Topological entropy of left-invariant magnetic flows on 2-step nilmanifolds

In this paper, we consider magnetic flows on 2-step nilmanifolds $M=\Gamma \setminus G$, where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to $g=T_{e}G$ vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when σ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high energy levels. (paper)