Immediate renormalization of cubic complex polynomials with empty rational lamination

TL;DR

This paper investigates the immediate renormalization of cubic polynomials with non-repelling fixed points, showing that under certain conditions, the critical point not in the Julia set is recurrent.

Contribution

It establishes a new connection between immediate renormalization and the recurrence of critical points in cubic polynomials with specific Julia set properties.

Findings

Critical point outside the renormalized set is recurrent.

Immediate renormalization is characterized by the structure of Julia sets.

No (pre)periodic cutpoints imply critical recurrence.

Abstract

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.