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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2023

A report on an ergodic dichotomy

Résumé

We establish (some directions) of a Ledrappier correspondence between H\"older cocycles, Patterson-Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson-Sullivan measures for $\theta$-Anosov representations over a local field and show that these are parametrized by the $\theta$-critical hypersurface of the representation. We use these Patterson-Sullivan measures to establish a dichotomy concerning directions in the interior of the $\theta$-limit cone of the representation in question: if $u$ is such a half-line, then the subset of $u$-conical limit points has either total-mass if $|\theta|\leq2$ or zero-mass if $|\theta|\geq4.$ The case $|\theta|=3$ remains unsettled.
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Dates et versions

hal-03839001 , version 1 (11-11-2022)
hal-03839001 , version 2 (09-11-2023)

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Citer

Andrés Sambarino. A report on an ergodic dichotomy. Ergodic Theory and Dynamical Systems, 2023, ⟨10.1017/etds.2023.13⟩. ⟨hal-03839001v2⟩
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