Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms

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 ${\mathbb{T}}^2$.