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Similarity at Misiurewicz Maps in the Cubic Parameter Curves

Araceli Bonifant, Brady Young

TL;DR

The paper proves a cubic-parameter analogue of Tan Lei's similarity at Misiurewicz points by establishing asymptotic self-similarity of the connectedness locus $\mathscr{C}(\mathcal{S}_p)$ and the filled Julia set $K_F$ under local perturbations near a Misiurewicz map on $\mathcal{S}_p$. Using the local parametrization $\mathbf{t}$ of $\mathcal{S}_p$, the authors construct a non-constant entire Poincaré function $\phi$ and scale sequences $\rho_k \to 0$ so that both the dynamical set $K_F$ and the parameter locus converge to the same limit model $\mathscr{K}=\phi^{-1}(K_F)$ in the Hausdorff topology. A key component is showing the convergence of perturbations in both spaces via a common limit, supported by a transversality analysis and a Poincaré-function construction, echoing Tan Lei and Kawahira. The results extend the well-known quadratic case to the cubic family $\mathcal{S}_p$, clarifying the geometry of cubic parameter spaces and providing tools (external rays, Böttcher coordinates, Hurwitz-type arguments) to control perturbations and landing behavior.

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.

Similarity at Misiurewicz Maps in the Cubic Parameter Curves

TL;DR

The paper proves a cubic-parameter analogue of Tan Lei's similarity at Misiurewicz points by establishing asymptotic self-similarity of the connectedness locus and the filled Julia set under local perturbations near a Misiurewicz map on . Using the local parametrization of , the authors construct a non-constant entire Poincaré function and scale sequences so that both the dynamical set and the parameter locus converge to the same limit model in the Hausdorff topology. A key component is showing the convergence of perturbations in both spaces via a common limit, supported by a transversality analysis and a Poincaré-function construction, echoing Tan Lei and Kawahira. The results extend the well-known quadratic case to the cubic family , clarifying the geometry of cubic parameter spaces and providing tools (external rays, Böttcher coordinates, Hurwitz-type arguments) to control perturbations and landing behavior.

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 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.

Paper Structure

This paper contains 5 sections, 6 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Similarity and The Quadratic Case
  3. The Curve $\mathcal{S}_p$
  4. Similarity between $\mathscr{C}(\mathcal{S}_p)$ and $K_F$ for a Misiurewicz Map
  5. Transversality of Repelling Periodic Points

Key Result

Theorem 1.1

Let $F \in \mathcal{S}_p$ be a Misiurewicz map. Assume that the curve $\mathcal{S}_p$ has been parametrized locally near $F$ by a parameter $\textbf{t}$. Then there exists a non-constant entire function $\phi$ on $\mathbb{C}$, a sequence $\rho_k \to 0$, and a constant $q \neq 0$ such that if we set as $k \to \infty$ in the Hausdorff topology.

Figures (3)

  • Figure 1: Left: The parameter curve $\mathcal{S}_2$ near a Misiurewicz map. Right: The filled Julia set of the Misiurewicz map near the co-critical point $2a$
  • Figure 2: Left: The Julia set of the Misiurewicz map $z^2 - i$. Right: The filled Julia set of a Misiurewicz map in $\mathcal{S}_2$, with the critical and co-critical points labeled.
  • Figure 3: The canonical projections of $\mathcal{S}_1$ and $\mathcal{S}_2$ into $\mathbb{C}$ and $\mathbb{C}^*$ respectively.

Theorems & Definitions (14)

  • Conjecture 1
  • Theorem 1.1: Main Theorem
  • Definition 2.1
  • Theorem 2.2: Tan Lei, 1990
  • Definition 3.1
  • Lemma 4.1
  • Theorem 4.2
  • proof : Proof of Lemma \ref{['l-1']}
  • proof : Proof of Theorem \ref{['t-main']}
  • Theorem 5.1
  • ...and 4 more