Topological Entropy and Diffeomorphisms of Surfaces with Wandering Domains

TL;DR

This paper proves that smooth surface diffeomorphisms permuting a dense collection of domains with bounded geometry have zero topological entropy, linking geometric properties with dynamical complexity.

Contribution

It establishes that diffeomorphisms with certain dense domain permutations and smoothness conditions necessarily have zero topological entropy, a new connection between geometry and dynamics.

Findings

Diffeomorphisms with dense domain permutations have zero topological entropy.

Smoothness condition $C^{1+eta}$ is crucial for the result.

Bounded geometry of domains is essential for the entropy conclusion.

Abstract

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.