Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied by Guivarc'h and Raugi '86, Gol'dsheid and Margulis '89, Bonatti-Viana '04, Avila-Viana '07 and very recently by Mauricio Poletti.
My* questions are: What is the importance of having simplicity of the spectrum? Is it there any physical property that translates into simplicity of the spectrum?
In order to be specific about the question, I include a formal theorem of the simplicity of spectrum.
Let $(M,\mathcal B, \mu)$ be a complete separable probability space. Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, i.e. $F(x,v):=(f(x),A(x)v)$, defined by a measurable function $A:M\to GL(d)$ over a measurable function $f:M\to M$ that preserves $\mu.$ By Oseledets' theorem we have that for $\mu$-a.e. $x\in M$ there is $k=k(x),$ numbers $\lambda_1(x)>\cdots>\lambda_k(x)$ and flags $\mathbb{R}^d=V_x^1\supsetneq \cdots \supsetneq V_x^k \supsetneq \{0\},$ such that for all $i=1,\ldots,k:$
- k(f(x))=k(x) and $\lambda_i(f(x))=\lambda_i(x)$ and $A(x)V_x^i=V_{f(x)}^i;$
- the map $x\mapsto k(x)$ and $x\mapsto \lambda_i(x)$ and $x\mapsto V_x^i$ are measurable; and
- $\lim_n \frac{1}{n}\log\|A^n(x)v\|=\lambda_i(x)$ for all $v\in V_x^i\setminus V_x^{i+1}.$
Formal theorem: Under a $\star$ condition, that depends on $(M,\mu,F),$ for $\mu$-a.e. $x\in M,$ we have that $d=k(x).$
*The first question was originally asked yesterday, in other terms, by Mario Ponce. I had this question at almost the same time, but I am still curious about the answer, this is the reason I dediced to ask it here.
