X2 and X3 invariant measures and entropy

Abstract

Let p and q be relatively prime natural numbers. Define T 0 and S 0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1). Let be a borel measure invariant for both T 0 and S 0 and ergodic for the semigroup they generate. We show that if is not Lebesgue measure, then with respect to both T 0 and S 0 have entropy zero. Equivalently, both T 0 and S0 are -almost surely invertible. © 1990, Cambridge University Press. All rights reserved.