×2 and ×3 invariant measures and entropy

Abstract

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 μ be a borel measure invariant for both T 0 and S 0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T 0 and S 0 have entropy zero. Equivalently, both T 0 and S 0 are μ-almost surely invertible.

Locations

• Ergodic Theory and Dynamical Systems - View - PDF