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

Abstract

In this article, we derive an inequality of Łojasiewicz-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 $\mathring{K}\ne \varnothing$ satisfies the so-called ŁS condition $G_A\ge c\cdot dist(\cdot ,K{)}^{c^{\prime }}$ in a neighborhood of $K$, for some constants $c,c^{\prime }>0$. Relatively few examples of compact sets satisfying the ŁS condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. The fact that filled Julia sets satisfy the ŁS 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 ŁS condition. We also prove, in dimension $n\ge 1$, 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.

Cited by in corpus (1)

• Pluripotential theory on the support of closed positive currents and applications to dynamics in ${\mathbb{C}}^n$