Hyperbolic components of cosine family with a fixed critical point
Complex Variables
2026-03-12 v1
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.
Cite
@article{arxiv.2603.10554,
title = {Hyperbolic components of cosine family with a fixed critical point},
author = {Weiyuan Qiu and Lingrui Wang},
journal= {arXiv preprint arXiv:2603.10554},
year = {2026}
}