Flows with uncountable but meager group of self-similarities
Let \((T_t)_{t\in\mathbb R}\) be an ergodic free measure-preserving flow on a standard probability space. Denote by \(T\circ s\) a flow \((T_{st})_{t\in\mathbb R}\). Let NEWLINE\[NEWLINEI(T)=\{s\in\mathbb R: s\neq0, \,T\circ s\text{ is isomorphic to }T\}.NEWLINE\]NEWLINE These are called self-similarities of \(T\). If \(I(T)\neq\{1,-1\}\), \(T\) is called self-similar. In this article, the author proves the following theorem: There exist weakly mixing Gaussian flows with uncountable but meager group of self-similarities. Given a probability measure \(\tau\) on a locally compact second countable abelian group, denote by \(H_G(\tau)\) the set of all \(g\in G\) such that the translation of \(\tau\) by \(g\) is equivalent to \(\tau\). Let NEWLINE\[NEWLINEI_{\mathrm{Po}}(T)=\{s\in\mathbb R: s\neq0,\, T\circ s\text{ is isomorphic to \(T\) via a Poisson transformation}\}.NEWLINE\]NEWLINE He also proves the following theorem: For each probability measure \(\kappa\) on \(\mathbb R_+\), there is a weakly mixing Poisson flow \(\tilde T\) with \(I_{\mathrm{Po}}(\tilde T)=H_{\mathbb R_+}(\kappa)\).NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].
scientific article; zbMATH DE number 6077498