Topological Entropy and Diffeomorphisms of Surfaces with Wandering Domains

Let $M$ be a closed surface and $f$ a diffeomorphism of $M$. A diffeomorphism is said to permute a dense collection of domains, if the union of the domains are dense and the iterates of any one domain are mutually disjoint. In this note, we show that if $f\in {\text{diff}}^{1+\alpha }(M)$, with $\alpha >0$, and permutes a dense collection of domains with bounded geometry, then $f$ has zero topological entropy.