Structure of Julia sets for post-critically finite endomorphisms on $\mathbb{p}^2$

TL;DR

This paper investigates the structure of Julia sets for post-critically finite endomorphisms on the complex projective plane, revealing their decomposition, properties of Fatou disks, and implications for dynamical behavior and invariant sets.

Contribution

It characterizes the Julia set structure for PCF maps on , clarifies the non-wandering set, and provides new proofs and characterizations related to Green currents and expansion properties.

Findings

J_1J_2 is contained in basins of critical cycles and super-saddle cycles.

For points in J_2 not in stable manifolds, no Fatou disk contains them.

Under smoothness assumptions, no sporadic super-saddle cycles exist, simplifying the Julia set structure.

Abstract

Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of $PC(f)$ are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case $J_1\setminus J_2$ is contained in the union of the basins of critical component cycles, and for every $x\in J_2$ there is no Fatou disk containing $x$. As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on $\mathbb{P}^2$ with no sporadic super-saddle cycles. 2. We give a new proof of de Th\'elin's laminarity of the Green current in $J_1\setminus J_2$ for PCF maps on $\mathbb{P}^2$. 3. We show that for PCF maps on $\mathbb{P}^2$ an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on $\mathbb{P}^2$ which are expanding on $J_2$ or satisfy Axiom A.