A Non-Autonomous Model for Parabolic Implosion

TL;DR

This paper investigates non-autonomous perturbations of parabolic M"obius maps through orthogonal polynomials, revealing instances of non-autonomous parabolic implosion and demonstrating almost sure convergence in a random perturbative setting.

Contribution

It introduces a novel non-autonomous model for parabolic implosion using orthogonal polynomials associated with perturbed M"obius transformations.

Findings

Orthogonal polynomials linked to non-autonomous parabolic maps are characterized.

Convergence results are established for a random perturbative regime.

The work provides new insights into non-autonomous dynamical systems and their stability.

Abstract

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