Moscow Mathematical Journal

Volume 23, Issue 4, October–December 2023 pp. 441–461.

Immediate Renormalization of Cubic Complex Polynomials with Empty Rational Lamination

Authors: Alexander Blokh (1), Lex Oversteegen (2), and Vladlen Timorin (3)
Author institution:(1) Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170
(2) Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170
(3) Faculty of Mathematics, HSE University, 6 Usacheva St., 119048 Moscow, Russia;
Independent University of Moscow, Bolshoy Vlasyevskiy Per. 11, 119002 Moscow, Russia

Summary:

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL invariant filled Julia set $K^*$ such that $b\in K^*$. In that case, exactly one critical point of $P$ does not belong to $K^*$. We show that if, in addition, the Julia set of $P$ has no (pre)periodic cutpoints, then this critical point is recurrent.

2020 Math. Subj. Class. Primary: 37F20; Secondary: 37C25, 37F10, 37F50

Keywords: Complex dynamics, Julia set, Mandelbrot set.

Contents Full-Text PDF