
Let $A_1,\ldots, A_k$ be $SL(2,\mathbb{R})$ matrices. Given a
sequence $\omega \in \{1,\ldots, k\}^\mathbb{N}$, define the Lyapunov
exponent
\[
\chi(\omega) = \lim_{n\to\infty} \frac 1n \log
A_{\omega_n}\cdot\ldots\cdot A_{\omega_1}
\]
(if such a limit exists). Let
\[
L(\alpha) = \{\omega; \chi(\omega)=\alpha\}.
\]
The function $\alpha \to f(\alpha) = h_{\rm top}(L(\alpha))$ is called
Lyapunov spectrum of the matrix cocycle $(A_1,\ldots, A_k)$.
We give a description of the Lyapunov spectrum for generic
$SL(2,\mathbb{R})$ matrix cocycles. Of more interesting properties: it is a
concave function, a Legendre transform of some properly defined pressure
function, it has a maximum $\log k$ which is achieved only at one point, for
an elliptic cocycle generically we have $f(0)\in (0,\log k)$. It is a joint
work with Lorenzo Diaz and Katrin Gelfert.
