Exponential mixing and essential spectral gaps for Anosov subgroups

TL;DR

The paper analyzes translation flows attached to Zariski-dense $\Theta$-Anosov subgroups of a real semisimple group and proves exponential mixing for a large family of flows parametrized by $\mathsf{v}$ in the interior of the $\Theta$-limit cone, away from an exceptional cone where a Lie-theoretic local non-integrability condition fails. It foregrounds a Dolgopyat-type mechanism built from metric Anosov structure, Markov sections, and transfer operators, together with a precise Lie-theoretic LNIC to control oscillations uniformly in a family of directions. Consequences include an essential spectral gap for the Selberg zeta function and a prime orbit theorem with a power-saving error term, together with uniform exponential mixing on compact vector families. A central feature is the explicit dependence on $\mathsf{v}$ via the growth indicator $\psi_\Theta$ and the associated $\kappa_\Theta$, which quantify decay rates near the boundary of the limit cone and identify where the main dynamics change qualitatively. Overall, the work extends exponential mixing and spectral-gap phenomena to the broad setting of $\Theta$-Anosov subgroups using a robust synthesis of Lie theory, symbolic dynamics, and Dolgopyat’s analytic machinery, with a clear geometric/topological interpretation of the exceptional set and the resonance structure.

Abstract

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.

Theorems & Definitions (87)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10