The standard family (or Taylor-Chirikov standard family) is an example
of a family of dynamical systems having simple expressions but with
complicated dynamics. A famous conjecture of Sinai is that for large
parameter the standard map has positive entropy for the Lebesgue measure.
In this seminar, I will talk about a recent result which I obtain the
uniqueness of the measure of maximal entropy of the standard map for
sufficiently large parameters. Moreover, I obtain that such a measure is
Bernoulli and the periodic points whose Lyapunov exponents are bounded
away from zero equidistribute with respect to this measure. I can also
obtain estimates on the Hausdorff dimension of the measure and of the
support.