No round wandering domains for ${\bf C^1}$-diffeomorphisms of tori

TL;DR

This paper proves that for dimensions two and higher, $C^1$-diffeomorphisms of tori cannot have wandering domains that are geometric balls if they are semi-conjugate to minimal translations, extending previous higher-regularity results.

Contribution

It establishes a non-existence result for wandering domains of a specific geometric shape under $C^1$ regularity, improving prior results requiring higher smoothness.

Findings

No $C^1$-diffeomorphism of the torus semi-conjugate to a minimal translation has wandering domains that are geometric balls.

The result holds for all dimensions $n eq 1$, specifically for $n geq 2$.

Extends previous work that required $C^{n+1}$ regularity to the $C^1$ case.

Abstract

We prove that if $n\geq 2$, then there is no $C^1$-diffeomorphism $f$ of $n$-torus, such that $f$ is semi-conjugate to a minimal translation and its wandering domains are geometric balls. This improves a recent result of A. Navas, who proved it assuming $C^{n+1}$ regularity of $f$.