  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.
