Exceptional Sets for Nonuniformly Hyperbolic Diffeomorphisms

Abstract

For a surface diffeomorphism, a compact invariant locally maximal set W and some subset $$A\subset W$$ we study the A-exceptional set, that is, the set of points whose orbits do not accumulate at A. We show that if the Hausdorff dimension of A is smaller than the Hausdorff dimension d of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least d. Particular consequences occur when there is some a priori defined hyperbolic structure on W and, for example, if there exists an SRB measure.