The intrinsic normalising constants of transformations preserving infinite measures
Let \(T=(X_ T,{\mathcal B}_ T,m_ T,T)\) and \(S=(X_ S,{\mathcal B}_ S,m_ S,S)\) be measure preserving transformations of infinite \(\sigma\)-finite measure spaces. For \(0<c<\infty\), a c-map of T onto S is a measurable map \(\pi\) : \(X_ T\to X_ S\) with \(\pi T=S\pi\) and \(cm_ S=m_ T\circ \pi^{-1}.\) If such a map exists, S is called a c-factor of T. \(\Delta\) (T) denotes the sets of all c for which there exists a conservative ergodic measure preserving transformation U such that T is both a 1- factor and a c-factor of U. T and S are called similar if for suitable \(c,c'>0\), T is a c-factor of S and S a c'-factor of T. \(\Delta\) (T) is a multiplicative subgroup and an analytic subset of \({\mathbb{R}}^+\). If S and T are similar, then \(\Delta (T)=\Delta (S).\) Hajian, Ito, and Kakutani constructed transformations \(T_{\alpha}\) for which \(\Delta (T_{\alpha})=\{\alpha^ n: n\in {\mathbb{Z}}\}.\) Examples are given, where the group \(\Delta\) (T) is any countable subgroup of \({\mathbb{R}}^+\), and also an uncountable subgroup of any Hausdorff dimension. These latter groups are achieved by transformations which are not similar to their inverses. Rationally ergodic T have \(\Delta (T)=\{1\}\). This is connected with the existence of ``laws of large numbers''. Section 3 studies the ergodic self-joinings of countable group extensions of nonsingular group rotations. For example, ergodic \({\mathbb{Z}}^ m\times {\mathbb{Q}}^ n\)-extensions of a finite measure preserving group rotation has \(\Delta (T)=1\).
conservative ergodic measure preserving transformation
ergodic self-joinings of countable group extensions
scientific article; zbMATH DE number 4049937
scientific article