Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

2008/03/01 by Marcelo Viana · 22 citations
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.4007/annals.2008.167.643

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

Cited by

• Density of nonzero exponent of contraction for pinching cocycles in Hom(S1)
• Parametric Furstenberg Theorem on random products of S L ( 2 , R ) matrices
• Extremal Lyapunov exponents: an invariance principle and applications
• Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent
• Continuity of Lyapunov exponents in the C0 topology
• Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps
• COCYCLES WITH QUASI-CONFORMALITY I: STABILITY AND ABUNDANCE
• Invariance principle in dynamical systems
• Ledrappier-Young entropy formula for C¹ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle
• Physical Measure and Absolute Continuity for One-Dimensional Center Direction
• On Kanai's conjecture for frame flows over negatively curved manifolds
• Positive Lyapunov exponents for cocycles over time one anosov maps
• Holonomies and cohomology for cocycles over partially hyperbolic\n diffeomorphisms
• Generic simple cocycles over Markov maps
• Thermodynamic formalism of GL₂(\ℝ)-cocycles with canonical\n holonomies
• Cocycles with one exponent over partially hyperbolic systems
• What are Lyapunov exponents, and why are they interesting?
• Simple Lyapunov spectrum for certain linear cocycles over partially hyperbolic maps
• Invariance principle and rigidity of high entropy measures
• Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases
• Effective Gaps between singular values of non-stationary matrix products subject to non-degenerate noise
• On classification of higher rank Anosov actions on compact manifold

Related

• Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
• Lyapunov exponents with multiplicity 1 for deterministic products of matrices
• Positive Lyapunov exponents for symplectic cocycles
• Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases
• Continuity of Lyapunov exponents in the C0 topology