Exceptional sets for average conformal dynamical systems

Abstract

Let f : M → M be a C1+α local diffeomorphism/diffeomorphism of a compact Riemannian manifold M and μ be an expanding/hyperbolic ergodic f-invariant Borel probability measure on M. Assume f is average conformal expanding/hyperbolic on the support set W of μ and W is locally maximal. For any subset A ⊂ W with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the A-exceptional set and the limit A-exceptional set.