On the group of A-P diffeomorphisms and its exponential map

TL;DR

This paper introduces the group of almost periodic diffeomorphisms on Euclidean space and Lie groups, studies its exponential maps, and explores applications to fluid dynamics, revealing complex geodesic structures with conjugate points.

Contribution

It defines the almost periodic diffeomorphism group and analyzes its exponential maps, providing new insights into geometric structures relevant to fluid equations.

Findings

Existence of geodesics with conjugate points in the group of almost periodic diffeomorphisms.

Analysis of the properties of Riemannian and Lie group exponential maps.

Applications to understanding fluid equations through geometric methods.

Abstract

We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.