Exponential mixing and essential spectral gaps for Anosov subgroups

TL;DR

This paper proves exponential mixing and spectral gaps for certain flows associated with Zariski dense Anosov subgroups, using Lie theoretic techniques and Dolgopyat's method, with exceptions characterized by a specific cone.

Contribution

It establishes uniform exponential mixing and spectral gaps for translation flows of Anosov subgroups outside an explicitly described exceptional cone.

Findings

Exponential mixing holds for vectors outside the exceptional cone.

Essential spectral gap for the Selberg zeta function is proven.

Prime orbit theorem with power saving error term is derived.

Abstract

Let $\Gamma$ be a Zariski dense $\Theta$-Anosov subgroup of a connected semisimple real algebraic group for some nonempty subset of simple roots $\Theta$. 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 $\Theta$-limit cone $\mathcal{L}_\Theta$ of $\Gamma$, which are conjugate to reparametrizations of the Gromov geodesic flow. We prove that for all $\mathsf{v}$ outside an exceptional cone $\mathscr{E} \subset \operatorname{int}\mathcal{L}_\Theta$, 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.