Regularity and convergence rates for the Lyapunov exponents of linear cocycles

Wilhelm Schlag

doi:10.3934/jmd.2013.7.619

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2013, Volume 7, Issue 4: 619-637. Doi: 10.3934/jmd.2013.7.619

This issue Previous Article The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited Next Article

Regularity and convergence rates for the Lyapunov exponents of linear cocycles

Wilhelm Schlag^1,

1.
Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615

Received: July 31, 2013

Published: February 2014

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Abstract

We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto ( x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where $\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain distinct near $E$. Moreover, they depend in a Hölder fashion on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct. Similar results, with a weaker modulus of continuity, hold for higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also derive optimal statements about the rate of convergence of the finite-scale Lyapunov exponents to their infinite-scale counterparts. A key ingredient in our arguments is the Avalanche Principle, a deterministic statement about long finite products of invertible matrices, which goes back to work of Michael Goldstein and the author. We also discuss applications of our techniques to products of random matrices.

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Mathematics Subject Classification: 37D25.

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