Volume preserving diffeomorphisms as Poincare maps for volume preserving flows

TL;DR

This paper investigates conditions under which volume-preserving diffeomorphisms can be represented as Poincaré maps of volume-preserving flows on a product manifold, linking discrete and continuous dynamical systems.

Contribution

It establishes a framework for representing volume-preserving diffeomorphisms as Poincaré maps of volume-preserving flows, extending the understanding of their structural relationship.

Findings

Characterization of volume-preserving diffeomorphisms as Poincaré maps.

Conditions for representing diffeomorphisms via volume-preserving flows.

Insights into the interplay between discrete and continuous volume-preserving dynamics.

Abstract

Let $q:M\to M$ be a volume-preserving diffeomorphism of a smooth manifold $M$. We study the possibility to present $q$ as the Poincar\'e map, corresponding to a volume-preserving vector field on $\mathbb{T}\times M$, $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.