Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

Abstract

Given a partially hyperbolic diffeomorphism defined on a compact Riemannian manifold M, in this paper we define the concept of unstable topological entropy of f on a set not necessarily compact. Using recent results of Yang (2016 (arXiv:1601.05504)) and Hu et al (2017 Adv. Math. 321 31–68) we extend a theorem of Bowen (1973 Trans. Am. Math. Soc. 184 125–36) proving that, for an ergodic f -invariant measure , the unstable measure theoretical entropy of f is upper bounded by the unstable topological entropy of f on any set of positive -measure. We define a notion of unstable topological entropy of f using a Hausdorff dimension like characterization and we prove that this definition coincides with the definition of unstable topological entropy introduced in Hu et al (2017 Adv. Math. 321 31–68).