Anosov groups: local mixing, counting and equidistribution

By Samuel Edwards, Minju M. Lee, H. Oh. Geometry & Topology, 2020

60 citations recorded by the source corpus.

For a Zariski dense Anosov subgroup $\Gamma$ of a semisimple real Lie group $G$, we describe the asymptotic behavior of matrix coefficients $\Phi(g)=\langle g f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ for local functions $f_1, f_2\in C_c(\Gamma\backslash G)$. These asymptotics involve higher rank analogues of Burger-Roblin measures. As an application, for any symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$. The link between mixing and counting is provided by an equidistribution result for the translates $\Gamma\backslash \Gamma H a$ as $a\to \infty$ in $H\backslash G$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/27b1882c59fd15af9e78cba8134d6f826323779c

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Invariant Measures for Horospherical Actions and Anosov Groups (2020, background): “The works of Burger [11] and Quint [37] on a higher rank version of the PattersonSullivan theory supply a continuous family of maximal horospherically invariant Burger-Roblin measures, as was introduced in [15].”