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

TL;DR

This paper proves that certain smooth families of circle diffeomorphisms with strictly monotone rotation number are topologically conjugate to a linear Dehn twist, extending previous results where the rotation number was exactly equal to the parameter.

Contribution

It generalizes the conjugacy result from equal rotation numbers to strictly monotone rotation numbers in smooth families of circle diffeomorphisms.

Findings

Topological conjugacy to linear Dehn twist under specific conditions

Differentiability results for families with monotone rotation number

Extension of classical conjugacy results to broader rotation number assumptions

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 $(1&1 0&1)$. 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 $t$.