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 $\mathbb{T}^2$.