On the n-matings of polynomials

TL;DR

This paper introduces n-matings, a generalization of polynomial matings, establishing new links between polynomial and rational maps, classifying hyperbolic 2-matings, and exploring diverse n-matings in specific families.

Contribution

It defines n-matings, classifies hyperbolic 2-matings, and analyzes their dynamics and ubiquity, extending the classical polynomial mating concept.

Findings

Classified hyperbolic 2-matings based on orientation-reversing equators.

Described dynamics of rational maps with orientation-reversing equators via half polynomial matings.

Demonstrated the widespread occurrence of n-matings in specific polynomial families.

Abstract

We introduce the notion of n-mating in this work, which includes the classical mating of polynomials as a special case. The new notion brings further links between the polynomial world and the rational world than the classical one, as well as a natural classification of rational maps according to their n-unmatability. We classify the hyperbolic 2-matings according to the (non-)existence of orientation-reversing equators for them. For rational maps admitting orientation-reversing equators, we describe their dynamics via matings of half polynomials. There are diverse types of n-matings from the bicritical family and the degree-2 capture family exhibited in our explorations, which demonstrates the ubiquity of them. Finally we consider the postcritical realization programme of rational maps (among the atomic and mating family respectively). The compositive trick is exploited to deal with problems in the programme.