Measure rigidity for horospherical subgroups of groups acting on trees

Abstract

We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let G be a closed subgroup of the group of automorphisms of a biregular tree and Γ ≤ G a discrete subgroup. For a large class of groups G, we give a classification of the probability measures on G/Γ invariant under horospherical subgroups. When Γ is a cocompact lattice, we show the unique ergodicity of the horospherical action. Moreover, we prove Hedlund's theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

(Non)-escape of mass and equidistribution for horospherical actions on trees

“…This section is devoted to the proof of Theorem B which we deduce from Theorem A and our previous work [9].…”

“…This can be done using the Howe-Moore property, established in our setting in [21] and amenable ergodic theorem [19]. Our topological result in [9] says, however, that all points x ∈ X that do not lie in a compact G 0 η -orbit have dense orbits. Therefore, the immediate question arises whether every dense orbit equidistributes to the Haar measure on G/Γ.…”

“…Theorem B is proven by using Theorem A, the classification of G 0 η -orbits, given in [9] and the Howe-Moore property established in [21].…”

“…In our geometric setup, the role of Ad-unipotent subgroups in classical homogeneous dynamics is played by the horospherical subgroups G 0 η of G, for η ∈ ∂T . In the earlier work [9], the authors classified Borel probability measures on G/Γ invariant under G 0 η -action for large class of groups G and 2010 Mathematics Subject Classification. 22D40,20E08.…”

“…In the proof of Theorem A, using the underlying geometric setting, we translate the problem of understanding the distribution of G 0 η -orbit in G/Γ to the language of Markov chains, where it appears as a problem of controlling the distributions of a Markov chain with non-constant initial distributions. We then rely on two main ingredients: the first is a qualitative description of the behaviour of the discrete geodesic flow on G/Γ, as studied in [9]. This allows us to understand the behaviour of initial distributions of the Markov chain.…”