Flows with uncountable but meager group of self-similarities

TL;DR

This paper constructs specific ergodic flows with uncountable yet meager groups of self-similarities, revealing complex symmetry structures in dynamical systems.

Contribution

It introduces new examples of ergodic flows with uncountable but meager self-similarity groups and explores Poisson flows with prescribed quasi-invariance groups.

Findings

Constructed a weakly mixing Gaussian flow with uncountable, meager self-similarity group.

Developed Poisson flows with self-similarity groups characterized by a given measure.

Demonstrated the existence of zero-entropy Poisson flows with prescribed quasi-invariance groups.

Abstract

Given an ergodic probability preserving flow $T=(T_t)_{t\in\Bbb R}$, let $I(T):=\{s\in\Bbb R^*\mid T\text{is isomorphic to}(T_{st})_{t\in\Bbb R}\}$. A weakly mixing Gaussian flow $T$ is constructed such that $I(T)$ is uncountable and meager. For a Poisson flow $T$, a subgroup $I_{\text{Po}}(T)\subset I(T)$ of Poissonian self-similarities is introduced. Given a probability measure $\kappa$ on $\Bbb R^*_+$, a zero-entropy Poisson flow $T$ is constructed such that $I_{\text{Po}}(T)$ is the group of $\kappa$-quasi-invariance.