Dynamics on the space of 2-lattices in 3-space

2017/08/15 by Sargent, Oliver, Shapira, Uri · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.04464

Abstract

We study the dynamics of SL₃(ℝ) and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of ℝ³. We focus on the case where the acting group is Zariski dense in either SL₃(ℝ) or SO(2,1)(ℝ). Using techniques of Benoist and Quint we prove that for a compactly supported probability measure μ on SL₃(ℝ) whose support generates a group which is Zariski dense in SL₃(ℝ), there exists a unique μ-stationary probability measure on X. When the Zariski closure is SO(2,1)(ℝ) we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.

Cited by

• Limit distributions for SO(n,1) action on k-lattices in ℝⁿ⁺¹

Related

• Equidistribution of lattice orbits in the space of homothety classes of rank 2 sublattices in \mathbb R³
• Limiting distribution of dense orbits in a moduli space of rank m discrete subgroups in (m+1)-space
• The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions
• Measure rigidity for solvable group actions in the space of lattices
• Dynamics on spaces of quasimorphisms and applications to approximate lattice theory