On \(\mu\)-recurrent nonsingular endomorphisms
Let T be a non-singular invertible transformation acting on a \(\sigma\)- finite measure space (X,\(\mu)\). \textit{D. Maharam} [Fundam. Math. 56, 35-50 (1964; Zbl 0133.003)] introduced a skew product \(T^*: X\times {\mathbb{R}}^+\to X\times {\mathbb{R}}^+\) defined by \(T^*(x,y)=(Tx,y/w(x))\) where \(w(x)=\frac{d\mu T}{d\mu}(x)\). She showed that T is conservative if and only if \(T^*\) is conservative. The main result of this paper generalizes the above to non-invertible transformations. The author uses the idea of \(\mu\)-recurrence of T: T is \(\mu\) recurrent if for every measurable \(g\geq 0\), \(\sum_{i\geq 0}g(T^ ix)w_ i(x)\) only takes the values 0 or \(\infty\) a.e. (where \(w_ i(x)=w(x)...w(T^{i-1})).\) A non- singular endomorphism of a \(\sigma\)-finite space is then shown to be \(\mu\)-recurrent if and only if \(T^*\) is conservative. An extension of a theorem of \textit{W. Krieger} [Math. Ann. 223, 19-70 (1976; Zbl 0332.46045)] on the recurrence of the Radon-Nikodým derivatives of conservative non-singular automorphisms is given, followed by an application of the above to the construction of conservative ergodic invertible natural extensions for \(\mu\)-recurrent ergodic nonsingular endomorphisms.
\(\mu \)- recurrent ergodic nonsingular endomorphisms
scientific article; zbMATH DE number 4051946