Symmetry Reduction by Lifting for Maps

TL;DR

This paper investigates symmetry reduction techniques for maps with continuous symmetries, demonstrating how to reduce dimensionality and preserve volume, with applications to various types of maps and an extension of Noether's theorem.

Contribution

It introduces a method to reduce the dimensionality of maps with symmetries using lifting and Poincaré sections, applicable beyond symplectic cases, and refines Noether's theorem for symplectic maps.

Findings

Reduction yields skew-product form of maps

Volume preservation is maintained under reduction

Method compares favorably with traditional techniques

Abstract

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.