Representation of transformations preserving the Lebesgue measure in the form of a product of periodic transformations (Q1082468)

Let f denote an invertible measure-preserving transformation of the interval \([0,1]\) with Lebesgue measure and let \(f=g\) iff \(f(x)=g(x)\) for a.e. \(x\in [0,1]\), \(fg(x)=^{def}f(g(x)),x\in [0,1]\). f is called an involution if \(f^ 2=^{def}ff=id\) where id denots the identity mapping. The main result of the paper under review is the following Theorem 1. For arbitrary f the following statements are valid. 1) There exist three involutions a,b,c such that \(f=abc\); 2) For each \(m\geq 2\) and \(n\geq 3\) there exist two invertible measure-preserving transformations a and b such that \(f=ab\), \(a^ m=b^ n=id\). This improves some results of \textit{A. Fathi} [Isr. J. Math. 29, 302-308 (1978; Zbl 0375.28008.)].