Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

2008/03/01 by Marcelo Viana · 98 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Lyapunov exponent #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Nonlinear system #Pure mathematics #Quantum chaos and dynamical systems

paper · pdf · doi:10.4007/annals.2008.167.643

published in Annals of Mathematics 167(2), 643-680 (Princeton University)

openalex publication_date 2008/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We prove that for any s > 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.

Citations

• Products of Random Matrices
• Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture

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