Lyapunov spectrum properties and continuity of the lower joint spectral radius

2020/01/12 by Reza Mohammadpour, Mohammadpour, Reza · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #26A18 #37D30 #37D35 #37H15 #93D30 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation

paper · pdf · doi:10.48550/arxiv.2001.03958

openalex publication_date 2020/01/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

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(α_t)) → h_top(E(β(A)) for generic cocycles, where E(α)=\x∈ X: lim_{n→ ∞}(1)/(n)log ‖Aⁿ(x)‖=α\. 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.

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• Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications
• Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative t

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