The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system

Abstract

The author shows that if M is a compact Riemannian manifold and T:M to M is a transitive Anosov diffeomorphism then the Hausdorff dimension of the set of points with non-dense (full) orbit under T equals dim M. The same statement is proved for Anosov flows. For an expanding endomorphism T:M to M he also proves the same result replacing only full orbits by forward orbits.