Hyperbolic components of cosine family with a fixed critical point

Weiyuan Qiu; Lingrui Wang

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arXiv · 2603.10554

Hyperbolic components of cosine family with a fixed critical point

Weiyuan Qiu·Lingrui Wang

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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.

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Weiyuan Qiu, Lingrui Wang. 2026-03-11. Hyperbolic components of cosine family with a fixed critical point. https://arxiv.org/abs/2603.10554

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