2023/05/06 by Bersudsky, Michael, Xing, Hao
#37 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2305.04132
We study the distribution of orbits of a lattice Γ\leqSL(3,\mathbb R) in the moduli space X2,3 of covolume one rank-two discrete subgroups in \mathbb R3. Each orbit is dense, and our main result is the limiting distribution of these orbits with respect to norm balls, where the norm is given by the sum of squares. Specifically, we consider ΓT=\γ∈Γ:‖γ‖≤ T\ and show that, for any fixed x0∈ X2,3 and φ∈ Cc(X2,3), limT→∞(1)/(#ΓT)∑γ∈ΓTφ(x0⋅γ)=∫_X2,3φ(x)d νx0(x), where νx0 is an explicit probability measure on X2,3 depending on x0. To prove our result, we use the duality principle developed by Gorodnik and Weiss which recasts the above problem into the problem of computation of certain volume estimates of growing skewed balls in H and proving ergodic theorems of the left action of the skewed balls on SL(3,ℝ)/Γ. The ergodic theorems are proven by applying theorems of Shah building on the linearisation technique. The main contribution of the paper is the application of the duality principle in the case where H has infinitely many non-compact connected components.