Rays to renormalizations

Genadi Levin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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=∂Kf 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.

Original languageEnglish
Pages (from-to)133-149
Number of pages17
JournalBulletin of the Polish Academy of Sciences, Mathematics
Volume68
Issue number2
DOIs
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2020.

Keywords

  • external rays
  • Julia set
  • renormalization

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