Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909792002375680 |
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| author | Kim, Dongryul M. Oh, Hee Wang, Yahui |
| author_facet | Kim, Dongryul M. Oh, Hee Wang, Yahui |
| contents | Let $G$ be a connected semisimple real algebraic group. The class of transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup $Γ$, we show that the $Γ$-action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen-Margulis-Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf-Tsuji-Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06846 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups Kim, Dongryul M. Oh, Hee Wang, Yahui Dynamical Systems Group Theory Geometric Topology Let $G$ be a connected semisimple real algebraic group. The class of transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup $Γ$, we show that the $Γ$-action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen-Margulis-Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf-Tsuji-Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well. |
| title | Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups |
| topic | Dynamical Systems Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2306.06846 |