On topological entropy of group actions on 𝑆¹

Abstract

In this paper, we show that a surface group action on S 1 {S^1} with a non-zero Euler number has a positive topological entropy. We also show that if a surface group action on S 1 {S^1} has a Euler number which attains the maximal absolute value in the inequality of Milnor-Wood, then the topological entropy of the action equals the exponential growth rate of the group.