Locally homogeneous Axiom A flows II: geometric structures for Anosov subgroups
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arXiv
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| Format: | Preprint |
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2025
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| author | Delarue, Benjamin Monclair, Daniel Sanders, Andrew |
| author_facet | Delarue, Benjamin Monclair, Daniel Sanders, Andrew |
| contents | Given a non-compact semisimple real Lie group $G$ and an Anosov subgroup $Γ$, we utilize the correspondence between $\mathbb R$-valued additive characters on Levi subgroups $L$ of $G$ and $\mathbb R$-affine homogeneous line bundles over $G/L$ to systematically construct families of non-empty domains of proper discontinuity for the $Γ$-action. If $Γ$ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the $\mathbb R$-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally homogeneous Axiom A flows II: geometric structures for Anosov subgroups Delarue, Benjamin Monclair, Daniel Sanders, Andrew Geometric Topology Differential Geometry Dynamical Systems Given a non-compact semisimple real Lie group $G$ and an Anosov subgroup $Γ$, we utilize the correspondence between $\mathbb R$-valued additive characters on Levi subgroups $L$ of $G$ and $\mathbb R$-affine homogeneous line bundles over $G/L$ to systematically construct families of non-empty domains of proper discontinuity for the $Γ$-action. If $Γ$ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the $\mathbb R$-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of $G$. |
| title | Locally homogeneous Axiom A flows II: geometric structures for Anosov subgroups |
| topic | Geometric Topology Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2502.20195 |