Fatou directions along the Julia set for endomorphisms of CP^k

2010/06/04 by Romain Dujardin, Dujardin, Romain · 1 citation
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS

paper · pdf · doi:10.48550/arxiv.1006.0882

Final, shorter version, to appear in J. Math. Pures Appl

arxiv created 2012/03/26 · arxiv updated 2012/03/28

Abstract

Not much is known about the dynamics outside the support of the maximal entropy measure μ for holomorphic endomorphisms of \mathbbCP^k. In this article we study the structure of the dynamics on the Julia set, which is typically larger than Supp(μ). The Julia set is the support of the so-called Green current T, so it admits a natural filtration by the supports of the exterior powers of T. For 1≤ q ≤ k, let J_q= Supp(T^q). We show that for a generic point of J_q∖ J_q+1 there are at least (k-q) "Fatou directions" in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions.

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