On self-similarities of ergodic flows

TL;DR

This paper investigates the structure of self-similarities in ergodic flows, revealing their topological properties, existence of flows with specific self-similarity sets, and constructing examples with prescribed symmetry groups.

Contribution

It establishes the Borel nature of the set of self-similarities, constructs flows with particular self-similarity sets, and explores the spectral and rigidity properties related to these symmetries.

Findings

$I(T)$ is a Borel subset of $R^*$

Existence of mixing flows with uncountable meager $I(T)$

Construction of flows with prescribed $I(T)$ for various groups

Abstract

Given an ergodic flow $T=(T_t)_{t\in\Bbb R}$, let $I(T)$ be the set of reals $s\ne 0$ for which the flows $(T_{st})_{t\in\Bbb R}$ and $T$ are isomorphic. It is proved that $I(T)$ is a Borel subset of $\Bbb R^*$. It carries a natural Polish group topology which is stronger than the topology induced from $\Bbb R$. There exists a mixing flow $T$ such that $I(T)$ is an uncountable meager subset of $\Bbb R^*$. For a generic flow $T$, the transformations $T_{t_1}$ and $T_{t_2}$ are spectrally disjoint whenever $|t_1|\ne |t_2|$. A generic transformation (i) embeds into a flow $T$ with $I(T)=\{1\}$ and (ii) does not embed into a flow with $I(T)\ne \{1\}$. For each countable multiplicative subgroup $S\subset\Bbb R^*$, it is constructed a Poisson suspension flow $T$ with simple spectrum such that $I(T)=S$. If $S$ is without rational relations then there is a rank-one weakly mixing rigid flow $T$ with $I(T)=S$.