Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

Abstract

We prove a strengthenedCr-closing lemma (r≥1) 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 strengthenedCr-closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤk on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus.