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Patterson-Sullivan theory for coarse cocycles

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arxiv 2404.09713 v2 pith:F2IOHKMZ submitted 2024-04-15 math.DS math.DGmath.GT

Pierre-Louis Blayac , Richard Canary , Feng Zhu , Andrew Zimmer This is my paper · ORCID
classification math.DSmath.DGmath.GT
keywords associatedcoarsemeasurespatterson-sullivancocyclecocyclesgroupstheory
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In this paper we develop a theory of Patterson--Sullivan measures associated to coarse cocycles of convergence groups. This framework includes Patterson-Sullivan measures associated to the Busemann cocycle on the geodesic boundary of a Gromov hyperbolic metric spaces and Patterson-Sullivan measures on flag manifolds associated to Anosov (or more general transverse) subgroups of semisimple Lie groups, as well as more examples. Under some natural geometric assumptions on the coarse cocycle, we prove existence, uniqueness, and ergodicity results.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Classification of horospherical invariant measures in higher rank: The Full Story

    math.DS 2026-01 accept novelty 8.0 of 10

    For Zariski dense Anosov subgroups of arbitrary semisimple real algebraic groups, every horospherical-invariant ergodic Radon measure on the minimal set is a Burger–Roblin measure.

  2. Strict concavity of the growth indicator function for relatively Anosov groups

    math.DG 2026-07 accept novelty 7.0 of 10

    Strict concavity of the growth indicator function is proved for Zariski dense relatively theta-Anosov groups by establishing C^1 regularity of the Manhattan hypersurface.

  3. Free Semigroups of Large Critical Exponent

    math.GR 2025-02 conditional novelty 7.0 of 10

    For convergence groups with expanding coarse-cocycles, the author builds free subsemigroups with critical exponent arbitrarily close to the ambient critical exponent, with applications to transverse groups and Anosov ...

  4. Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

    math.GR 2026-07 conditional novelty 6.0 of 10

    Relatively Anosov groups have exact-dimensional Patterson–Sullivan measures and C1 Manhattan manifolds, yielding a strictly concave C1 growth indicator.

  5. Orbital counting for relatively Anosov groups

    math.DS 2026-07 conditional novelty 5.0 of 10

    Relatively Anosov subgroups satisfy orbital counting and equidistribution with exponential growth rate e^{δR}, generalizing Sambarino's Anosov theorem.

  6. Geometry and Dynamics of Transverse Groups

    math.DS 2025-02 conditional

    A survey of recent work showing that Patterson-Sullivan theory, including shadow lemmas and the Hopf-Tsuji-Sullivan dichotomy, extends to transverse subgroups of SL(d,R), with applications to Anosov and relatively Ano...

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