Reeb orbits trapped by Denjoy minimal sets

Let $\phi$ be any flow on $T^n$ obtained as the suspension of a diffeomorphism of $T^{n-1}$ and let $\mathcal{A}$ be any compact invariant set of $\phi$. We realize $(\mathcal{A},\phi {\mathrm{\mid }}_{\mathcal{A}})$ up to reparametrization as an invariant set of the Reeb flow of a contact form on ${\mathbb{R}}^{2n+1}$ equal to the standard contact form outside a compact set and defining the standard contact structure on all of ${\mathbb{R}}^{2n+1}$. This generalizes the construction of Geiges, R"ottgen and Zehmisch.