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arXiv · 2607.03655

The Rotation Number for a Generic $C^1$ Family of Circle Diffeomorphisms is Not H\"older

Xuzheng Lang
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Abstract

We study the regularity of the rotation number as a function of the parameter in monotone one-parameter families of circle diffeomorphisms. We prove that for an open and dense set of $C^2$ monotone families, the optimal H\"older exponent is exactly $1/2$, showing therefore that the earlier result by J. Graczyk is sharp. For a $C^1-$ dense set of $C^{1+\alpha}$ families, we show that the rotation number cannot be more regular than $\alpha/(1+\alpha)-$H\"older. As a consequence, we get that the rotation number for a generic $C^1$ family of circle diffeomorphisms is not H\"older.

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Xuzheng Lang. 2026-07-04. The Rotation Number for a Generic $C^1$ Family of Circle Diffeomorphisms is Not H\"older. https://arxiv.org/abs/2607.03655

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