No bounded geometry wandering domains for sufficiently regular automorphisms

TL;DR

The paper proves that certain regular automorphisms of manifolds, including quasiconformal and smooth homeomorphisms, cannot have wandering domains with controlled geometry, answering a longstanding question in dynamics.

Contribution

It establishes the non-existence of bounded geometry wandering domains for sufficiently regular automorphisms on specific manifolds, under various analytic and geometric assumptions.

Findings

No bounded geometry wandering domains for quasiconformal automorphisms of n-tori and hyperbolic surfaces.

Wandering domains with controlled geometry cannot exist under minimal set measure zero or invariant conformal structures.

Uniform relative separation of wandering domains also precludes their existence in these settings.

Abstract

A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of $n$-tori, $n\ge2$, and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.