ON THE MULTIPLICATIVE ERGODIC THEOREM FOR UNIQUELY ERGODIC SYSTEMS

We consider the question of uniform convergence in the multiplicative ergodic theorem lim(n-->infinity) 1/n.log parallel to A(T(n-1)x)...A(x )parallel to = Lambda(A)for continuous function A : X --> GL(d)(R), wh ere (X,T) is a uniquely ergodic system. We show that the inequality li m sup(n-->infinity) n(-1).log parallel to A(T(n-1)x)...A(x)parallel to less than or equal to Lambda(A) holds uniformly on X, but it may happ en that for some exceptional zero measure set E subset of X of the sec ond Baire category: lim inf(n-->infinity) n(-1).log parallel to A(T(n- 1)x)...A(x)parallel to < Lambda(A). We call such A a non-uniform funct ion. We give sufficient conditions for A to be uniform, which turn out to be necessary in the two-dimensional case. More precisely, A is uni form iff either it has trivial Lyapunov exponents, or A is continuousl y cohomologous to a diagonal function. For equicontinuous system (X, T ), such as irrational rotations, we identify the collection of nan-uni form matrix functions as the set of discontinuity of the functional La mbda on the space C(X, GL(2)(R)), thereby proving, that the set of all uniform matrix functions forms a dense G(delta)-set in C(X, GL(2)(R)) .It follows, that M. Herman's construction of a non-uniform matrix fun ction on an irrational rotation, gives an example of discontinuity of A on C(X, GL(2)(R)).