Ergodic average of typical orbits and typical functions

Xiaobo Hou; Wanshan Lin; Xueting Tian

doi:10.3934/dcds.2023029

Article Contents

2023, Volume 43, Issue 7: 2781-2811. Doi: 10.3934/dcds.2023029

This issue Previous Article Propagation dynamics of a nonlocal reaction-diffusion system Next Article Upper capacity entropy and packing entropy of saturated sets for amenable group actions

Ergodic average of typical orbits and typical functions

1.
School of Mathematical Sciences, Fudan University, China

^*Corresponding author: Xueting Tian
^*Corresponding author: Xueting Tian

Received: April 1, 2022

Revised: January 1, 2023

Early access: March 17, 2023

Published online: July 1, 2023

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Abstract

In this article we mainly aim to know what kind of asymptotic behavior of typical orbits can display. For example, we show in any transitive system, the empirical measures of a typical orbit can cover all empirical measures of dense orbits and can intersect some physical-like measures. In particular, if the union set of empirical measures of all dense orbits is not singleton, then the typical orbit will display historic behavior simultaneously for typical continuous functions and the limit set of ergodic average along every continuous function equals to a closed interval composed by the union of limit sets of ergodic average on all dense orbits. Moreover, if the union set of empirical measures of all dense orbits contains all ergodic measures, the above interval equals to the rotation set. These results are not only suitable for systems with specification-like properties or minimal systems, but also suitable for many other systems including all general (not assumed uniformly hyperbolic) nontrivial homoclinic classes and Bowen eyes. Moreover, we introduce a new property called $ m $-$ g $-product property weaker than classical specification property and minimal property, and nontrivial examples are constructed.

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Mathematics Subject Classification: Primary: 37B05, 37B10; Secondary: 37B65.

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Figure 1. Relationships between uniquely ergodic property, transitivity, approximate product property, minimal property and m-g-product property

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