Polynomials in molecules

TL;DR

This paper explores the relationship between polynomials and molecules in complex dynamics, providing criteria for polynomials to belong to molecules and conditions for their placement within hyperbolic components.

Contribution

It offers a new characterization of polynomials in molecules based on critical points and Fatou chains, and clarifies the structure of hyperbolic components in parameter space.

Findings

Polynomials of degree ≥ 2 are in a molecule iff all critical points are in maximal Fatou chains.

Distinct molecules are mutually disjoint.

Necessary and sufficient conditions for subhyperbolic polynomials to be on the closures of hyperbolic components.

Abstract

This paper characterizes polynomials within molecules. We show that a geometrically finite polynomial of degree $d\geq2$ lies in a molecule if and only if all its critical points belong to maximal Fatou chains, and show that distinct molecules are mutually disjoint. We also establish a necessary and sufficient condition for subhyperbolic polynomials to be on the closures of bounded hyperbolic components.