Expanding Cone and Applications to Homogeneous Dynamics

Abstract

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.