Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915907086843904 |
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| author | Kim, Dongryul M. Zimmer, Andrew |
| author_facet | Kim, Dongryul M. Zimmer, Andrew |
| contents | In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity Kim, Dongryul M. Zimmer, Andrew Geometric Topology Dynamical Systems Group Theory Probability In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets. |
| title | Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity |
| topic | Geometric Topology Dynamical Systems Group Theory Probability |
| url | https://arxiv.org/abs/2505.16556 |