Similarity at Misiurewicz Maps in the Cubic Parameter Curves

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.

19 pages, 3 figures