Gluing polynomials along the circle

TL;DR

This paper demonstrates that two non-renormalizable polynomials with a super-attracting fixed point can be glued into a rational map along their basin boundary, expanding understanding of polynomial dynamics and complex structure manipulation.

Contribution

It introduces a novel gluing construction for polynomials with super-attracting fixed points, enabling the creation of rational maps from separate polynomial dynamics.

Findings

Gluing polynomials along the basin boundary produces rational maps.

Applicable to non-renormalizable polynomials with the same degree.

Extends the toolkit for complex dynamical systems construction.

Abstract

Gluing is a cut and paste construction where the dynamics of a map in a given domain is replaced by a different one, under the condition that the two agree along the gluing curve. Here we consider two polynomials with a finite super-attracting fixed point of the same degree. We prove that any two such non-renormalizable polynomials can be glued into a rational map along the Jordan boundary of the immediate basin of the super-attracting fixed point.