Center Specification Property and Entropy for Partially Hyperbolic Diffeomorphisms

TL;DR

This paper explores the relationship between topological entropy, entropy on the center foliation, and periodic center leaves for partially hyperbolic diffeomorphisms, establishing bounds and equalities under certain conditions.

Contribution

It introduces the center specification property for partially hyperbolic systems and derives entropy bounds and equalities involving center foliation and periodic leaves.

Findings

If a compact locally maximal invariant center set is center topologically mixing, then the system has the center specification property.

The topological entropy of the system is bounded above by the sum of the entropy on the center foliation and the growth rate of periodic center leaves.

For one-dimensional center foliations, the topological entropy equals the growth rate of periodic center leaves.

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 $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ 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)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.