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Rays to renormalizations

Levin Genadi

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 2

133--149

Let KP be the filled Julia set of a polynomial P, and Kf the filled Julia set of a renormalization f of P. We show, loosely speaking, that there is a finite-to-one function λ from the set of P-external rays having limit points in Kf onto the set of f-external rays to Kf such that R and λ(R) share the same limit set. In particular, if a point of the Julia set Jf = δKf of a renormalization is accessible from C \ Kf then it is accessible through an external ray of P (the converse is obvious). Another interesting corollary is that a component of KP \ Kf can meet Kf only in a single (pre-)periodic point. We also study a correspondence induced by λ on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set Kp ) of some results of Levin and Przytycki (1996), Blokh et al. (2016) and Petersen and Zakeri (2019) where it is assumed that Kp is disconnected and Kf is a periodic component of Kp .