Foliations and minimality for twist maps

Abstract

In this paper we consider area-preserving twist maps $f:{\mathbf{T}}^2\to {\mathbf{T}}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a unique $f$-invariant foliation ${\mathcal{F}}_G=\{\mathrm{Graph}({\varphi }_t){\}}_{t\in {\mathbf{T}}^1}$, where each function ${\varphi }_t:{\mathbf{T}}^1\to {\mathbf{T}}^1$ is $K$-Lipschitz for some constant $K=K(f)$. Moreover, we also prove that in case ${\mathcal{F}}_G$ is a Lipschitz foliation, then $f$ is minimal.