Ergodic measure preserving transformations of finite type
Let T be an invertible ergodic measure preserving (e.m.p.) transformation of an infinite \(\sigma\)-finite Lebesgue measure space (X,\({\mathcal B},m)\). A set W is called weakly wandering (w.w.) if there exists a strictly increasing sequence \((n_ i)\) of integers such that \(T^{n_ i}W\cap T^{n_ j}W=\emptyset\) for \(i\neq j\); and \((n_ i)\) is called w.w. if there exists such a W. W (and \((n_ i))\) are called exhaustive (exh.) w.w. if \(\cup T^{n_ i}W=X.\) It was shown by Jones and the reviewer that exh. w.w. sets exist for all T with the above properties. Eigen and Hajian showed that there exist transformations T and exh. w.w. sequences \((n_ i)\) for which W must be of finite measure. T is then called of finite type. Theorem 1 asserts that exh. w.w. sets W, \(W^ 1\) for the same \((n_ i)\) and T must have the same measure, finite or infinite. It is also shown that there exist transformations T such that \(\mu (W)=\infty\) for all exh. w.w. \((n_ i)\) (called of infinite type). The authors also study a particular class of e.m.p. transformations T constructed from direct sum decompositions \({\mathbb{N}}=A\oplus B\) by Eigen and Hajian.
invertible ergodic measure preserving transformation
scientific article; zbMATH DE number 4105281