Hyperbolic components of cosine family with a fixed critical point

TL;DR

This paper investigates the structure of hyperbolic components in the parameter space of cosine functions with a fixed critical point, revealing their classification, boundary properties, and geometric nature.

Contribution

It classifies hyperbolic components into three types, proves their boundedness and simple connectivity, and establishes that their boundaries are Jordan curves or quasidisks.

Findings

Hyperbolic components are bounded and simply connected.

Type-A component has an isolated boundary point at 0.

Boundaries of hyperbolic components are Jordan curves or quasidisks.

Abstract

We studied the parameter plane of the cosine functions with a fixed critical point. The hyperbolic components can be classified into three types: A, C and D. All the hyperbolic components are bounded and simply connected, except for the unique type-A component, which contains 0 as an isolated boundary point. Using the method of para-puzzle, we constructed a phase-parameter transfer mapping and proved that the boundaries of hyperbolic components are Jordan curves. By a similar idea, the hyperbolic components of type C are quasidisks.