Lyapunov Spectrum Properties and Continuity of the Lower Joint Spectral Radius

Abstract

AbstractIn this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that $$h_{top}(E(\alpha _{t}))\ \rightarrow h_{top}(E(\beta ({\mathcal {A}}))$$ h top ( E ( α t ) ) → h top ( E ( β ( A ) ) , where $$E(\alpha )=\{x\in X: \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert {\mathcal {A}}^{n}(x)\Vert =\alpha \}$$ E ( α ) = { x ∈ X : lim n → ∞ 1 n log ‖ A n ( x ) ‖ = α } , for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.