# American Institute of Mathematical Sciences

April  2019, 39(4): 2059-2075. doi: 10.3934/dcds.2019086

## Q-entropy for general topological dynamical systems

 1 School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, China 2 Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou 215006, Jiangsu, China 3 Department of Applied Mathematics, Chinese Culture University, Yangmingshan, Taipei, 11114, Taiwan

* Corresponding author: Chih-Chang Ho

Received  April 2018 Revised  September 2018 Published  January 2019

The aim of this paper is to extend the $q$-entropy from symbolic systems to a general topological dynamical system. Using a (weak) Gibbs measure as the reference measure, this paper defines $q$-topological entropy and $q$-metric entropy, then studies basic properties of these entropies. In particular, this paper describes the relations between $q$-topological entropy and topological pressure for almost additive potentials, and the relations between $q$-metric entropy and local metric entropy. Although these relations are quite similar to that described in [19], the methods used here need more techniques from the theory of thermodynamic formalism with almost additive potentials.

Citation: Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
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