Ergodic Theory and Dynamical Systems

Research Article

×2 and ×3 invariant measures and entropy

Daniel J. Rudolpha1

a1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Abstract

Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).

Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.

(Received September 08 1987)

(Revised October 22 1989)