The closing lemma for piecewise diffeomorphic maps of the circle (Q5951109)

This paper deals with the closing lemma for piecewise diffeomorphic maps of the circle. Indeed, let \(f\) be a piecewise \(C^r\)-diffeomorphic map of the circle \(S^1\), \(r\geq 1\). It is shown that if \(x\in S^1\) is a nontrivially recurrent point, then under a certain condition on the character of recurrence at this point, one can transform it into a periodic point by a \(C^r\)-small perturbations. As a consequence, the authors obtain the closure lemma for certain flows with infinitely many fixed points on orientable closed surfaces of genus \(g\geq 2\).