Semiconjugacy of Quasiperiodic Flows and Finite Index Subgroups of Multiplier Groups

Abstract

It will be shown that if $\varphi$ is a quasiperiodic flow on the $n$-torus that is algebraic, if $\psi$ is a flow on the $n$-torus that is smoothly conjugate to a flow generated by a constant vector field, and if $\varphi$ is smoothly semiconjugate to $\psi$, then $\psi$ is a quasiperiodic flow that is algebraic, and the multiplier group of $\psi$ is a finite index subgroup of the multiplier group of $\varphi$. 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 $n$.

Submitted to the Proceedings of the AIMS' Fifth International Conference on Dynamical Systems and Differential Equations