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

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 $f$.