Minimal Homeomorphisms on Low-Dimension Tori

TL;DR

This paper characterizes minimal homeomorphisms on low-dimensional tori, showing their linear parts are quasi-unipotent with roots of unity eigenvalues, and constructs examples with specific linear parts.

Contribution

It establishes a link between minimality and quasi-unipotent linear parts for tori of dimension less than five, and provides conditions for conjugacy to affine transformations.

Findings

Minimal homeomorphisms on $T^n$, $n<5$, have quasi-unipotent linear parts.

If the linear part has 1 as an eigenvalue and is quasi-unipotent, a minimal skew-product diffeomorphism exists.

The results are not known for dimensions greater than 4.

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 $\phi $ of $T^{n}$ is the linear mapping $L$ induced by $\phi $ on the first homology group of $T^{n}$. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of $L$ if $\phi $ minimal. We show that if $\phi $ is minimal and $n<5$ then $L$ is quasi-unipontent, i.e., all the eigenvalues of $L$ are roots of unity and conversely if $L\in GL(n,\Z)$ is quasi-unipotent and 1 is an eigenvalue of $L$ then there exists a $ C^{\infty}$ minimal skew-product diffeomorphism $\phi $ 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.