Local unstable entropies of partially hyperbolic diffeomorphisms

Abstract

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.