Partially hyperbolic random dynamics on Grassmannians

TL;DR

This paper studies how small random perturbations of a fixed partially hyperbolic matrix induce a random dynamics on Grassmannians, showing concentration near stable fixed points and deriving bounds on Lyapunov exponents.

Contribution

It provides new concentration bounds for the invariant measure of the random dynamics on Grassmannians under weak conditions, highlighting the effect of perturbation strength.

Findings

High probability the dynamics remains near stable fixed points

Bounds on sums of Lyapunov exponents derived

Concentration results depend on perturbation strength

Abstract

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure showing that with high probability the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.