Intermediate Lyapunov exponents for systems with periodic orbit gluing property

Xueting Tian; Shirou Wang; Xiaodong Wang

doi:10.3934/dcds.2019042

Article Contents

2019, Volume 39, Issue 2: 1019-1032. Doi: 10.3934/dcds.2019042

This issue Previous Article Topological entropy of free semigroup actions for noncompact sets Next Article Continuous shift commuting maps between ultragraph shift spaces

Intermediate Lyapunov exponents for systems with periodic orbit gluing property

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada
3.
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
4.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Xiaodong Wang
* Corresponding author: Xiaodong Wang

Received: March 2018

Revised: August 2018

Published: February 2019

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Abstract

We prove that the average Lyapunov exponents of asymptotically additive functions have the intermediate value property provided the dynamical system has the periodic gluing orbit property. To be precise, consider a continuous map $f \colon X\rightarrow X$ over a compact metric space $X$ and an asymptotically additive sequence of functions $\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$. If $f$ has the periodic gluing orbit property, then for any constant $a$ satisfying

$\inf\limits_{\mu\in \mathcal M_{inv} (f,X)} \chi_\Phi(\mu) <a<\sup\limits_{\mu\in\mathcal M_{inv} (f,X)} \chi_\Phi(\mu)$

where $\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$, and the infimum and supremum are taken over the set of all $f$-invariant probability measures, there is an ergodic measure $\mu_a\in \mathcal M_{inv} (f,X)$ such that $\chi_\Phi(\mu_a) = a$ and ${\rm{supp}}(\mu)=X.$

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Mathematics Subject Classification: 37A25, 37D25, 54H20.

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