Ergodic decompositions of geometric measures on Anosov homogeneous spaces

2020/10/21 by Lee, Minju, Oh, Hee · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2010.11337

Abstract

Let G be a connected semisimple real algebraic group and Γ a Zariski dense Anosov subgroup of G with respect to a minimal parabolic subgroup P. Let N be the maximal horospherical subgroup of G given by the unipotent radical of P. We describe the N-ergodic decompositions of all Burger-Roblin measures as well as the A-ergodic decompositions of all Bowen-Margulis-Sullivan measures on Γ\backslash G. As a consequence, we obtain the following refinement of the main result of [LO]: the space of all \it non-trivial N-invariant ergodic and P^∘-quasi-invariant Radon measures on Γ\backslash G, up to constant multiples, is homeomorphic to \mathbb R^{rank G-1}× \1, ⋯, k\ where k is the number of P^∘-minimal subsets in Γ\backslash G.

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• Classification of horospherical invariant measures in higher rank
• Dynamics and Rigidity through the Lens of Circles

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