Livšic Theorem for matrix cocycles

2011/03/01 by Boris Kalinin · 2 citations
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #advanced mathematical theories

paper · pdf · doi:10.4007/annals.2011.173.2.11

Abstract

We prove the Livic Theorem for arbitrary GL(m, R) cocycles. We consider a hyperbolic dynamical system f : X X and a Hlder continuous function A : X GL(m, R). We show that if A has trivial periodic data, i.e. A(f n-1 p) A(f p)A(p) = Id for each periodic point p = f n p, then there exists a Hlder continuous function C : X GL(m, R) satisfying A(x) = C(f x)C(x) -1 for all x X. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Hlder function A.

Cited by

• Normal forms in a neighborhood of hyperbolic periodic orbits for flows in dimension 3
• Cohomology of Lipschitz-valued cocycles
• The generalized Berger-Wang formula and the spectral radius of linear cocycles
• A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

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