Nonuniform measure rigidity

Abstract

Abstract. We consider an ergodic invariant measure µ for a smooth action α of Z k , k ≥ 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R k , k ≥ 2 on a (2k + 1)-dimensional manifold. We prove that if µ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then µ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.

The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

“…So we only consider the case (2). In that case, unless the entropy function is identically zero, the measure µ is absolutely continuous [7]. To summarize, the (deep) results of [7] and the above arguments provide the following dichotomy for ergodic invariant measures for maximal rank actions of rank k ≥ 2:…”

Normal forms for non-uniform contractions

“…The theory, however, was not developed for quite a while. The linearization of a C 1+α diffeomorphism along a one-dimensional nonuniformly contracting foliation was constructed in [KKt07] and used in the study of measure rigidity in [KKt07,KKtR11]. Similar results for higher dimensional foliations with pinched exponents were obtained by Katok and Rodriguez Hertz in [KtR15].…”

A note on two types of Lyapunov exponents and entropies for Zk-actions

“…Remark 3.2. In Kalinin, Katok and Hertz's paper [7] which investigates the nonuniform measure rigidity for Z k -actions, Lyapunov exponents for a Z k action on M are extended to a R k action on the suspension manifold S, which is a bundle over T k with fiber M . We can see that for any v = (v 1 , · · · , v k ) ∈ S k−1 , the Lyapunov exponents χ j ( v) in Proposition 2.1 of [7] is just the direction Lyapunov exponents λ v j in our case.…”