{\L}S condition for filled Julia sets in $\mathbb{C}$

TL;DR

This paper establishes a {}Lojasiewicz-Siciak condition for filled Julia sets in complex dynamics, revealing that these irregular sets satisfy a specific inequality relating the Green function and Euclidean distance.

Contribution

It proves that filled Julia sets in complex dynamics satisfy the {}Lojasiewicz-Siciak condition, a property previously known for few classes of sets, and introduces the concept of obstruction points.

Findings

Filled Julia sets satisfy the {}Lojasiewicz-Siciak condition.

Obstruction points to the {}L condition are rare in polynomially convex, L-regular sets.

The result applies to a broad class of complex dynamical sets, expanding understanding of their geometric properties.

Abstract

In this article, we derive an inequality of {\L}ojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by $dist$ the euclidian distance in $\mathbb{C}$, we show that the Green function $G_K$ of the filled Julia set $K$ of a polynomial such that $\mathring{K}\neq \emptyset$ satisfies the so-called {\L}S condition $\displaystyle G_A\geq c\cdot dist(\cdot, K)^{c'}$ in a neighborhood of $K$, for some constants $c,c'>0$. Relatively few examples of compact sets satisfying the {\L}S condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. The fact that filled Julia sets satisfy the {\L}S condition may seem surprising, since they are in general very irregular. In order to prove our main result, we define and study the set of obstruction points to the {\L}S condition. We also prove, in dimension $n\geq 1$, that for a polynomially convex and L-regular compact set of non empty interior, these obstruction points are rare, in a sense which will be specified.