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

TL;DR

This paper investigates the topological entropy of magnetic flows on 2-step nilmanifolds with left-invariant structures, showing conditions for zero entropy and constructing examples with positive entropy at high energy levels.

Contribution

It provides conditions under which magnetic flows have zero topological entropy and constructs examples with positive entropy at high energies on 2-step nilmanifolds.

Findings

Magnetic flows with rational cohomology class and vanishing restriction have zero topological entropy.

Existence of magnetic fields with positive topological entropy at high energy levels.

Conditions linking magnetic field properties to entropy behavior.

Abstract

We consider magnetic flows on 2-step nilmanifolds $M = \Gamma \backslash G$, where the Riemannian metric $g$ and the magnetic field $\sigma$ are left-invariant. Our first result is that when $\sigma$ 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 $\sigma$ 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.