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February  2013, 33(2): 905-920. doi: 10.3934/dcds.2013.33.905

## Divergence points in systems satisfying the specification property

 1 Department of Mathematics, South China University of Technology, Guangzhou, 510641, China

Received  July 2011 Revised  February 2012 Published  September 2012

Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets: $$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$ where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys., 23 (2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.
Citation: Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
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##### References:
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