Real-analytic realization of Uniform Circular Systems and some applications

Abstract

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth ergodic diffeomorphisms of certain compact manifolds are unclassifiable up to measure-isomorphism. In this paper we show that the uniform circular systems used in the work of Foreman—Weiss admit real-analytic realizations on the two-dimensional torus. As a consequence we obtain the same anti-classification result for real-analytic ergodic diffeomorphisms on the torus. In another application we show the existence of an uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms on the torus.