On the topological entropy of induced transformations for free semigroup action

Abstract

The aim of this paper is to reveal the relationship between the topological entropy of a free semigroup action and that of its induced transformations. More precisely, we prove that the topological entropy of a free semigroup action vanishes if and only if the topological entropy of its induced system is zero; if the topological entropy of is positive, then that of its induced systems , are infinite. This in particular extends the results for classical topological entropy done by Bauer and Sigmund (1975 Monatsh. Math. 79 81–92), Glasner and Weiss (1995 J. Amer. Math. Soc. 8 665–686).