Group Actions with Hyperbolicity and Measure Rigidity

Stationary probability measures on projective spaces

28 mai 2024, 09:30

1h

Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Orateur

Çagri Sert (University of Zurich)

Description

We give a description of stationary probability measures on projective spaces for an iid random walk on ${\mathrm{PGL}}_d(\mathbb{R})$ without any algebraic assumptions. This is done in two parts. In a first part, we study the case (non-critical or block-dominated case) where the random walk has distinct deterministic exponents in the sense of Furstenberg--Kifer--Hennion. In a second part (critical case), we show that if the random walk has only one deterministic exponent, then any stationary probability measure on the projective space lives on a subspace on which the ambient group of the random walk acts semisimply. This connects the critical setting with the work of Guivarc'h--Raugi and Benoist--Quint. Combination of all these works allow to get a complete description. Joint works with Richard Aoun.