Combinatorial structure of the parameter plane of the family $\lambda \tan z^2$

TL;DR

This paper analyzes the combinatorial structure of the parameter plane for the family of functions λ tan z^2, identifying hyperbolic components, their connectivity, and the bifurcation locus.

Contribution

It characterizes the hyperbolic components of the parameter space, proving they are mostly bounded, simply connected, and describes the structure of the bifurcation locus.

Findings

All hyperbolic components are bounded except four period-one components.

Hyperbolic components are all simply connected.

The bifurcation locus is the complement of hyperbolic components in the parameter space.

Abstract

In this article we will discuss combinatorial structure of the parameter plane of the family $ \mathcal F = \{ \lambda \tan z^2: \lambda \in \mathbb C^*, \ z \in \mathbb C\}.$ The parameter space contains components where the dynamics are conjugate on their Julia sets. The complement of these components is the bifurcation locus. These are the hyperbolic components where the post-singular set is disjoint from the Julia set. We prove that all hyperbolic components are bounded except the four components of period one and they are all simply connected.