Antipode Preserving Cubic Maps: the Fjord Theorem

TL;DR

This paper investigates a family of antipode-preserving cubic rational maps, focusing on the structure of fjords in the parameter space that relate to hyperbolic components and rotation numbers.

Contribution

It introduces the concept of fjords in the parameter plane of antipode-preserving cubic maps and analyzes their role in the map's hyperbolic components.

Findings

Fjord structures decompose the parameter space into regions with distinct rotation numbers.

The study characterizes the hyperbolic component boundaries using fjord geometry.

The work enhances understanding of the parameter space topology for these maps.

Abstract

This note will study a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points. We focus particularly on the fjords, which are part of the central hyperbolic component but stretch out to infinity. These serve to decompose the parameter plane into subsets, each of which is characterized by a corresponding rotation number.