Quasiperiodic Flows and Algebraic Number Fields

Abstract

We classify a quasiperiodic flow as either algebraic or transcendental. For an algebraic quasiperiodic flow on the n-torus, we prove that an absolute invariant of the smooth conjugacy class of this flow, known as the multiplier group, is a subgroup of the group of units of the ring of integers in a real algebraic number field of degree n over Q. We also prove that for any real algebraic number field F of degree n over Q, there exists an algebraic quasiperiodic flow on the n-torus whose multiplier group is exactly the group of units of the ring of integers in F. We support and formulate the conjecture that the multiplier group distinguishes the algebraic quasiperiodic flows from the transcendental ones.