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Misiurewicz points and subhyperbolicity in unicritical algebraic correspondences

Carlos Siqueira

Abstract

We provide the first definition of \emph{Misiurewicz parameter} for the unicritical family of algebraic correspondences $ z^r + c$, with $ r > 1$ rational, and prove that, at every Misiurewicz parameter, the correspondence uniformly expands the canonical orbifold metric on a neighborhood of the Julia set. This is achieved using Thurston's ideas on postcritically finite rational maps, regular branched coverings, and orbifolds, viewing the correspondence as a global analytic multifunction. This result provides the necessary tools for further investigations into the fine structure of the parameter space near Misiurewicz points, particularly in exploring similarities between the local geometry of the parameter space and the Julia sets at such parameters. Finally, we present both rigorous examples and empirical evidence suggesting that Misiurewicz parameters are abundant and may be detected by identifying increasingly small copies of the Multibrot set nested within itself: the smaller the copy, the closer it is likely to be to a Misiurewicz parameter.

Misiurewicz points and subhyperbolicity in unicritical algebraic correspondences

Abstract

We provide the first definition of \emph{Misiurewicz parameter} for the unicritical family of algebraic correspondences , with rational, and prove that, at every Misiurewicz parameter, the correspondence uniformly expands the canonical orbifold metric on a neighborhood of the Julia set. This is achieved using Thurston's ideas on postcritically finite rational maps, regular branched coverings, and orbifolds, viewing the correspondence as a global analytic multifunction. This result provides the necessary tools for further investigations into the fine structure of the parameter space near Misiurewicz points, particularly in exploring similarities between the local geometry of the parameter space and the Julia sets at such parameters. Finally, we present both rigorous examples and empirical evidence suggesting that Misiurewicz parameters are abundant and may be detected by identifying increasingly small copies of the Multibrot set nested within itself: the smaller the copy, the closer it is likely to be to a Misiurewicz parameter.

Paper Structure

This paper contains 22 sections, 27 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results.
  3. Key developments on algebraic correspondences
  4. Application to asymptotic similarity.
  5. On critically finite correspondences.
  6. Definitions and main results
  7. Branched coverings and orbifolds
  8. The sheaf $\mathfrak{S}_{U}$
  9. Repelling cycles.
  10. Outline.
  11. The sheaf of germs of analytic functions
  12. Germs.
  13. Global analytic functions.
  14. Analytic continuation.
  15. Holomorphic multifunctions.
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $(S, \nu)$ be a Riemann surface orbifold that is conformally isomorphic to the complex plane. Suppose that $\nu$ is non-trivial, that is, we have at least two ramified points with different ramification indices. Then there exists a regular branched covering $\varphi: \tilde{S}_{\nu} \to (\mathbb

Figures (1)

  • Figure 1: The Multibrot set $M_{4,2}$ and related copies, illustrating the similarity between $M_{4,2}$ and the filled Julia set $K_{-2}$ near the parameter $-2$. There exists an infinite sequence of Misiurewicz points $c_n$, each located near a small copy of $M_{4,2}$, and converging to $-2$. See Example \ref{['asdgalkhcjsdfs']}. The point $-2$ is not exceptional; similar patterns occur near infinitely many copies of $M_{4,2}$, as well as in other Multibrot sets $M_{p,q}$, as discussed in siqueira2025quadratic.

Theorems & Definitions (67)

  • Definition 2.1: Misiurewicz point
  • Definition 2.2
  • Definition 2.3: Multibrot set
  • Example 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Thurston, Douady and Hubbard
  • proof : Sketch of Proof and References
  • Definition 2.4: Orbifold metric
  • Definition 2.5: Ramified points of the correspondence
  • ...and 57 more