On super-exponential divergence of periodic points for partially hyperbolic systems

Xiaolong Li; Katsutoshi Shinohara

doi:10.3934/dcds.2021169

Article Contents

2022, Volume 42, Issue 4: 1707-1729. Doi: 10.3934/dcds.2021169

This issue Previous Article Rigidity, weak mixing, and recurrence in abelian groups Next Article Nonlinear nonlocal reaction-diffusion problem with local reaction

On super-exponential divergence of periodic points for partially hyperbolic systems

Xiaolong Li^1, , and
Katsutoshi Shinohara^2,

1.
Graduate School of Mathematics and Statistics, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan, China
2.
Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan

^* Corresponding author: Xiaolong Li
^* Corresponding author: Xiaolong Li

Received: December 31, 2020

Revised: August 31, 2021

Early access: November 2021

Published: April 2022

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Abstract

We say that a diffeomorphism $ f $ is super-exponentially divergent if for every $ b>1 $ the lower limit of $ \#\mbox{Per}_n(f)/b^n $ diverges to infinity, where $ \mbox{Per}_n(f) $ is the set of all periodic points of $ f $ with period $ n $. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any $ n $-dimensional smooth closed manifold $ M $ where $ n\ge 3 $, there exists a non-empty open subset $ \mathcal{O} $ of $ \mbox{Diff}^1(M) $ such that diffeomorphisms with super-exponentially divergent property form a dense subset of $ \mathcal{O} $ in the $ C^1 $-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a $ C^r $-residual subset of $ \mbox{Diff}^r(M)\ (1\le r\le \infty) $ is also shown.

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Mathematics Subject Classification: Primary: 37C20, 37C25, 37C29, 37D30.

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