Topological Entropy of Left-Invariant Magnetic Flows on 2-Step Nilmanifolds

We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, 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 $\mathfrak{g} = T_eG$ 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.

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