Lyapunov Spectrum Properties and Continuity of the Lower Joint Spectral Radius

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