Regularity and convergence rates for the Lyapunov exponents of linear cocycles
Wilhelm Schlag - Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615, United States (email) 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.
Keywords: Multiplicative Ergodic Theorem, shift dynamics, Lyapunov exponents.
Received: August 2013; Available Online: March 2014. |