Dynamique des applications polynomiales semi-regulieres

Abstract

For any proper polynomial map $f:C^k\longrightarrow C^k$ define the function \alpha as $$\alpha(z):=\limsup_{n\to\infty} \frac{\log^+\log^+|f^n(z)|}{n} where \log^+:=\max(\log, 0).$$ Let f=(P_1,...,P_k) be a proper polynomial map. We define a notion of s-regularity using the extension of f to P^k. When f is (maximally) regular we show that the function \alpha is l.s.c and takes only finitely many values: 0 and d_1, ..., d_k, where d_i:=deg P_i. We then describe dynamically the sets (\alpha\leq d_i). If d_i>1, this allows us to construct the equilibrium measure \mu associated to f as a generalized intersection of positive currents. We then gives an estimate of the Hausdorff dimension of \mu. This is a special case of our results. We extend the approach to the larger class of (\pi,s)-regular maps. This gives an understanding of the biggest values of \alpha. The results can be applied to construct dynamically interesting measures for automorphisms.