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On the measure-theoretic entropy and topological pressure of free semigroup actions

Published online by Cambridge University Press:  26 July 2016

XIAOGANG LIN ,
DONGKUI MA  and
YUPAN WANG
XIAOGANG LIN
Affiliation:
School of Business Administration, South China University of Technology, Guangzhou 510641, PR China email [email protected]
DONGKUI MA
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou 510641, PR China email [email protected]
YUPAN WANG
Affiliation:
School of Computer Science and Engineering, South China University of Technology, Guangzhou 510641, PR China
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Abstract

In this paper we introduce the notions of topological pressure and measure-theoretic entropy for a free semigroup action. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action we assign a skew-product transformation whose fiber topological pressure is taken to be the topological pressure of the initial action. Some properties of these two notions are given, followed by two main results. One is the relationship between the topological pressure of the skew-product transformation and the topological pressure of the free semigroup action, the other is the partial variational principle about the topological pressure. Moreover, we apply this partial variational principle to study the measure-theoretic entropy and the topological entropy of finite affine transformations on a metrizable group.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 686 - 716
Copyright
© Cambridge University Press, 2016 

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