Rays to renormalizations

Genadi Levin¹

¹ Institute of Mathematics The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel <a href="https://orcid.org/0000-0003-0763-4711">ORCID: 0000-0003-0763-4711</a>

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 68 (2020) no. 2, pp. 133-149.

Résumé

Let $K_P$ be the filled Julia set of a polynomial $P$, and $K_f$ the filled Julia set of a renormalization $f$ of $P$. 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 $\mathbb C\setminus K_f$ then it is accessible through an external ray of $P$ (the converse is obvious). Another interesting corollary is that a component of $K_P\setminus K_f$ can meet $K_f$ only in a single (pre-)periodic point. We also study 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 and Przytycki (1996), Blokh et al. (2016) and Petersen and Zakeri (2019) where it is assumed that $K_P$ is disconnected and $K_f$ is a periodic component of $K_P$.

Détail

DOI : 10.4064/ba210129-19-2

Keywords: filled julia set polynomial filled julia set renormalization nbsp loosely speaking there finite to one function lambda set p external rays having limit points set f external rays lambda share limit set particular point julia set partial renormalization accessible mathbb setminus accessible through external ray converse obvious another interesting corollary component setminus meet only single pre periodic point study correspondence induced lambda arguments rays these results generalizations polynomials covering notably connected julia set nbsp results levin przytycki blokh petersen zakeri where assumed nbsp disconnected periodic component nbsp