Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Strongly Recurrent Transformation Groups

Version 1 : Received: 17 December 2021 / Approved: 20 December 2021 / Online: 20 December 2021 (10:28:50 CET)

How to cite: Amini, M.; Swid, J. Strongly Recurrent Transformation Groups. Preprints 2021, 2021120304 (doi: 10.20944/preprints202112.0304.v1). Amini, M.; Swid, J. Strongly Recurrent Transformation Groups. Preprints 2021, 2021120304 (doi: 10.20944/preprints202112.0304.v1).

## Abstract

We define the notions of strong and strict recurrency for actions of countable ordered groups on $\sigma$-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

## Keywords

ordered groups; $\sigma$-finite spaces; strongly recurrent actions; strictly recurrent actions; weakly wandering sets

## Subject

MATHEMATICS & COMPUTER SCIENCE, Analysis

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