  We study the dynamics of polynomial-like mappings in several variables. A
special case of our results is the following theorem. Let f be a proper
holomorphic map from an open set U onto a Stein manifold V, $U\subset\subset
V$. Assume f is of topological degree d_t>1. Then there is a probability
measure \mu supported on $\bigcap_{n\geq 0}f^{-n}(V)$ satisfying the following
properties. 1. The measure \mu is invariant, K-mixing, of maximal entropy \log
d_t. 2. If J is the Jacobian of f with respect to a volume form then $\int \log
J \d \mu \geq \log d_t$. 3. For every probability measure \nu on V with no mass
on pluripolar sets $d_t^{-n} (f^n)^*\nu$ converges to $\mu$. 4. If the p.s.h.
functions on V are \mu-integrables (\mu is PLB) then (a) The Lyapounov
exponents for \mu are strictly positive. (b) \mu is exponentially mixing. (c)
There is a proper analytic subset E of V such that for $z\not\in\E$,
$\mu^z_n:=d_t^{-n} (f^n)^*\delta_z$ converges to \mu. (d) The measure \mu is a
limit of Dirac masses on the repelling periodic points. The condition \mu is
PLB is stable under small pertubation of f. This gives large families where it
is satisfied.
