Julia sets with a wandering branching point

TL;DR

This paper demonstrates the existence of cubic Julia sets with wandering branching points, challenging the classical Thurston theorem applicable to quadratic Julia sets, by constructing specific polynomial limits.

Contribution

It reestablishes that higher degree Julia sets can have wandering branching points, providing explicit polynomial constructions as limits of particular critical point behaviors.

Findings

Existence of cubic Julia sets with wandering branching points

Construction of such Julia sets as limits of specific polynomial sequences

Counterexample to the quadratic case Thurston theorem in higher degrees

Abstract

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to a repelling fixed point.