Foliations and minimality for twist maps

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 $f$-invariant foliation ${\mathcal{F}}_G=\{\mathrm{Graph}({\phi }_t){\}}_{t\in {\mathbf{T}}^1}$, where each function ${\phi }_t:{\mathbf{T}}^1\to {\mathbf{T}}^1$ is $K$-Lipschitz for some constant $K=K(f)$. Moreover, we also prove that in case ${\mathcal{F}}_G$ is a Lipschitz foliation and the leaf dynamics, which is always a transitive circle homeomorphism of rotation number $\alpha$, is bi-Lipschitz conjugate to the irrational rotation $R_{\alpha }$, then $f$ is minimal.