Exceptional sets for nonuniformly expanding maps

Sara Campos, Katrin Gelfert

Published 2015-06-01Version 1

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the map is expansive. Furthermore, if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy.