Exponential mixing of frame flows for convex cocompact locally symmetric spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912240382246912 |
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| author | Chow, Michael Sarkar, Pratyush |
| author_facet | Chow, Michael Sarkar, Pratyush |
| contents | Let $G$ be a connected center-free simple real algebraic group of rank one and $Γ< G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $Γ\backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14737 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Exponential mixing of frame flows for convex cocompact locally symmetric spaces Chow, Michael Sarkar, Pratyush Dynamical Systems Differential Geometry Spectral Theory 37D40, 22E40, 37A25 Let $G$ be a connected center-free simple real algebraic group of rank one and $Γ< G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $Γ\backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case. |
| title | Exponential mixing of frame flows for convex cocompact locally symmetric spaces |
| topic | Dynamical Systems Differential Geometry Spectral Theory 37D40, 22E40, 37A25 |
| url | https://arxiv.org/abs/2211.14737 |