'October, 2005. Revised: June, 2006.'
title: 'Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for $\mathit{SL}(2,\mathbb{R})$ cocycles'
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Introduction and statement of the results
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*Throughout this paper let ${\mathbb{G}}= \mathit{SL}(2,\mathbb{R})$.*

Let $\mu$ be a finite positive measure on a measurable space $X$, $T \colon X \to X$ be a $\mu$-preserving map, and $A\colon X\to{\mathbb{G}}$ be a measurable map. The pair $(T,A)$ is called a cocycle. It induces a skew-product map $F_{T,A} \colon X \times {\mathbb{R}}^2 \to X \times {\mathbb{R}}^2$ defined by $F_{T,A}(x,v) = (T(x), A(x)v)$. Denote, for $n \in {\mathbb{N}}$, $$A_T^n(x) = A(T^{n-1}(x)) \cdots A(T(x)) A(x),
\quad \text{so that} \quad
F^n_{T,A}(x,v) = (T^n(x), A^n_T(x)v).$$ The group ${\mathbb{G}}$ also acts on ${\mathbb{P}}^1 = {\mathbb{P}}({\mathbb{R}}^2)$, so $(T,A)$ also induces a skew-product map on $X \times {\mathbb{P}}^1$. By simplicity we use the same notations in both ${\mathbb{R}}^2$ and ${\mathbb{P}}^1$ cases.

Provided $\log\|A\| \in L^1(\mu)$, the upper Lyapunov exponent of the cocycle $(T,A)$ at $x\in X$, given by $$\lambda(T,A,x) = \lim_{n \to \infty} \frac{1}{n}\log\|A_T^n(x)\|,$$ exists for $\mu$-almost every $x \in X$. (See e.g. [@LArnold] for basic facts about Lyapunov exponents.) We denote also $${\mathit{LE}}(A,T) = \int_X \lambda(T,A,x)\,d\mu(x).$$

A cocycle $(T,A)$ where $A$ is essentially bounded is called *uniformly hyperbolic* if there exists, for 