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

TL;DR

This paper investigates the structure of the Julia set for holomorphic endomorphisms of complex projective space, revealing the existence of Fatou directions and providing estimates on expansion rates in these complex dynamical systems.

Contribution

It introduces the concept of Fatou directions along the Julia set and analyzes their properties for generic points, extending understanding of complex dynamics outside the maximal entropy measure support.

Findings

Existence of at least (k-q) Fatou directions at generic points in certain Julia set subsets.

Provides estimates for the rate of expansion transverse to Fatou directions.

Characterizes the structure of the Julia set via supports of exterior powers of the Green current.

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 $1\leq q \leq k$, 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.