On the multiplicative ergodic theorem for uniquely ergodic systems

Abstract

We consider the question of uniform convergence in the multiplicative ergodic theorem lim n→∞ 1 n ·log‖A(T n−1x)…A(x)‖=⋀(A) for continuous function A : X → GL d ( R ), where ( X, T) is a uniquely ergodic system. We show that the inequality lim sup n→∞ n −1 · log ∥ A( T n−1 x) ⋯ A( x) ∥ ≤ Λ( A) holds uniformly on X, but it may happen that for some exceptional zero measure set E ⊂ X of the second Baire category: lim inf n→∞ n −1 · log ∥ A( T n−1 x) ⋯ A( x)∥ < Λ( A). We call such A a non-uniform function. We give sufficient conditions for A to be uniform, which turn out to be necessary in the two-dimensional case. More precisely, A is uniform iff either it has trivial Lyapunov exponents, or A is continuously cohomologous to a diagonal function. For equicontinuous system ( X, T), such as irrational rotations, we identify the collection of non-uniform matrix functions as the set of discontinuity of the functional A on the space C( X, GL 2( R )), thereby proving, that the set of all uniform matrix functions forms a dense G δ -set in C( X, GL 2( R )). It follows, that M. Herman's construction of a non-uniform matrix function on an irrational rotation, gives an example of discontinuity of A on C( X, GL 2( R )).