Some properties on topological entropy of free semigroup action

ABSTRACT

The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined using open covers. Furthermore, a quantitative analogue of Bowen's theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Biś. Finally, a formula for the entropy of skew-product transformation with respect to the subshift is obtained.

Keywords:

• Topological entropy
• free semigroup action
• power rule
• skew-product transformation

2010 AMS Subject Classification Codes:

• 37B40
• 37C85

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Erratum

Acknowledgments

This work was conducted when Wen-Chiao Cheng visited Department of Mathematics, South China University of Technology. Wen-Chiao Cheng sincerely appreciates the warm hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

National Science Council (NSC) grant [NSC 103-2115-M-034-001]; National Natural Science Foundation of China (NSFC) [11671151]; Guangdong Natural Science Foundation [2014A030313230].