Remarks on J. Langer and D. A. Singer decomposition theorem for diffeomorphisms of the circle (Q1087194)

The author gives a simple proof that all \(C^ 4\) diffeomorphisms of the torus can be factorized into a finite number of diffeomorphisms commuting with reflection. In one dimension, \(C^ 3\) suffices and even \(C^ 2\) can yield that the factors are almost diffeomorphisms. The derivatives of the function and the inverse are in \(L^ 1\) and are positive. The main result is: there exists a number \(\rho\) such that all \(C^ r\) (r\(\geq 2)\) orientation-preserving diffeomorphisms of the circle can be written as the composition of a finite number of factors that commute either with \(R_ 0\) or \(R_{\rho}\). When \(r>2\), the factors can be chosen among \(C^{r-2}\) diffeomorphisms, and when \(r=2\) they can be chosen so that they and their inverses have positive weak derivatives in \(L^ p\), \(1\leq p<\infty\).