Minimal Homeomorphisms on Low-Dimension Tori

Abstract

In this article we study minimal homeomorphisms(all orbits are dense) of the tori $T^n,$ $n<5.$ The linear part of a homeomorphism $\varphi$ of $T^n$ is the linear mapping $L$ induced by $\varphi$ on the first homology group of $T^n$. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of $L$ if $\varphi$ minimal. We show that if $\varphi$ is minimal and $n<5$ then $L$ is quasi-unipontent, i.e., all the eigenvalues of $L$ are roots of unity and conversely if $L\in GL(n,\mathbb{Z})$ is quasi-unipotent and 1 is an eigenvalue of $L$ then there exists a $C^{\infty }$ minimal skew-product diffeomorphism $\varphi$ of $T^n$ whose linear part is precisely $L.$ We do not know if these results are true for $n>4$. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

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