On the topological entropy of induced transformations for free semigroup action.

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 (X , F) vanishes if and only if the topological entropy of its induced system (M (X) , F) is zero; if the topological entropy of (X , F) is positive, then that of its induced systems (M (X) , F) , (K (X) , F) 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). [ABSTRACT FROM AUTHOR]