Exponential mixing and essential spectral gaps for Anosov subgroups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912241136173056 |
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| author | Chow, Michael Sarkar, Pratyush |
| author_facet | Chow, Michael Sarkar, Pratyush |
| contents | Let $Γ$ be a Zariski dense $Θ$-Anosov subgroup of a connected semisimple real algebraic group for some nonempty subset of simple roots $Θ$. In the Anosov setting, there is a natural compact metric space $\mathcal{X}$ equipped with a family of translation flows $\{a^{\mathsf{v}}_t\}_{t \in \mathbb R}$, parameterized by vectors $\mathsf{v}$ in the interior of the $Θ$-limit cone $\mathcal{L}_Θ$ of $Γ$, which are conjugate to reparametrizations of the Gromov geodesic flow. We prove that for all $\mathsf{v}$ outside an exceptional cone $\mathscr{E} \subset \operatorname{int}\mathcal{L}_Θ$, which is a smooth image of the linear spans of the walls of the Weyl chamber, the translation flow is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure associated to $\mathsf{v}$. Moreover, the exponential rate is uniform for a compact set of such vectors. We also obtain an essential spectral gap for the Selberg zeta function and a prime orbit theorem with a power saving error term. Our proof relies on Lie theoretic techniques to prove the crucial local non-integrability condition (LNIC) for the translation flows and thereby implement Dolgopyat's method in a uniform fashion. The exceptional cone $\mathscr{E}$ arises from the failure of LNIC for those vectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential mixing and essential spectral gaps for Anosov subgroups Chow, Michael Sarkar, Pratyush Dynamical Systems Differential Geometry Number Theory 37A17, 37D20, 37A25, 37C30, 11M36, 11N45 Let $Γ$ be a Zariski dense $Θ$-Anosov subgroup of a connected semisimple real algebraic group for some nonempty subset of simple roots $Θ$. In the Anosov setting, there is a natural compact metric space $\mathcal{X}$ equipped with a family of translation flows $\{a^{\mathsf{v}}_t\}_{t \in \mathbb R}$, parameterized by vectors $\mathsf{v}$ in the interior of the $Θ$-limit cone $\mathcal{L}_Θ$ of $Γ$, which are conjugate to reparametrizations of the Gromov geodesic flow. We prove that for all $\mathsf{v}$ outside an exceptional cone $\mathscr{E} \subset \operatorname{int}\mathcal{L}_Θ$, which is a smooth image of the linear spans of the walls of the Weyl chamber, the translation flow is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure associated to $\mathsf{v}$. Moreover, the exponential rate is uniform for a compact set of such vectors. We also obtain an essential spectral gap for the Selberg zeta function and a prime orbit theorem with a power saving error term. Our proof relies on Lie theoretic techniques to prove the crucial local non-integrability condition (LNIC) for the translation flows and thereby implement Dolgopyat's method in a uniform fashion. The exceptional cone $\mathscr{E}$ arises from the failure of LNIC for those vectors. |
| title | Exponential mixing and essential spectral gaps for Anosov subgroups |
| topic | Dynamical Systems Differential Geometry Number Theory 37A17, 37D20, 37A25, 37C30, 11M36, 11N45 |
| url | https://arxiv.org/abs/2408.11274 |