Dynamique des applications polynomiales semi-regulieres
Abstract
Description
For any proper polynomial map $f:C^k\longrightarrow C^k$ define the function αas $$α(z):=\limsup_{n\to\infty} \frac{\log^+\log^+|f^n(z)|}{n} where \log^+:=\max(\log, 0).$$ 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 αis 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 (α\leq d_i). If d_i>1, this allows us to construct the equilibrium measure μassociated to f as a generalized intersection of positive currents. We then gives an estimate of the Hausdorff dimension of μ. This is a special case of our results. We extend the approach to the larger class of (π,s)-regular maps. This gives an understanding of the biggest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.
29 pages, nouvelle version
29 pages, nouvelle version