A Necessary and Sufficient Condition for a Self-Diffeomorphism of a Smooth Manifold to be the Time-1 Map of the Flow of a Differential Equation

TL;DR

This paper establishes a precise criterion to determine when a self-diffeomorphism of a smooth manifold can be realized as the time-1 map of a differential equation's flow, bridging topological dynamics and global analysis.

Contribution

It provides a necessary and sufficient condition characterizing when a self-diffeomorphism is the time-1 map of a differential equation's flow on a smooth manifold.

Findings

Derived a criterion linking diffeomorphisms to differential equation flows

Bridged concepts between topological dynamics and differential equations

Enhanced understanding of the structure of flow maps on manifolds

Abstract

In topological dynamics, one considers a topological space $X$ and a self-map $f: X \to X$ of $X$ and studies the self-map's properties. In global analysis, one considers a smooth manifold $M^n$ and a differential equation $\xi: M \to TM$ on $M$ and studies the flow $\Phi_t: M \times \mathbb{R} \to M$ of the differential equation. In this paper, we consider a necessary and sufficient condition for a self-diffeomorphism $f$ of a manifold $M$ to be the time-1 map $\Phi_1$ of the flow of a differential equation on $M$.