Properties of ergodic flows associated to product odometers
The paper deals with a property of group actions called approximate transitivity (AT). Let G be a Borel group acting on a standard space (X,\({\mathcal B},\mu)\). Roughly speaking, the action is said to be AT if for any finite number of finite measures \(\nu_ 1,...,\nu_ k\) equivalent to \(\mu\) there exists another finite measure \(\nu\) equivalent to \(\mu\) such that the linear hull of \(G\nu\) contains the set \(\{\nu_ 1,...,\nu_ k\}.\) This notion had been introduced and investigated by \textit{A. Connes} and \textit{E. J. Woods} [Ergodic Theory Dyn. Syst. 5, 203- 236 (1985; Zbl 0606.46041)]. The present paper has two main results. Firstly, it is proved that if a second countable locally compact group acts transitively on a locally compact Hausdorff space \((X,{\mathcal B},\mu)\) with \(\mu (U)>0\) for all open \(U\subset X\) then the action is AT. Secondly, a short ergodic theoretic proof for the Poincaré flow of a product odometer to be AT is provided, thus reproving part of a result of Connes and Woods.
scientific article; zbMATH DE number 4214301