Dimension, recurrence via entropy and Lyapunov exponents for C-1 map with singularities

摘要

Let f : M - M be a C-1 self-map of a smooth Riemannian manifold M and mu be an f-invariant ergodic Borel probability measure with a compact support Lambda. We prove that if f is Holder mild on the intersection of the singularity set and 3, then the pointwise dimension of mu can be controlled by the Lyapunov exponents of mu 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,C- alpha continuity and Holder mildness conditions. Consequently, this shows that the Holder mildness condition is much weaker than the C-1,C- alpha continuity condition. As applications of our result, if we study the recurrence rate of f instead of the pointwise dimension of mu, then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.