# 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.

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