Nonuniform measure rigidity

2008/03/20 by Boris Kalinin, Kalinin, Boris, Anatole Katok +3
Mathematics · Physics and Astronomy · #37C40 #37C85 #37D25 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.0803.3094

openalex publication_date 2008/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an ergodic invariant measure μ for a smooth action 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 of the action 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.

Citations

• Invariant measures for higher-rank hyperbolic abelian actions

Related

• Measure and cocycle rigidity for certain non-uniformly hyperbolic\n actions of higher rank abelian groups
• Entropy, Lyapunov exponents, and rigidity of group actions
• Invariant measures for higher-rank hyperbolic abelian actions
• Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori
• Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’