Dynamics of polynomial-like mappings

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem: Let f : U --> V be a proper holomorphic map from an open set U subset of V onto a Stein manifold V. Assume f, is of. topological degree d(t) greater than or equal to 2. Then there is a probability measure mu supported on K:= boolean AND(n) greater than or equal to0 f(-n) (V) satisfying the following properties: (1) The measure A is invariant, K-mixing, of maximal entropy log dt. (2) If J is the Jacobian of f with respect to a volume form Omega then integral log J dmu greater than or equal to log d(t). (3) For every probability measure nu on V with no mass on pluripolar sets d(t)(-n)(f(n))*nu 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 epsilon(o) of V such that f(-1) (epsilon(0)) subset of epsilon(o) and for z is not an element of epsilon, mu(n)(z) d(t)(-n)(f(n))*delta(z) -mu where epsilon = boolean OR(ngreater than or equal to0)f(n)(epsilon(0)); (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. (C) 2003 Editions scientifiques et medicales Elsevier SAS. Tous droits reserves.