Immediate renormalization of complex polynomials

TL;DR

This paper investigates the conditions under which a cubic polynomial with a non-repelling fixed point is immediately renormalizable, showing that under certain Julia set conditions, the critical point outside the quadratic-like set is recurrent.

Contribution

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

Findings

Critical point outside the quadratic-like set is recurrent.

Immediate renormalization implies specific Julia set structures.

Recurrent critical points are characterized under no periodic cutpoints.

Abstract

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be \emph{immediately renormalizable} if there exists a (connected) quadratic-like 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.