Strengthening the $C^r$-closing lemma for dynamical systems and foliations on the torus

We prove a strengthened $C^r$ -closing lemma ($r\ge1$) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthened $C^r$ -closing lemma ($r\ge1$) is proved for a special class of infinitely smooth actions of the integer lattice $\mathbb Z^k$ on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus.