Invariant measures for higher-rank hyperbolic abelian actions

1996/08/01 by A. Katok, Ralf Spatzier, R. J. Spatzier · 5 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · doi:10.1017/s0143385700009081

openalex publication_date 1996/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23

Abstract

Abstract We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝ k , ℤ k and We show that they are either Haar measures or that every element of the action has zero metric entropy.

Cited by

• Nonuniform measure rigidity
• Rigidity of U-Gibbs measures near conservative Anosov diffeomorphisms on \mathbbT³
• Quantitative Density under Higher Rank Abelian Algebraic Toral Actions
• Ratner's theorem on SL(2,R)-invariant measures
• Multi-invariant measures and subsets on nilmanifolds

Related

• Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’
• Measure and cocycle rigidity for certain non-uniformly hyperbolic\n actions of higher rank abelian groups
• Nonuniform measure rigidity
• Slow entropy for abelian actions
• Entropy, Lyapunov exponents, and rigidity of group actions