Rays to renormalizations

TL;DR

This paper explores the relationship between external rays of a polynomial and its renormalizations, establishing a finite-to-one correspondence that preserves limit sets and accessibility properties of Julia sets.

Contribution

It generalizes previous results by establishing a finite-to-one correspondence between external rays of a polynomial and its renormalizations for all polynomials, including connected Julia sets.

Findings

Established a finite-to-one function between external rays of P and f with shared limit sets.

Proved accessibility of Julia set points from external rays of P via renormalization.

Showed that components of K_P extbackslash K_f meet K_f at most at a single pre-periodic point.

Abstract

Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function \lambda from the set of P-external rays having limit points in K_f onto the set of f-external rays to K_f such that R and \lambda(R) share the same limit set. In particular, if a point of the Julia set J_f=\partial K_f of a renormalization is accessible from C\setminus K_f then it is accessible through an external ray of P (the inverse is obvious). Another interesting corollary is that: a component of K_P\setminus K_f can meet K_f only at a single (pre-)periodic point. We study also a correspondence induced by \lambda on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set K_P) of some results of Levin-Przytycki, Blokh-Childers-Levin-Oversteegen-Schleicher and Petersen-Zakeri where the case is considered when K_P is disconnected and K_f is a periodic component of K_P.