respect to $f$. If $\chi ^1_\mu >0$, then we give a lower bound of the lower pointwise dimension of $\mu$ in terms of $\chi ^1_\mu$ and of the entropy $h_\mu (f)$. Moreover, if $Df_{\{\cdot \}}$ is non-degenerate on $\Lambda$ and $\chi ^s_\mu >0$, then we give an upper bound of the upper pointwise dimension of $\mu$ in terms of $\chi ^s_\mu$ and of the entropy $h_\mu (f)$. Furthermore, if $f$ is $C^{1+\alpha }$ for some $\alpha >0$, then the non-degeneracy condition can be removed.

As direct applications of the above results, we also give the corresponding lower and upper bounds of some classical characteristics of dimensional type of $\mu$ in terms of the Lyapunov exponents and of the entropy $h_\mu (f)$.