Poincar\'{e} functions with spiders' webs

TL;DR

This paper investigates the structure of Poincaré functions (linearizers) associated with polynomials at repelling fixed points, showing that under certain Julia set conditions, their escaping sets form connected spider's web structures.

Contribution

It classifies linearizers of polynomials based on the spider's web structure of their fast escaping sets, linking Julia set properties to the topology of escaping sets.

Findings

Fast escaping sets of linearizers form spider's webs when Julia set components are singleton.

Connectedness of escaping sets is established under specific Julia set conditions.

Provides a classification of polynomial linearizers based on escaping set topology.

Abstract

For a polynomial p with a repelling fixed point w, we consider Poincar\'{e} functions of p at w, i.e. entire functions L which satisfy L(0)=w and p(L(z))=L(p'(w)*z) for all z in the complex plane. We show that if the component of the Julia set of p that contains w equals {w}, then the (fast) escaping set of L is a spider's web; in particular it is connected. More precisely, we classify all linearizers of polynomials with regards to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point.