The conditional variational principle for maps with the pseudo-orbit tracing property

Zheng Yin; Ercai Chen

doi:10.3934/dcds.2019019

Article Contents

2019, Volume 39, Issue 1: 463-481. Doi: 10.3934/dcds.2019019

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The conditional variational principle for maps with the pseudo-orbit tracing property

Zheng Yin^1, and
Ercai Chen^2,3, ,

1.
School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China
2.
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, China
3.
Center of Nonlinear Science, Nanjing University, Nanjing 210093, Jiangsu, China

* Corresponding author.
* Corresponding author.

Received: February 28, 2018

Published: December 2018

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Abstract

Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X \to X$ is a continuous map. We define $n$-ordered empirical measure of $x \in X$ by

$\mathscr{E}_n(x) = \frac{1}{n}\sum\limits_{i = 0}^{n-1}δ_{f^ix},$

where $δ_y$ is the Dirac mass at $y$. Denote by $V(x)$ the set of limit measures of the sequence of measures $\mathscr{E}_n(x)$. In this paper, we obtain conditional variational principles for the topological entropy of

$\Delta_{sub}(I): = \left\{ {x \in X:V(x)\subset I} \right\},$

and

$\Delta_{cap}(I): = \left\{ {x \in X:V(x)\cap I≠\emptyset } \right\}.$

in a dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.

Keywords:

Topological entropy,
conditional variational principle.

Mathematics Subject Classification: Primary: 37B40; Secondary: 37C45.

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References

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