Dynamique des applications polynomiales semi-regulieres

Abstract

For any proper polynomial map $f:C^k⟶C^k$ define the function αas $$\alpha (z):=\underset{n\to \infty }{\mathrm{limsup}}\frac{{\log }^{+}{\log }^{+}|f^n(z)|}{n}where{\log }^{+}:=\max (\log ,0).$$ 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