Semiconjugacy of Quasiperiodic Flows and Finite Index Subgroups of Multiplier Groups

Abstract

It will be shown that if $ϕ$ is a quasiperiodic flow on the $n$-torus that is algebraic, if $ψ$ is a flow on the $n$-torus that is smoothly conjugate to a flow generated by a constant vector field, and if $ϕ$ is smoothly semiconjugate to $ψ$, then $ψ$ is a quasiperiodic flow that is algebraic, and the multiplier group of $ψ$ is a finite index subgroup of the multiplier group of $ϕ$. This will partially establish a conjecture that asserts that a quasiperiodic flow on the $n$-torus is algebraic if and only if its multiplier group is a finite index subgroup of the group of units of the ring of integers in a real algebraic number field of degree $n$.