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

2015/12/08 by Jonathan I. Epstein, Jonathan Epstein, Epstein, Jonathan
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DG #math.DS

paper · pdf · doi:10.48550/arxiv.1512.02612

13 pages

arxiv created 2015/12/08 · openalex publication_date 2015/12/08 · arxiv updated 2015/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \mathfrakg = 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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