Elementary Spectral Invariants in Rabinowitz Floer Homology

• Elementary spectral invariants are action-level selectors defined in Floer theory that use minimax principles to extract critical energy values from possibly degenerate functionals.
• They are constructed within Rabinowitz Floer homology to analyze leaf-wise intersections and overcome the challenges of degenerate variational problems in symplectic topology.
• Their robust and locally Lipschitz properties under perturbations yield quantitative lower bounds and guarantee the persistence of critical points in complex Hamiltonian dynamics.

Elementary spectral invariants are action-level selectors defined in Floer-theoretic contexts, originally introduced to capture “quantitative” information in Hamiltonian dynamics via a mini-max principle applied to the action functional. In Rabinowitz Floer homology—the focus of the referenced work—these invariants are constructed analogously but in a framework where the variational functional is degenerate, and the critical points correspond to leaf-wise intersections under global Hamiltonian perturbations. Their robustness, even in degenerate situations, enables new quantitative existence results in symplectic topology that extend well beyond classical Morse-theoretic tools (Albers et al., 2010).

1. Emergence and Foundational Principles

Spectral invariants in Hamiltonian Floer homology originated with the construction of action selectors by Viterbo, Oh, and Schwarz. In the standard Floer setting, given an action functional $\mathcal{A}_H$ associated to a Hamiltonian $H$ on a symplectic manifold, and a nonzero class $a$ in Floer homology, the spectral invariant $\rho_H(a)$ is defined (using a minimax over cycles representing $a$) to be the minimal action level at which $a$ appears in the filtered Floer complex. This value is always a critical value of the action, i.e., an “energy” of some periodic orbit.

In Rabinowitz Floer homology, the construction is generalized to functionals of the form

$$\mathcal{A}^{\mathfrak{M}}(v, \eta) = - \int_0^1 v^*\lambda - \int_0^1 H(v, t)\,dt - \eta \int_0^1 F(v, t)\,dt,$$

where $(v,\eta)$ are loop plus Lagrange multiplier variables, and $(F,H)$ comprise a Moser pair encoding the hypersurface and perturbation. The critical points correspond to leaf-wise intersections—a generalization of closed characteristics to perturbed settings. Importantly, the mini-max approach defining the spectral invariant can proceed even if the functional $\mathcal{A}^{\mathfrak{M}}$ is not Morse; this is crucial since in symplectic dynamics degeneracy is generic.

2. Analytical and Algebraic Framework

The setup centers on an exact symplectic manifold $(M, d\lambda)$ with a hypersurface of restricted contact type. The Moser pair $(F,H)$ is chosen such that $F(x, t) = \rho(t) f(x)$, with $f$ defining the energy level, and $H$ supported only on part of the time domain.

The Rabinowitz action functional’s critical points satisfy: $$\partial_t v = \eta X_F(v, t) + X_H(v, t), \quad \int_0^1 F(v, t)dt = 0.$$ The Rabinowitz Floer chain complex is generated by critical points of $\mathcal{A}^{\mathfrak{M}}$, even when $\mathcal{A}^{\mathfrak{M}}$ is degenerate. Analytical control is achieved (see Lemma 2.11) by bounding the Lagrange multiplier $\eta$, which is essential in infinite-dimensional and degenerate settings.

For a nonzero homology class $X$, the elementary spectral invariant is

$$\sigma^{\mathfrak{M}}(X) := \inf \{ \mathcal{A}^{\mathfrak{M}}(\xi) \mid [\xi] = X \},$$

where $\xi$ runs over cycles in the Floer complex. The core technical achievement is establishing that $\sigma^{\mathfrak{M}}(X)$ is well-defined and locally Lipschitz in the Moser pair $\mathfrak{M}$.

The construction closely follows the paradigm of Morse homology and Hamiltonian Floer homology: filtered chain complexes, minimax procedures, and continuation under perturbation. The finite-dimensional Morse case is treated first, then generalized to the infinite-dimensional Rabinowitz context.

3. Applications: Leaf-wise Intersections and Quantitative Existence Theorems

The main use of these elementary invariants lies in the paper of leaf-wise intersections—a phenomenon wherein the perturbed Hamiltonian flow returns points to their original characteristic leaf on a hypersurface, crucial in symplectic rigidity and dynamics. The key result (Theorem 1.2) states that if the homology of the free loop space of the base (for fiberwise star-shaped hypersurfaces in $T^*B$) is infinite, then there exist leaf-wise intersections with unbounded time shifts $\eta$. Spectral invariants provide lower bounds for the action/time shift of these critical points.

The construction is robust in degenerate settings—when the functional is not Morse—and the spectral invariant still selects a genuine critical value. Iteration inequalities and continuity results (Proposition 4.3 and Theorem 4.7) give quantitative estimates for the variation of spectral invariants under changes in the Moser pair, supporting the analysis of how intersections persist or bifurcate under perturbations. Via the infinite range of possible critical values, one concludes (with appropriate topological constraints) the existence of infinitely many leaf-wise intersections, or the existence of a periodic leaf-wise intersection in the degenerate case.

4. Continuity, Sensitivity, and Technical Properties

The local Lipschitz (continuity) property of the spectral invariant with respect to the Moser pair is established, ensuring robust behavior under perturbation and allowing the invariants to persist in practical, non-idealized (possibly degenerate) scenarios. This continuity is essential in controlling the dynamical consequences of global Hamiltonian perturbations.

The quantitative estimates established for spectral invariants include:

• Lipschitz bounds in the Moser pair.
• Iteration inequalities governing the change in the invariant under iterates of a given Moser pair (Appendix).

These properties not only provide analytical control but reveal sensitivity to the underlying symplectic geometry. This sensitivity makes the invariants suitable for investigating questions about the dynamics under large or small perturbations.

5. Robustness in Degenerate and Nonclassical Situations

A defining feature of these elementary spectral invariants is their applicability even when the Morse condition fails. In classical Morse homology, spectral invariants become ill-defined without non-degeneracy. By contrast, the construction in Rabinowitz Floer homology (and more broadly, in related Floer-theoretic settings) is based on minimax procedures and local continuity properties, allowing direct work with degenerate functionals.

This robustness underpins the primary applications:

• Existence proofs for critical points and leaf-wise intersections are not obstructed by degeneracy.
• The method does not require regularization or perturbation to Morse situations, streamlining both theoretical and practical efforts.

6. Further Directions and Broader Implications

The extension of spectral invariants to Rabinowitz Floer homology opens possibilities for studying:

• The interaction between spectral invariants and Floer-theoretic product structures (such as pair-of-pants products).
• More refined questions in symplectic rigidity, e.g., quantitative bounds for periodic orbits or non-displaceability.
• Generalization to other degenerate variational problems in symplectic topology.
• Behavior of invariants under strong Hamiltonian perturbations and for broader classes of energy hypersurfaces.

The framework provides a template for developing spectral invariants in other infinite-dimensional, potentially degenerate, variational problems where minimax-type selectors retain meaning and quantitative power under analytic delicacy.

Summary Table: Key Properties of Elementary Spectral Invariants in Rabinowitz Floer Homology

Property	Description	Context
Definition	Minimax value over cycles representing given homology class	$\sigma^{\mathfrak{M}}(X)$
Functional	Rabinowitz action functional on loops plus multiplier	$\mathcal{A}^{\mathfrak{M}}$
Critical Points	Solutions correspond to leaf-wise intersections	Non-Morse/degenerate allowed
Continuity	Locally Lipschitz in choice of Moser pair	Proposition 4.3, Theorem 4.7
Quantitative Results	Lower bounds on action/time shift for critical points; infinite range supports count	Theorem 1.2 and related
Iteration Inequality	Governs change under iteration of Moser pair	Appendix

These elementary invariants form a core tool in modern symplectic topology, enabling quantitative, robust results in global Hamiltonian dynamics and beyond (Albers et al., 2010).