Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909621509160960 |
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| author | Kim, Dongryul M. Oh, Hee |
| author_facet | Kim, Dongryul M. Oh, Hee |
| contents | For a geometrically finite Kleinian group $Γ$, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup $Γ$ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of $Γ$ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization Kim, Dongryul M. Oh, Hee Dynamical Systems Group Theory Geometric Topology For a geometrically finite Kleinian group $Γ$, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup $Γ$ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of $Γ$ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing. |
| title | Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization |
| topic | Dynamical Systems Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2404.09745 |