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# Arkiv för Matematik

## Volume 58 (2020)

### Number 1

### Periodic flows with global sections

Pages: 39 – 56

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a3

#### Author

#### Abstract

Let $G = \lbrace h_t \; \vert \; t \in \mathbb{R} \rbrace$ be a continuous flow on a connected $n$-manifold $M$. The flow $G$ is said to be strongly reversible by an involution $\tau$ if $h_{-t} = \tau h_t \tau$ for all $t \in \mathbb{R}$, and it is said to be periodic if $h_s = $ identity for some $s \in \mathbb{R}^\ast$. A closed subset $K$ of $M$ is called a global section for $G$ if every orbit $G(x)$ intersects $K$ in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if $G$ is periodic and strongly reversible by a reflection, then $G$ has a global section.

#### Keywords

periodic flow, strongly reversible, reflection, global section

#### 2010 Mathematics Subject Classification

37B05, 37B20, 54H20, 57S05, 57S10

Received 23 February 2019

Received revised 14 July 2019

Accepted 1 August 2019

Published 21 July 2022