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

Autor: Frédéric Protin
Rok vydání: 2018
Předmět:
Zdroj: Annali di Matematica Pura ed Applicata (1923 -). 197:1845-1854
ISSN: 1618-1891
0373-3114
DOI: 10.1007/s10231-018-0752-x
Popis: 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 Euclidean distance in $$\mathbb {C}$$ , we show that the Green function $$G_K$$ of the filled Julia set K of a polynomial such that $$\mathring{K}\ne \emptyset $$ satisfies the so-called ŁS condition $$\displaystyle G_A\ge c\cdot \hbox {dist}(\cdot , K)^{c'}$$ in a neighborhood of K, for some constants $$c,c'>0$$ . Relatively few examples of compact sets satisfying the ŁS condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. For instance, this is the case for the filled Julia sets of quadratic polynomials of the form $$z\mapsto z^2+a$$ , provided that the parameter a is parabolic, hyperbolic or Siegel. The fact that filled Julia sets satisfy the ŁS condition may seem surprising, since they are in general very irregular and sometimes they have cusps. However, we provide an explicit example of a curve which has a cusp and satisfies the ŁS condition. 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.
Databáze: OpenAIRE
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