Ergodic Automorphisms of $T^∞$ Are Bernoulli Transformations

Abstract

The purpose of this paper is to prove the fact stated by the title. After Ornstein and Friedman [5], [1], several authors studied the Bernoulli properties of various transformations. Especially Katznelson [3] proved that every ergodic automorphism of a finite-dimensional torus is a Bernoulli transformation extending the results by Sinai-Ornstein-Friedman [9], [1]. The present result is on the way towards the conjecture that every ergodic automorphism of a compact metrizable abelian group is a Bernoulli transformation. Let X be a compact metrizable group and /^ be its normalized Haar measure. Then (X, ft) is a Lebesgue space (cf. [10]). Let a be an (group) automorphism of X, then cr is an invertible measure-preserving transformation of (X, fji). Our problem is concerned with measure-theoretic properties of a. We call a a Bernoulli transformation if there exists a measurable partition f of X such that {o-^}_00<??/<00 are independent and V o-f=£ the partition of X into individual points. We assume X—T°° an infinite-dimensional torus i.e. X is a compact metrizable abelian group which is connected, locally connected and infinite-dimensional. Further we assume naturally that the automorphism a is ergodic i.e. any cr-invariant measurable set has measure 0 or 1. Our result is the following