Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909231954788352 |
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| author | Chow, Michael Sarkar, Pratyush |
| author_facet | Chow, Michael Sarkar, Pratyush |
| contents | Let $G$ be a connected semisimple real algebraic group and $Γ< G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb R\}$ on $Γ\backslash G$ for any interior direction $\mathsf{v}$ of the limit cone of $Γ$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf{u}$ in some fixed subspace transverse to $\mathsf{v}$. We also obtain a uniform bound for the correlation function which decays exponentially in $\|\mathsf{u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(Γ\backslash G)$ proved by Edwards--Lee--Oh. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_11377 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces Chow, Michael Sarkar, Pratyush Dynamical Systems Geometric Topology Spectral Theory 37A17, 37A25, 37C30, 37D20 Let $G$ be a connected semisimple real algebraic group and $Γ< G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb R\}$ on $Γ\backslash G$ for any interior direction $\mathsf{v}$ of the limit cone of $Γ$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf{u}$ in some fixed subspace transverse to $\mathsf{v}$. We also obtain a uniform bound for the correlation function which decays exponentially in $\|\mathsf{u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(Γ\backslash G)$ proved by Edwards--Lee--Oh. |
| title | Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces |
| topic | Dynamical Systems Geometric Topology Spectral Theory 37A17, 37A25, 37C30, 37D20 |
| url | https://arxiv.org/abs/2105.11377 |