Topological conjugacy of affine transformations of tori

Abstract

0. Introduction. Let Kn denote the n-dimensional torus. Let T be an affine transformation of KI onto Kn and S an affine transformation of Km onto Km for which there exists a continuous map h of Kn onto Km satisfying hT=Sh. R. L. Adler and R. Palais have proved that, in the special cases when m = n and T and S are ergodic automorphisms of Kn, all homeomorphism g of Kn, satisfying gT= Sg, are affine transformations [1]. The main purpose of this paper is to obtain necessary and sufficient conditions for the existence of a nonaffine continuous map g of Kn onto Km such that gT= Sg. (Theorem 2, ?4). The conditions obtained are unchanged under the requirements that m =n and T, S, h and g be invertible transformations of Kn. We obtain the above-mentioned result of Adler and Palais as a corollary of Theorem 2. We also prove that two automorphisms T and S of Kn must be ergodic if all homeomorphisms g of Kn such that gT= Sg are affine transformations. (We require, of course, that there exist homeomorphisms g such that gT= Sg.) Another corollary of Theorem 2 concerns affine transformations with quasi-discrete spectrum. A. H. M. Hoare and W. Parry have shown that every minimal affine transformation of a compact, connected, metric, abelian group X has quasidiscrete spectrum, and that if T and S are minimal affine transformations of X then all homeomorphisms g of X, such that gT= Sg, are affine [5], [6]. Our corollary is restricted to toroidal groups but extends Hoare and Parry's result in this case. Theorem 2 is also used to obtain a necessary and sufficient condition for a strong-mixing affine transformation of Kn to have a continuous pth root (Theorem 3, ?5). In Theorem 1 a necessary and sufficient condition for an affine transformation of a compact, connected, metric, abelian group to be ergodic is given. I should like to express my gratitude to Dr. W. Parry for supervising this work and also to Dr. D. E. Cohen and Dr. A. H. M. Hoare for supplying useful examples. My thanks are also due to the referee for some helpful suggestions.