[Paper Review] On the Conley Conjecture for Reeb Flows

Viktor L. Ginzburg, Başak Z. Gürel+1 more|arXiv (Cornell University)|Jul 7, 2014

Geometric and Algebraic Topology44 references4 citations

TL;DR

This paper establishes the contact Conley conjecture for pre-quantization circle bundles over aspherical manifolds, proving that every Reeb flow on such manifolds has infinitely many geometrically distinct closed orbits under index conditions. The proof uses cylindrical contact homology and the notion of symplectically degenerate maxima, with an application showing infinitely many periodic orbits for twisted geodesic flows on surfaces of genus at least two with non-vanishing magnetic fields.

ABSTRACT

In this paper we prove the existence of infinitely many closed Reeb orbits for a certain class of contact manifolds. This result can be viewed as a contact analogue of the Hamiltonian Conley conjecture. The manifolds for which the contact Conley conjecture is established are the pre-quantization circle bundles with aspherical base. As an application, we prove that for a surface of genus at least two with a non-vanishing magnetic field, the twisted geodesic flow has infinitely many periodic orbits on every low energy level.

Motivation & Objective

• To establish a contact analogue of the Hamiltonian Conley conjecture for Reeb flows on pre-quantization circle bundles with aspherical base.
• To demonstrate the existence of infinitely many geometrically distinct closed Reeb orbits under index conditions, despite bounded contact homology rank.
• To apply the result to twisted geodesic flows on surfaces of genus ≥2 with non-vanishing magnetic fields, proving infinitely many periodic orbits on low energy levels.
• To extend the Conley conjecture to contact manifolds where homological growth does not drive the result, relying instead on dynamical and topological invariants like symplectically degenerate maxima.

Proposed method

• Utilizes cylindrical contact homology and its invariance properties for pre-quantization circle bundles over aspherical manifolds.
• Applies the notion of symplectically degenerate maxima (SDM) to detect non-degenerate Reeb orbits with controlled Conley–Zehnder index.
• Employs a filling construction involving a high-energy level in the cotangent bundle and the pre-quantization bundle to apply Theorem 3.2 on contact homology invariants.
• Uses perturbation techniques to ensure non-degeneracy and control the mean index of closed orbits, particularly in the fiber homotopy class.
• Relies on the fact that iterates of a given orbit contribute only bounded amounts to contact homology, enabling detection of new orbits.
• Applies index bounds via the mean index and Conley–Zehnder index to show that for large prime iterations, orbits must be simple or come from positive-index SDMs.

Experimental results

Research questions

• RQ1 Does every Reeb flow on a pre-quantization circle bundle over an aspherical manifold have infinitely many geometrically distinct closed orbits?
• RQ2 Can the Conley conjecture be extended to the contact setting when contact homology does not grow with iteration?
• RQ3 Do twisted geodesic flows on surfaces of genus ≥2 with non-vanishing magnetic fields have infinitely many periodic orbits on low energy levels?
• RQ4 What topological and dynamical conditions ensure the existence of infinitely many closed Reeb orbits in the absence of homological growth?

Key findings

• The contact Conley conjecture holds for pre-quantization circle bundles over aspherical manifolds with atoroidal first Chern class, under index conditions.
• For every sufficiently large prime k, there exists a simple closed Reeb orbit in the k-th iterate of the fiber class, provided one simple orbit in the fiber class is a symplectically degenerate maximum.
• The Reeb flow on low energy levels of the twisted geodesic flow on a surface of genus ≥2 with non-vanishing magnetic field has infinitely many periodic orbits.
• The mean index of contractible Reeb orbits on the pre-quantization bundle satisfies Δ(x) < 2χ(M) + o(1) as ε → 0, ensuring bounded index growth.
• Long orbits (period ≥ T) have large negative mean index, ensuring their Conley–Zehnder index is less than 2χ(M) + 1 + o(1) < 0, which helps in detecting new orbits.
• The construction of a strong, exact filling with c₁(TW) = 0 and a non-trivial Reeb flow on one boundary component allows the application of Theorem 3.2 to detect orbits in the fiber class.