Center Specification Property and Entropy for Partially Hyperbolic Diffeomorphisms

Abstract

Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation ${\mathcal{W}}^c$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f,{\mathcal{W}}^c)$ and the growth rate of periodic center leaves $p^c(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\hat{I}›$ is center topologically mixing then $f{|}_{\hat{I}}›$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $h(f)\le h(f,{\mathcal{W}}^c)+p^c(f)$. Moreover, if the center foliation ${\mathcal{W}}^c$ is of dimension one, we obtain an equality $h(f)=p^c(f)$.