Center specification property and entropy for partially hyperbolic diffeomorphisms

Lin  Wang; Yujun Zhu

doi:10.3934/dcds.2016.36.469

Article Contents

2016, Volume 36, Issue 1: 469-479. Doi: 10.3934/dcds.2016.36.469

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Center specification property and entropy for partially hyperbolic diffeomorphisms

Lin Wang^1, and
Yujun Zhu^1,

1.
College of Mathematics and Information Science, and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024

Received: September 2014

Revised: April 2015

Published: May 2017

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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 $h(f)= p^{c}(f)$.

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Mathematics Subject Classification: Primary: 37D30, 37B40; Secondary: 37C50.

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