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The growth rate of closed prime magnetic geodesics on closed contact manifolds

Lina Deschamps, Levin Maier, Tom Stalljohann

TL;DR

The paper proves that on any closed contact manifold $(M,\\alpha)$ there exists an infinite-dimensional family of Riemannian metrics for which every energy level of the magnetic geodesic flow with field $d\\alpha$ contains at least as many embedded periodic orbits as there are Reeb orbits, via a correspondence where Reeb orbits become magnetic geodesics. By identifying this metric family with bundle metrics on the contact distribution, the authors leverage Mañé’s critical value to establish robust growth phenomena: quadratic growth of prime magnetic geodesics on $S^3$-type examples and, on closed 3-manifolds not graph manifolds, exponential growth and positive topological entropy on all energy levels for generic choices. The results are tied to, and sharpen, the framework of DMS25Contact by working within a more rigid metric class that still yields strong dynamical conclusions, including implications for the abundance of periodic orbits and the complexity of the flow across energy surfaces. Overall, the work broadens the scope of magnetic geodesic dynamics on contact manifolds, linking Reeb dynamics, variational methods, and entropy considerations in a unified growth theory.

Abstract

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such metric, and for all energy levels, the number of embedded periodic orbits of the corresponding magnetic geodesic flow grows at least as fast as the number of geometrically distinct periodic Reeb orbits of period less than $t$. As a corollary, we deduce that for every closed 3-manifold which is not a graph manifold, there exists an open $C^1$-neighborhood of the set of nondegenerate contact forms such that for each contact form in this neighborhood, there exists an infinite-dimensional space of Riemannian metrics as above. For the corresponding magnetic systems, the number of prime closed magnetic geodesics grows at least exponentially on all energy levels. Consequently, the restriction of the magnetic geodesic flow to any energy surface has positive topological entropy.

The growth rate of closed prime magnetic geodesics on closed contact manifolds

TL;DR

The paper proves that on any closed contact manifold there exists an infinite-dimensional family of Riemannian metrics for which every energy level of the magnetic geodesic flow with field contains at least as many embedded periodic orbits as there are Reeb orbits, via a correspondence where Reeb orbits become magnetic geodesics. By identifying this metric family with bundle metrics on the contact distribution, the authors leverage Mañé’s critical value to establish robust growth phenomena: quadratic growth of prime magnetic geodesics on -type examples and, on closed 3-manifolds not graph manifolds, exponential growth and positive topological entropy on all energy levels for generic choices. The results are tied to, and sharpen, the framework of DMS25Contact by working within a more rigid metric class that still yields strong dynamical conclusions, including implications for the abundance of periodic orbits and the complexity of the flow across energy surfaces. Overall, the work broadens the scope of magnetic geodesic dynamics on contact manifolds, linking Reeb dynamics, variational methods, and entropy considerations in a unified growth theory.

Abstract

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such metric, and for all energy levels, the number of embedded periodic orbits of the corresponding magnetic geodesic flow grows at least as fast as the number of geometrically distinct periodic Reeb orbits of period less than . As a corollary, we deduce that for every closed 3-manifold which is not a graph manifold, there exists an open -neighborhood of the set of nondegenerate contact forms such that for each contact form in this neighborhood, there exists an infinite-dimensional space of Riemannian metrics as above. For the corresponding magnetic systems, the number of prime closed magnetic geodesics grows at least exponentially on all energy levels. Consequently, the restriction of the magnetic geodesic flow to any energy surface has positive topological entropy.
Paper Structure (8 sections, 4 theorems, 22 equations)

This paper contains 8 sections, 4 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Illustrations of \ref{['Thm: growth rate of magnetic geodesics on contact mnfds']}
  4. Related results.
  5. Preliminaries: magnetic systems and Mañé's critical value
  6. Intermezzo: magnetic systems
  7. Mañé's critical value.
  8. The Proof of \ref{['Thm: growth rate of magnetic geodesics on contact mnfds']}.

Key Result

Theorem 1.1

Let $(M, \alpha)$ be a closed contact manifold with Reeb vector field ${R_{\alpha}}$. We define an infinite-dimensional space $\mathcal{G}$ of Riemannian metrics on $M$, namely the space of all Riemannian metrics $g$ such that Then, for each $g \in \mathcal{G}$, the exact magnetic system $(M, g, \mathrm{d}\alpha)$ satisfies the following properties:

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3
  • Corollary 2
  • Corollary 3
  • Remark 4
  • Remark 5
  • ...and 1 more