On self-similarities of ergodic flows (Q2881017)

The authors consider the well-known problems of self-similarity in ergodic theory. A flow \(T=(T_t)_{t \in \mathbb{R}}\) is said to be totally self-similar if, for any \(s \in \mathbb{R}_{+}^*\), \((T_t)\) and \((T_{st})\) are isomorphic. Precisely, they study the properties of the set NEWLINE\[NEWLINEI(T)=\Big\{s \in\mathbb{R}^*~~:~~(T_t) {\text{~and~}} (T_{st}) {\text{~are~isomorphic}}\Big\}.NEWLINE\]NEWLINE Obviously, this set contains 1. The authors establish that \(I(T)\) has the following properties:NEWLINENEWLINE1) \(I(T)\) is Borel multiplicative subgroup of \(\mathbb{R}^*\).NEWLINENEWLINE2) \(I(T)\) is a Polish topological group with a stronger topology than the topology induced from \(\mathbb{R}\).NEWLINENEWLINE3) There exists a mixing flow \((T_t)\) such that \(I(T)\) is an uncountable meager subset of \(\mathbb{R}^*\).NEWLINENEWLINE4) A generic transformation from the Polish group of transformations on a on a Polish probability measure space can be embedded in a flow \((T_t)\) on this space with \(I(T) = \{1\}\).NEWLINENEWLINE5) For each countable multiplicative subgroup \(S \subset \mathbb{R}^*\), one can construct a Poisson suspension flow \((T_t)\) with simple spectrum such that \(I(T) = S\) and a measure of the maximal spectral type of \((T_t)\) is singular with respect to a measure of the maximal spectral type of \((T_{st})\) for each positive \(s \not \in S\).NEWLINENEWLINE6) If \(S\) contains no rational relations then there is a rank-one weakly mixing rigid flow \((T_t)\) with \(I(T) = S\).