Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

Given a partially hyperbolic diffeomorphism f : M -> M defined on a compact Riemannian manifold M, in this paper we define the concept of unstable topological entropy off on a set Y subset of M 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 it , the unstable measure theoretical entropy off is upper bounded by the unstable topological entropy off on any set of positive mu-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). (AU)