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

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

2010-06-04

arxiv.org/abs/1006.0882v2

Mathematics

Dynamical Systems

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

Scientific paper

Not much is known about the dynamics outside the support of the maximal entropy measure $\mu$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(\mu)$. 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 $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ 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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