Dynamique des applications polynomiales semi-régulières

Abstract

For any proper polynomial map f: C^k→C^k define the function α as $\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$ Let f=(P₁,..., 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 α is lower semicontinuous and takes only finitely many values: 0 and d₁,..., d_k, where d_i:=degP_i. We then describe dynamically the sets {α≤d_i}. We give a concrete description of regular maps. If d_i >1, this allows us to construct the equilibrium measure μ associated with f as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π, s)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.