Structure of Group Invariants of a Quasiperiodic Flow

TL;DR

This paper investigates the algebraic structure of invariants associated with quasiperiodic flows on the n-torus, revealing their composition as semidirect products involving algebraic integer groups and unimodular matrices.

Contribution

It introduces a new group invariant based on the multiplier representation, linking the flow's symmetry group to algebraic integers and unimodular matrices.

Findings

Group invariants are semidirect products of algebraic integer groups and unimodular matrices.

Subgroups of the multiplier group correspond to algebraic integers of degree at most n.

The structure provides a classification of quasiperiodic flows based on their symmetry groups.

Abstract

The multiplier representation of the generalized symmetry group of a quasiperiodic flow on the n-torus defines, for each subgroup of the multiplier group of the flow, a group invariant of the smooth conjugacy class of that flow. This group invariant is the internal semidirect product of a subgroup isomorphic to the n-torus by a subgroup isomorphic to that subgroup of the multiplier group. Each subgroup of the multiplier group is a multiplicative group of algebraic integers of degree at most n, which group is isomorphic to an abelian group of n by n unimodular matrices.