An Experimental View of Herman Rings for Dianalytic Maps of $\mathbb{RP}^2$

TL;DR

This paper experimentally investigates the existence and properties of Herman rings in a family of degree 3 dianalytic maps on $ ext{RP}^2$, analyzing their dynamical behavior and parameter space.

Contribution

It provides the first experimental analysis of Herman rings in dianalytic maps on $ ext{RP}^2$, focusing on their existence, properties, and parameter space.

Findings

Herman rings exist in certain parameter regions of the studied maps.

The paper characterizes the properties of these Herman rings.

It maps the parameter space where Herman rings are present.

Abstract

We provide an experimental study of the existence of Herman rings in a parametrized family of rational maps preserving antipodal points, and a discussion of their properties on $\mathbb{RP}^2$. We study analytic maps of the sphere that project to dianalytic maps on the nonorientable surface, $\mathbb{RP}^2$. They have a known form and we focus on a subset of degree $3$ dianalytic maps and explore their dynamical properties. In particular, we focus on maps for which the Fatou set has a Herman ring. We appeal to dynamical properties of particular maps to justify our assertion that these Fatou components are Herman rings and analyze the parameter space for this family of maps.