[Papers] [Alex Furman]

On the multiplicative ergodic theorem for uniquely ergodic systems

Consider the question of uniform convergence in the multiplicative ergodic theorem $\lim n^{-1}\log\|A(T^{n-1}x)\cdots A(x)\|=\Lambda (A)$ for continuous function $A:X\to GL(d,R)$, where $(X,T)$ is a uniquely ergodic system. We show that the inequality $\limsup_{n\to\infty} n^{-1}\cdot \log\|A(T^{n-1}x)\cdots A(x)\|\le\Lambda(A)$ holds uniformly on $X$, but it may happen that for some exceptional zero measure set $E\subset X$ of the second Baire category: $\liminf_{n\to\infty} n^{-1}\cdot \log\|A(T^{n-1}x)\cdots A(x)\|<\Lambda(A)$. 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 $\Lambda$ 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 function on an irrational rotation, gives an example of discontinuity of $\Lambda$ on $C(X,GL(2,R))$.

Authors: A. Furman,
Bibliographical: Ann. Inst. H. Poincare Probab. Statist. 33 (1997), no. 6, 797--815.