On a question of Kwakkel and Markovic on existence of wandering domains with bounded geometry

TL;DR

This paper proves that certain smooth surface diffeomorphisms with dense permuted domains of bounded geometry cannot exist on closed surfaces of genus at least one, regardless of entropy.

Contribution

It provides a negative answer to a question about the existence of specific diffeomorphisms with dense permuted domains on closed surfaces.

Findings

Such diffeomorphisms do not exist on closed surfaces of genus ≥ 1.

The result holds regardless of the topological entropy of the diffeomorphisms.

The proof addresses a question posed by Kwakkel and Markovic.

Abstract

A question of F. Kwakkel and V. Markovic on existence of C^1-diffeomorphisms of closed surfaces that permute a dense collection of domains with bounded geometry is answered in the negative. In fact, it is proved that for closed surfaces of genus at least one such diffeomorphisms do not exist regardless of whether they have positive or zero topological entropy.