Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms

TL;DR

This paper investigates the rigidity of Lyapunov spectra in random dynamical systems on circles and tori, showing conditions under which systems can be simultaneously linearized and establishing positive exponent rigidity.

Contribution

It establishes Lyapunov spectrum rigidity for random walks of expanding maps and Anosov diffeomorphisms, and proves positive Lyapunov exponent rigidity for certain matrix actions.

Findings

Lyapunov spectrum rigidity implies simultaneous linearization.

Rigidity results hold for systems with matching Lyapunov spectra.

Positive Lyapunov exponent rigidity is proven for specific matrix actions.

Abstract

In this paper we study the Lyapunov spectrum rigidity for random walks of expanding maps on unit circle $\mathbb{S}^1$ and Anosov diffeomorphisms on $d$-torus $\mathbb{T}^d$. Let $\nu$ be a probability supported on the set of expanding maps on $\mathbb{S}^1$ or a neighborhood of a generic Anosov automorphisms on $\mathbb{T}^d$. If the Lyapunov spectrum of the $\nu$-stationary SRB-measure coincides with the Lyapunov spectrum of the algebraic action, then we can simultaneously linearize $\nu$ almost every system to an affine action. Moreover, we prove the positive Lyapunov exponent rigidity for random walks of irreducible positive matrices acting on $\mathbb{T}^2$.