Versions of the closing lemma for certain dynamical systems on tori (Q1431455)

The authors prove the strengthened \(C^r\)-closing lemma \((r\geq 1)\) for wandering chain recurrent points of diffeomorphisms of the circle \(S^1\) and for wandering chain recurrent points of a flow without fixed points on \(S^2\). Moreover, the authors consider the so-called Herman actions of \(\mathbb{Z}^k\) on \(S^1\) and prove the strengthened \(C^\infty\)-closing lemma for them. They use this result for foliations without compact leaves on \(S^3\).