A reducible representation of the generalized symmetry group of a quasiperiodic flow

Abstract

The generalized symmetry group of a quasiperiodic flow on a $n$-torus is the group theoretic normalizer, within the group of diffeomorphisms of the $n$-torus, of the one parameter abelian group of diffeomorphisms generated by the flow. Up to conjugacy, the generalized symmetry group of a quasiperiodic flow is determined by a system of uncoupled first order partial differential equations. New types of symmetries (other than the classical types of symmetries or time-reversing symmetries) may exist depending on certain algebraic relationships being satisfied by pair wise ratios of the frequencies of the quasiperiodic flow. These new symmetries, when they exist, are a dominant feature of a reducible linear representation of the generalized symmetry group in the de Rham cohomology of the $n$-torus.