Dynamique des applications polynomiales semi-régulières

Abstract

For any proper polynomial mapf: C ^k→C ^k define the function α as

$$\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$$

Letf=(P ₁,...,P _k ) be a proper polynomial map. We define a notion ofs-regularity using the extension off to P^k. Whenf is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 andd ₁,...,d _k , whered _i :=degP _i . We then describe dynamically the sets {α≤d _i }. We give a concrete description of regular maps. Ifd _i >1, this allows us to construct the equilibrium measure μ associated withf 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.