Wandering domains for diffeomorphisms of the k-torus: a remark on a theorem by Norton and Sullivan

TL;DR

This paper proves that for certain smooth diffeomorphisms of the k-torus, wandering domains with all iterates as Euclidean balls cannot exist if the map is semiconjugate to a minimal translation.

Contribution

It establishes a non-existence result for wandering domains with Euclidean ball iterates in smooth torus diffeomorphisms semiconjugate to minimal translations.

Findings

No C^{k+1} diffeomorphism of the k-torus has a wandering domain with all iterates as Euclidean balls under semiconjugacy to a minimal translation.

The result extends understanding of the structure of dynamical systems on the torus.

Provides a remark on a theorem by Norton and Sullivan regarding wandering domains.

Abstract

We show that there is no C^{k+1} diffeomorphism of the k-torus which is semiconjugate to a minimal translation and has a wandering domain all of whose iterates are Euclidean balls.