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Dynamics of Complex Quadratic Families under Holomorphic and Nonholomorphic Singular Perturbations

Haitao Shang

TL;DR

The paper surveys the dynamics of real quadratic maps, emphasizing ergodic and topological perspectives and detailing the two-dimensional Hénon case. It then investigates complex quadratic maps under holomorphic and nonholomorphic singular perturbations, analyzing m=1, m=2, and m≥3 regimes to reveal a spectrum of Julia-set topologies (unit-disk limits, Cantor sets, Sierpiński curves, McMullen domains) and contrasting parameter-plane structures. Through this, it connects invariant measures, hyperbolicity, and renormalization with intricate fractal dynamics, highlighting both established results (e.g., ACIMs, exponential mixing) and rich perturbative phenomena. The work underscores the distinct behaviors induced by holomorphic versus nonholomorphic perturbations and points to substantial avenues for further theoretical development and generalization.

Abstract

This work reviews the topological and statistical properties of real quadratic maps and investigates the complex quadratic maps under holomorphic and nonholomorphic singular perturbations.

Dynamics of Complex Quadratic Families under Holomorphic and Nonholomorphic Singular Perturbations

TL;DR

The paper surveys the dynamics of real quadratic maps, emphasizing ergodic and topological perspectives and detailing the two-dimensional Hénon case. It then investigates complex quadratic maps under holomorphic and nonholomorphic singular perturbations, analyzing m=1, m=2, and m≥3 regimes to reveal a spectrum of Julia-set topologies (unit-disk limits, Cantor sets, Sierpiński curves, McMullen domains) and contrasting parameter-plane structures. Through this, it connects invariant measures, hyperbolicity, and renormalization with intricate fractal dynamics, highlighting both established results (e.g., ACIMs, exponential mixing) and rich perturbative phenomena. The work underscores the distinct behaviors induced by holomorphic versus nonholomorphic perturbations and points to substantial avenues for further theoretical development and generalization.

Abstract

This work reviews the topological and statistical properties of real quadratic maps and investigates the complex quadratic maps under holomorphic and nonholomorphic singular perturbations.

Paper Structure

This paper contains 13 sections, 23 theorems, 40 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Topological and Statistical Properties of Dynamics of Real Quadratic Maps
  3. Preliminary
  4. Dynamics of a Real Quadratic Family $F_{a}: \mathbb{R} \rightarrow \mathbb{R}, F_{a} (x)= 1-ax^{2}$
  5. Dynamics of the Hénon Map
  6. The Dynamics of Complex Quadratic Maps under Singular Perturbations $F_{\lambda , m}(z) = z^{2}+ \dfrac{\lambda }{{z} ^{m}}$ and $G_{\beta ,m}(z) = z^{2} + \dfrac{\beta }{\bar{z} ^{m}}$, where $m \in \mathbb{N}$ and $\lambda ,\beta \in \mathbb{C}$.
  7. Preliminary
  8. Singular Perturbation of Real Quadratic Family when $m=1$
  9. Singular Perturbations of Complex Quadratic Family when $m=1$
  10. Singular Perturbations of Quadratic Family when $m=2$
  11. Singular Perturbations of Quadratic Family when $m \geq 3$ and $m \in \mathbb{N}$
  12. Simple Comparison of Holomorphic and Nonholomorphic Singular Perturbations
  13. Summary

Key Result

Theorem 2.1

Given a measurable set $A \in \mathcal{B}$ with $\mu (A) > 0$ in a probability space $(X, \mathcal{B}, \mu)$, we have ∎

Figures (15)

  • Figure 1: An orbit for a of Hénon Map for Different Parameters a and b
  • Figure 2: Self-Similarity of the Hénon Attractor ($a = 1.4$ and $b = 0.3$)
  • Figure 3: The Bifurcation Diagrams of Hénon Map when $b = 0.3$, $x_{0} = 0$ and $y_{0} = 0$.
  • Figure 4: The Orbit Diagram of $F_{c}: \mathbb{R}\rightarrow\mathbb{R}$, where $F_{c}(x) = x^{2} + \dfrac{c}{x}$, in which $c \in \mathbb{R}$.
  • Figure 5: Analysis of the Dynamics when $c>0$. In this figure, purple line represents the fixed points as a function of parameter c's; Red line represents the critical points as a function of parameter c's; Green line represents the straight line $c=4/27$, which helps to see the saddle-node point at $(c ,x) = (4/27, 2/3)$.
  • ...and 10 more figures

Theorems & Definitions (56)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2.1
  • Theorem 2.2
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • ...and 46 more