Lyapunov exponents tell us the rate of divergence of nearby trajectories – a key component of chaotic dynamics. For one dimensional maps Lyapunov exponent at a point is a Birkhoff average of $\log |f'|$ along the trajectory of this point. For a typical point for an ergodic invariant measure it is equal to the average of $\log |f'|$ with respect to this measure.

In this talk we deal with Lyapunov exponents of products of random i.i.d. $2\times 2$ matrices of determinant $ \pm1$. Let $SL(2, \mathcal{R})$ denote the group of such matrices. Let $\mu $ be a probability measure in $SL(2, \mathcal{R})$ which satisfies the integrability $$\int_{SL(2, \mathcal{R})} \log||M||d\mu(M)<\infty.$$ If $ Y_{1}, Y_{2}, \dots $ are random independent matrices with distribution $\mu$, then the limit $$\gamma= \lim_{n\to\infty}\frac{1}{n} \log|| Y_{n}\dots Y_{1}||$$ (the upper Lyapunov exponent) exists a.s. and is constant, by the subadditive ergodic theorem. We have $\gamma \geq 0$.

The talk will follow a survey by Avila and Bochi and will be devoted to:

  1. Furstenberg's Theorem;
  2. Zero Lyapunov Exponents.