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 $C(X,GL(2,**R**))$.

Bibliographical: Ann. Inst. H. Poincare Probab. Statist.

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[Papers] [Alex Furman]