Dichotomy and measures on limit sets of Anosov groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911013927911424 |
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| author | Lee, Minju Oh, Hee |
| author_facet | Lee, Minju Oh, Hee |
| contents | Let $G$ be a connected semisimple real algebraic group. For any Zariski dense Anosov subgroup $Γ<G$, we show that a $Γ$-conformal measure is supported on the limit set of $Γ$ if and only if its "dimension" is $Γ$-critical. This implies the uniqueness of a $Γ$-conformal measure for each critical dimension. We deduce this from a higher rank analogue of the Hopf-Tsuji-Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_06794 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dichotomy and measures on limit sets of Anosov groups Lee, Minju Oh, Hee Dynamical Systems Geometric Topology Let $G$ be a connected semisimple real algebraic group. For any Zariski dense Anosov subgroup $Γ<G$, we show that a $Γ$-conformal measure is supported on the limit set of $Γ$ if and only if its "dimension" is $Γ$-critical. This implies the uniqueness of a $Γ$-conformal measure for each critical dimension. We deduce this from a higher rank analogue of the Hopf-Tsuji-Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups. |
| title | Dichotomy and measures on limit sets of Anosov groups |
| topic | Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2203.06794 |