Existence of periodic points of endomorphisms of a circle (Q1283075)

The paper deals with generalizations of the Block and Franke theorem on the existence of periodic points for a map of a circle to itself. More precisely, let \(S^1\) be a circle. By \(\text{End}(S^1)\) denote the set of continuous maps \(f:\mathbb{R} \to\mathbb{R}\) satisfying \(f(x+1)= f(x)+1\) with the usual \(C^0\) topology. Let \(f\in\text{End}(S^1)\) be a \(C^1\) map. Denote by \(\Sigma(f)\) the critical set of \(f\). \(x\in\Sigma(f)\) is called a fold if \(x\) is isolated in \(\Sigma(f)\) and \(D(f)\) changes sign at \(x\). Block and Franke proved the following theorem. Let \(f\in\text{End}(S^1)\) be a \(C^1\) map with first derivative of bounded variation and the critical set have a nonzero number of folds. Then \(f\) has a periodic point.