On the self-similarity problem for Gaussian-Kronecker flows

Abstract

It is shown that a countable symmetric multiplicative subgroup G = −H ∪ H with H ⊂ R * + is the group of self-similarities of a Gaussian-Kronecker flow if and only if H is additively Q-independent. In particular, a real number s = ±1 is a scale of self-similarity of a Gaussian-Kronecker flow if and only if s is transcendental. We also show that each countable symmetric subgroup of R * can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foiaş-Stratila property.

Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems

“…Finally, we point out that this presentation focused on -actions, eventhough the notions of Gaussian systems and Poisson suspensions extend naturally to more general group actions. For flows (Ê-actions), Gaussian systems and Poisson suspensions provide interesting examples of flows for which the self-similarity set I(T ) := {s ∈ Ê : (T st ) t∈Ê is isomorphic to (T t ) t∈Ê } can be fully described (see [9,3,10]). For other groups, the generalization of some topics presented in the present survey is not always obvious.…”

Spectral disjointness of rescalings of some surface flows

“…Their result was heavily influenced by the work of del Junco in [15] on similar properties, but for automorphisms. We would also like to refer the reader to the papers [9, 13] and [18] where the authors describe the set of self‐similarities for various classes of flows. Some of the results in this paper base on the constructions given in [3] where authors were focusing on relation of some special flows with their inverses, that is rescaling by .…”