×2 and ×3 invariant measures and entropy

Abstract

Let p and q be relatively prime natural numbers. Define T o and S o to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).Let (x. be a borel measure invariant for both T o and S o and ergodic for the semigroup they generate. We show that if /* is not Lebesgue measure, then with respect to fx. both T o and S o have entropy zero. Equivalently, both T o and S o are H-almost surely invertible.

Invariant measures for higher-rank hyperbolic abelian actions

“…He proved that the only invariant measure which makes the multiplications exact endomorphisms is Lebesgue 21]. D. Rudolph and A. Johnson strengthened this result by replacing the exactness condition with positive e n tropy for some and hence all elements of the action 26,8]. At the heart of their arguments lies a symbolic version of the natural extension of a Z 2 + -action to a Z 2 -action.…”

Invariant sets and measures of nonexpansive group automorphisms

“…This scarcity of invariant measures was conjectured by H. Furstenberg and is still open, though there are important partial results by several authors including D. Rudolph [9] for the one-dimensional case and A. Katok and R. Spatzier [5] in the higher-dimensional case.…”

Rigidity of measures—The high entropy case and non-commuting foliations

“…Measure rigidity, low entropy, and high entropy. Rudolph's result [33] has subsequently been proved using slightly different methods by J. Feldman [10] and W. Parry [26] but positive entropy remained a crucial assumption. A further extension was then given by B.…”

“…D. Rudolph [33] and A. Johnson [15] weakened this assumption considerably, and proved that µ must be the Lebesgue measure provided that ×m (or ×n) has positive entropy with respect to µ.…”