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

Abstract

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 $1\le q\le k$, 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.

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

References in corpus (1)

• Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

Cited by in corpus (1)

• Parametric Lyapunov exponents