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

Frédéric Protin
2018 Annali di Matematica Pura ed Applicata  
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 C, we show that the Green function G_K of the filled Julia set K of a polynomial such that K≠∅ satisfies the so-called ŁS 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 ŁS condition are known. Our
more » ... esult 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≥ 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.
doi:10.1007/s10231-018-0752-x fatcat:dxuivw4hprc3ngzvfndzjhuvsi