Lyapunov spectrum properties and continuity of the lower joint spectral radius

Abstract

We study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show the continuity of the entropy spectrum at the boundary of Lyapunov spectrum in the sense that $h_{top}(E(\alpha_{t}))\ \rightarrow h_{top}(E(\beta(\mathcal{A}))$ for generic cocycles, where $E(\alpha)=\{x\in X: \lim_{n\rightarrow \infty}\frac{1}{n}\log \|\mathcal{A}^{n}(x)\|=\alpha\}$. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles over mixing subshifts of finite type. Moreover, we prove the restricted variational principle for such cocycles. We prove the continuity of the lower joint spectral radius for general cocycles under the assumption that linear cocycles satisfy a cone condition.