LS condition for filled Julia sets in C

In this article we derive an inequality of Lojasiewicz-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 C, we show that the Green function G(K) of the filled Julia set K of a polynomial such that K(sic) not equal empty set satisfies the so-called LS condition G (A) >= c.dist(., K)(c)(') in a neighborhood of K, for some constants c, c' > 0. Relatively few examples of compact sets satisfying the LS 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 bar right arrow z(2) + a, provided that the parameter a is parabolic, hyperbolic or Siegel. The fact that filled Julia sets satisfy the LS 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 LS condition. In order to prove our main result, we define and study the set of obstruction points to the LS condition. We also prove, in dimension n >= 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.