Tongues in Degree 4 Blaschke Products

TL;DR

This paper explores the structure and bifurcations of tongue-like sets in the parameter space of degree 4 Blaschke products, revealing their topological properties and how they extend beyond initial domains.

Contribution

It provides a detailed topological analysis of tongue-like sets in Blaschke products and examines bifurcations near their tips, extending understanding of their parameter space.

Findings

Topological properties of tongue-like sets established

Bifurcation behaviors near tongue tips analyzed

Period one tongue shown to extend beyond natural domain

Abstract

The goal of this paper is to investigate the family of Blasche products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$, which is a rational family of perturbations of the doubling map. We focus on the tongue-like sets which appear in its parameter plane. We first study their basic topological properties and afterwords we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the period one tongue extends beyond its natural domain of definition.