Rays to renormalizations
Tom 68 / 2020
Streszczenie
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 $K_P$.
