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A Non-Autonomous Model for Parabolic Implosion

Katelynn Huneycutt, Samantha Sandberg-Clark, Liz Vivas

TL;DR

This work analyzes non-autonomous parabolic implosion for the model map $f(z)=\frac{z}{1-z}$ by studying compositions of Möbius transformations and their connection to orthogonal polynomials. It develops a recurrence-based orthogonal-polynomial framework with $q_{k+1}= (1+\rho_k-\epsilon_k^2) q_k-\rho_k q_{k-1}$ and compares to the autonomous rotation sequence $T_k=\frac{1-\rho^k}{1-\rho}$ to derive sharp convergence results under various perturbation regimes, including purely multiplicative, combined multiplicative-additive, and random additive perturbations. The authors establish convergence to the identity (and to Lavaurs-type behavior in the autonomous vs non-autonomous setting) under explicit conditions, and show almost-sure convergence in the random-additive case using martingale techniques. They also explore differences with the additive framework and illustrate the relevance to skew-product dynamics, suggesting robustness of non-autonomous parabolic implosion under stochastic perturbations and potential extensions to higher dimensions. Key contributions include explicit convergence criteria (via $a_k,b_k$ and $c_k$ sequences), a detailed Chebyshev/orthogonal-polynomial analysis, and applications to non-autonomous dynamics and random perturbations.

Abstract

Orthogonal polynomials appear naturally in the study of compositions of Möbius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic Möbius map. Our results can be viewed as instances of non-autonomous parabolic implosion, including a random perturbative regime in which convergence holds almost surely.

A Non-Autonomous Model for Parabolic Implosion

TL;DR

This work analyzes non-autonomous parabolic implosion for the model map by studying compositions of Möbius transformations and their connection to orthogonal polynomials. It develops a recurrence-based orthogonal-polynomial framework with and compares to the autonomous rotation sequence to derive sharp convergence results under various perturbation regimes, including purely multiplicative, combined multiplicative-additive, and random additive perturbations. The authors establish convergence to the identity (and to Lavaurs-type behavior in the autonomous vs non-autonomous setting) under explicit conditions, and show almost-sure convergence in the random-additive case using martingale techniques. They also explore differences with the additive framework and illustrate the relevance to skew-product dynamics, suggesting robustness of non-autonomous parabolic implosion under stochastic perturbations and potential extensions to higher dimensions. Key contributions include explicit convergence criteria (via and sequences), a detailed Chebyshev/orthogonal-polynomial analysis, and applications to non-autonomous dynamics and random perturbations.

Abstract

Orthogonal polynomials appear naturally in the study of compositions of Möbius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic Möbius map. Our results can be viewed as instances of non-autonomous parabolic implosion, including a random perturbative regime in which convergence holds almost surely.
Paper Structure (14 sections, 26 theorems, 150 equations, 1 figure)

This paper contains 14 sections, 26 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-autonomous perturbations
  3. Our results
  4. Compositions of Linear Transformations
  5. Orthogonal Polynomials
  6. Quadratic errors
  7. Cubic errors
  8. Explicit condition for convergence
  9. Combined rotation and translation
  10. Different initial conditions
  11. Non-Autonomous Implosions
  12. Difference with the Additive Condition
  13. Bifurcations for Skew-Products
  14. Random Compositions

Key Result

Theorem 1

Let $f$ be defined in a neighborhood $V$ of the origin and be of the form $f(z)=z+z^{2}+O(z^{3})$. Consider the perturbation of f as follows: given $\epsilon>0$ let $f_\epsilon(z) \coloneqq f(z)+\epsilon^2$. If we take a sequence of numbers $N_\epsilon \to \infty$ and $\epsilon \to 0$, such that $N_ uniformly on compacts on the basin of attraction of $f$. Here $\mathcal{L}_f$ is the Lavaurs map of

Figures (1)

  • Figure 1: Graph of $T_k$ and $|T_k|$ for $N=10,000$.

Theorems & Definitions (59)

  • Theorem 1: Lavaurs
  • Theorem A: Multiplicative perturbations
  • Theorem B: Combined multiplicative and additive perturbations
  • Remark
  • Theorem C: Difference between additive and multiplicative
  • Theorem D: Random perturbations
  • Remark
  • Lemma 1
  • proof
  • Remark
  • ...and 49 more