On the multiplicative ergodic theorem for uniquely ergodic systems

Abstract

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The Ten Martini Problem

“…The numerators in (9-6) can be bounded uniformly in  [Jit99], [Fur97]. Namely, for every E 2 ‫;ޒ‬ > 0, there exists a k 2 .…”

The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents

“…A proof of it in the case of continuous functions is given in [12]; the proof for upper semi-continuous functions is exactly the same.…”

Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

“…Proof. The argument is almost the same as the proof of [23,Lemma 5], but for the completeness, we give the details.…”

“…Theorem 4 is also proved in this section based on Theorem 3 and using a genericity theorem of uniformity borrowed from [23]. In fact, we will consider linear cocycles over a compact system (W, θ ) which need not be invertible and we will generalize a uniformity theorem of Walters (Theorem 6.14 below) by using natural extension.…”

“…Recently, this topic becomes more and more interesting; see [44,23,5,12,41,46,13,14,20,19] amongst others. In this paper, we consider a linear skew-product system…”