On a conjecture of Fomin

Abstract

Recently P. R. Halmos [5] mentioned a question raised by S. V. Fomin as to whether it follows that two measure preserving transformations are conjugate if they are unitarily equivalent and possess the same entropy. The purpose of this work is to concoct a counterexample to this conjecture. Let X be the unit interval with Borel measurability and Lebesgue measure. Let XXX be the unit square and XXXXX the unit cube with the usual direct product measurabilities and measures. Finally let T., T,,a, and T^,a,4, denote the following measure preserving transformations defined respectively on the unit interval, unit square, and unit cube by