[Paper Review] On the Generic Existence of Periodic Orbits in Hamiltonian Dynamics

Viktor L. Ginzburg, Başak Z. Gürel|ArXiv.org|Aug 23, 2009

Quantum chaos and dynamical systems18 references3 citations

TL;DR

This paper establishes $C^\infty$-generic existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms on complex projective spaces and Grassmannians, and for Reeb flows on fillable contact manifolds, using resonance relations in Floer homology. It proves that symplectomorphisms of the 2-torus with irrational flux have infinitely many periodic orbits if one exists and all are non-degenerate, via Floer–Novikov homology. The key contribution is a generic existence result relying on the fragility of resonance relations under $C^\infty$-small perturbations.

ABSTRACT

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that such a symplectomorphism necessarily has infinitely many periodic orbits whenever it has one and all periodic points are non-degenerate.

Motivation & Objective

• To establish $C^\infty$-generic existence of infinitely many periodic orbits in Hamiltonian dynamics on symplectic manifolds that are not symplectically aspherical, such as complex projective spaces and Grassmannians.
• To extend generic existence results to Reeb flows on fillable contact manifolds, including unit cotangent bundles and spheres with standard contact structures.
• To analyze the periodic orbit structure of symplectomorphisms on the 2-torus with irrational flux, showing that existence of one non-degenerate periodic orbit implies infinitely many.
• To unify and generalize prior results on periodic orbits using resonance relations in Floer homology and the fragility of such relations under $C^\infty$-small perturbations.
• To clarify the role of topological obstructions (e.g., odd homology) and dynamical conditions (e.g., non-degeneracy, non-hyperbolicity) in generic existence theorems.

Proposed method

• Utilizes resonance relations in Floer homology that must be satisfied if a Hamiltonian diffeomorphism or Reeb flow has only finitely many periodic orbits.
• Applies the principle that such resonance relations are fragile and can be destroyed by $C^\infty$-small perturbations, forcing the creation of infinitely many periodic orbits.
• Employs Floer–Novikov homology to analyze symplectomorphisms of the 2-torus with irrational flux, proving that non-degenerate periodic points imply infinite orbit growth.
• Relies on the fact that when $\operatorname{H}_{\text{odd}}(M;\mathbb{Z}) \neq 0$, non-degenerate Hamiltonian diffeomorphisms must have periodic points of even Conley–Zehnder index, which contradicts Floer homology unless infinitely many orbits exist.
• Uses the Weinstein conjecture as a necessary assumption in the Reeb flow case to ensure the existence of at least one closed characteristic, enabling the construction of a residual set of forms with infinitely many orbits.
• Applies the Birkhoff–Moser fixed point theorem in the context of non-hyperbolic periodic points to generate infinitely many orbits via generic perturbations.

Experimental results

Research questions

• RQ1 Under what generic conditions does a Hamiltonian diffeomorphism on a non-symplectically-aspherical manifold like $\mathbb{C}P^n$ have infinitely many periodic orbits?
• RQ2 Can the existence of a single non-degenerate periodic orbit in a symplectomorphism of the 2-torus with irrational flux imply the existence of infinitely many such orbits?
• RQ3 How do resonance relations in Floer homology constrain the number of periodic orbits, and why are they destroyed under $C^\infty$-small perturbations?
• RQ4 To what extent can the $C^\infty$-generic existence of infinitely many periodic orbits be established for Reeb flows on fillable contact manifolds?
• RQ5 What topological or dynamical conditions are necessary to ensure that a Hamiltonian diffeomorphism has infinitely many periodic orbits generically?

Key findings

• A $C^\infty$-generic Hamiltonian diffeomorphism on $\mathbb{C}P^n$ or a Grassmannian has infinitely many periodic orbits, despite the existence of diffeomorphisms with only finitely many.
• A symplectomorphism of the 2-torus with irrational flux has infinitely many periodic orbits if it has at least one and all periodic points are non-degenerate.
• The Reeb flow of a $C^\infty$-generic contact form on a fillable contact manifold has infinitely many periodic orbits, provided the Weinstein conjecture holds for the contact structure.
• The set of contact forms with infinitely many periodic orbits is residual in the $C^\infty$-topology on the space of non-degenerate contact forms.
• Resonance relations in Floer homology that constrain periodic orbits are fragile under $C^\infty$-small perturbations, which forces the creation of infinitely many orbits when such relations are violated.
• When $\operatorname{H}_{\text{odd}}(M;\mathbb{Z}) \neq 0$, a strongly non-degenerate Hamiltonian diffeomorphism on $M$ must have infinitely many periodic orbits, due to parity constraints on Conley–Zehnder indices.