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 ∈ Diff^(1+ α)(M), with α > 0, and permutes a dense collection of domains with bounded geometry, then f has zero topological entropy.