  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 $f \in \diff^{1+\alpha}(M)$, with $\alpha>0$, and
permutes a dense collection of domains with bounded geometry, then $f$ has zero
topological entropy.
