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

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

2011-09-14

arxiv.org/abs/1109.3214v1

Mathematics

Dynamical Systems

11 pages

Scientific paper

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 $(1&1 0&1)$. 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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