Unstable Entropy of Partially Hyperbolic Diffeomorphisms and Generic Points of Ergodic Measures

Gabriel Ponce

Published 2018-08-21Version 1

Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a compact Riemannian manifold $M$, in this paper we define the concept of unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily compact and we extend a theorem of R. Bowen proving that, for an ergodic $f$-invariant measure $\mu$, the unstable measure theoretical entropy of $f$ is upper bounded by the unstable topological entropy of $f$ on any set of full $\mu$-measure. We also define a notion of unstable topological entropy of $f$ using a Hausdorff dimension like characterization and we show that the unstable topological entropy of the set of generic points of an ergodic invariant measure $\mu$ is equal to the unstable metric entropy with respect to $\mu$.