Dimension, recurrence via entropy and Lyapunov exponents for $C^{1} map with singularities
Let $f:M\rightarrow M$ be a $C^{1} self map of a smooth Riemannian manifold $M$ and \unicode[STIX]{x1D707} be an $f$ invariant ergodic Borel probability measure with a compact support \unicode[STIX]{x1D6EC} We prove that if $f$ is Holder mild on the intersection of the singularity set and \unicode[STIX]{x1D6EC} then the pointwise dimension of \unicode[STIX]{x1D707} can be controlled by the Lyapunov exponents of \unicode[STIX]{x1D707} with respect to $f$ and the entropy of $f$ Moreover, we establish the distinction of the Hausdorff dimension of the critical points sets of maps between the $C^{1,\unicode[STIX]{x1D6FC} continuity and Holder mildness conditions. Consequently, this shows that the Holder mildness condition is much weaker than the $C^{1,\unicode[STIX]{x1D6FC} continuity condition. As applications of our result, if we study the recurrence rate of $f$ instead of the pointwise dimension of \unicode[STIX]{x1D707} then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.
Ergodic Theory and Dynamical Systems