Orbit Portraits of Unicritical Anti-polynomials

TL;DR

This paper extends the concept of orbit portraits to unicritical anti-polynomials, providing a detailed combinatorial description and realization theorem, which enhances understanding of their parameter spaces called multicorns.

Contribution

It offers an explicit description of orbit portraits for unicritical anti-polynomials and proves a realization theorem, advancing the combinatorial understanding of their parameter spaces.

Findings

Restricted set of orbit portraits compared to holomorphic case

Explicit characterization via characteristic angles

Established a realization theorem for these combinatorial objects

Abstract

Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical anti-polynomials and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial objects. The results obtained in this paper serve as a combinatorial foundation for a detailed understanding of the combinatorics and topology of the parameter spaces of unicritical anti-polynomials and their connectedness loci, known as the multicorns.