Similarity at Misiurewicz Maps in the Cubic Parameter Curves

TL;DR

This paper proves a conjecture about the similarity between the connectedness locus of certain cubic parameter curves at Misiurewicz parameters and their filled Julia sets, extending known results in complex dynamics.

Contribution

It provides a proof confirming the conjectured similarity in cubic parameter spaces, generalizing previous results from quadratic cases.

Findings

Confirmed the conjecture by Bonifant and Milnor.

Extended Tan Lei's similarity proof to cubic maps.

Established local similarity near free co-critical points.

Abstract

We present a proof of the conjecture by Bonifant and Milnor (see arXiv:2503.08868) regarding the similarity between the connectedness locus of the curve $\mathcal{S}_p$ at Misiurewicz parameters and their corresponding filled Julia sets in a neighborhood of the corresponding free co-critical point. The proof is in parallel with the generalization of Tan Lei's proof of similarity in the Mandelbrot set developed by Kawahira.