We have seen in Chapter 9 that the Lyapunov exponents may depend in a complicated way on the underlying linear cocycle. The theme, in the context of products of random matrices, of the present chapter is that this dependence is always continuous.

Let G(d) denote the space of compactly supported probability measures p on GL(d), endowed with the following topology: p' is close to p if it is close in the weak*-topology and supp p' is contained in a small neighborhood of supp p. Let λ+(p) and λ−(p) denote the extremal Lyapunov exponents of the product of random matrices associated with a given p ∈ G (d), in the sense of Section 2.1.1. In other words, λ±(p) = λ±(A, μ), where A : GL(d)ℕ→ GL(d), (αk)k ↦ α0 and μ = pℕ. A probability measure η on ℙℝd will be called pstationary if it is stationary for this cocycle. We are going to prove:

Theorem 10.1 (Bocker and Viana) The functions G(2) → R, p ↦ λ±(p) are continuous at every point in the domain.

Avila, Eskin and Viana announced recently that this statement remains true in arbitrary dimension. Even more, for any d ≥ 2, all the Lyapunov exponents depend continuously on the probability distribution p ∈ G (d). The proof will appear in [12].