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One-parameter families of circle diffeomorphisms with strictly monotone rotation number

Author: Kiran Parkhe
Journal: Proc. Amer. Math. Soc. 141 (2013), 4327-4337
MSC (2010): Primary 37C15, 37E10, 37E45; Secondary 37C05
Published electronically: August 22, 2013
MathSciNet review: 3105874
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Abstract: We show that if $ f \colon S^1 \times S^1 \to S^1 \times S^1$ is $ C^2$, with $ f(x, t) = (f_t(x), t)$, and the rotation number of $ f_t$ is equal to $ t$ for all $ t \in S^1$, then $ f$ is topologically conjugate to the linear Dehn twist of the torus $ \left ( \begin {smallmatrix}1&1\\ 0&1 \end{smallmatrix} \right )$. We prove a differentiability result where the assumption that the rotation number of $ f_t$ is $ t$ is weakened to say that the rotation number is strictly monotone in $ t$.

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Additional Information

Kiran Parkhe
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730

Keywords: Rotation number, strict monotonicity, topological conjugacy
Received by editor(s): December 12, 2011
Received by editor(s) in revised form: February 7, 2012, and February 14, 2012
Published electronically: August 22, 2013
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.