On the intersections of projected Hamiltonian orbits in cotangent bundles

Abstract: We study the generic behavior of Hamiltonian trajectories on a regular level set in the cotangent bundle, after projection to the base. We prove that for a generic submersive level set, projected trajectories have discrete (self-)intersections. Additionally, fixing end-point fibers, we prove that all intersections can be perturbed away if the base has dimension at least three. In particular, this applies to periodic orbits, and both results hold for Reeb flows on fiber-wise star-shaped hypersurfaces, including non-reversible Finsler flows, which answers a question of Rademacher. In the proof we make use of a multi-jet transversality theorem.

• The paper proves that, for residual Hamiltonians on regular submersive energy hypersurfaces, projected flow lines intersect only at discrete times, with stronger non-intersection results for dimensions at least three.
• The authors use multijet transversality and local Hamiltonian perturbations to eliminate finitely many intersections among chords and periodic orbits, overcoming the failure of tangency arguments for non-reversible systems.
• The results apply to star-shaped and Finsler hypersurfaces and, combined with filtered symplectic homology, convert homological growth into genuine counts of distinct Hamiltonian chords, including linear or exponential growth.

Overview

This paper by Dahinden and de Pooter studies the generic behavior of Hamiltonian trajectories on a regular energy hypersurface $\Sigma = H^{-1}(0) \subset T^*Q$, after projection to the base manifold $Q$. Its main results establish that, for a residual set of Hamiltonians with submersive level set, projected flow lines intersect each other and themselves only in discrete sets of times; moreover, when $\dim Q \geq 3$, all intersections between chords with fixed endpoint fibers (and between periodic orbits) can be perturbed away. This generalizes theorems of Rademacher on geodesics without self-intersections from Riemannian and reversible Finsler metrics to arbitrary Hamiltonians in the class $Ham_s$ — those whose zero level set is regular and projects submersively to $Q$ — thereby answering a question of Rademacher concerning non-reversible Finsler metrics.

The obstruction that Rademacher's argument faces in the non-reversible setting is that tangent projected geodesic segments need not coincide: without reversibility or an injectivity-radius mechanism, tangency does not force equality of curves. The present paper circumvents this by weakening the requirement from "no tangencies" to "discrete intersections," which suffices for the finite-resolution perturbation argument.

Main results

The paper proves two theorems. The first is the discreteness statement.

Theorem (discrete intersections). For $\dim Q \geq 3$, there is a subset of $Ham_s$ that is $C^7$-residual in both the weak and strong Whitney topologies such that any two distinct Hamiltonian flow lines on $\Sigma$, up to time shift, have discrete intersection times after projection to $Q$. For $Q$0 the analogous statement holds for a $Q$1-residual set.

Discreteness is deliberately weak: it asserts no lower bound on the distance between consecutive intersection times, and the authors exhibit an explicit example of lines in $Q$2 realizing arbitrarily close intersections while remaining projections of Hamiltonian flow lines. Consequently, no uniform separation of intersections is claimed or achievable generically.

Theorem (main). For $Q$3 and fixed points $Q$4, there is a $Q$5-residual set of Hamiltonians in $Q$6 such that: distinct flow lines connecting the fibers $Q$7 and $Q$8 project to curves meeting only at the endpoints; such a line has injective projection except at coincident endpoints; geometrically distinct periodic orbits have disjoint projections; and no periodic orbit that is not a multiple cover self-intersects under projection.

A corollary specializes these results to two important open subsets of $Q$9: fiberwise star-shaped hypersurfaces (whose flows are reparametrized contact flows on the spherization $\dim Q \geq 3$0) and fiberwise analytically convex hypersurfaces (whose flows are precisely Finsler geodesic flows, with the Finsler norm recovered as $\dim Q \geq 3$1). Since residual sets intersect open sets residually, the theorems hold verbatim in these settings, including for non-reversible Finsler flows.

As an application, the authors combine their genericity result with filtered symplectic homology / positive Rabinowitz–Floer homology for exactly fillable compact hypersurfaces: for a $\dim Q \geq 3$2-residual set of such hypersurfaces, the number of geometrically distinct chords between two fibers grows in length at least as fast as the dimension of symplectic homology filtered by length; for fiberwise star-shaped hypersurfaces the growth is at least linear, and exponential growth of loop-space homology in degree yields exponentially many geometrically distinct chords. Because the homological lower bounds count generators that may fail to be geometrically distinct, the non-intersection theorem is what upgrades them to genuine counts of distinct curves.

Method

The proof structure mirrors Rademacher's three-step scheme: (1) non-degeneracy of the relevant periodic orbits and chords holds on a $\dim Q \geq 3$3-residual set, so only finitely many orbits of bounded length need be considered, and they vary continuously with the Hamiltonian; (2) discreteness of intersections makes the set of intersections finite; (3) local smooth perturbations resolve the finitely many intersections one at a time.

Step (3) is carried out by an explicit construction: near an isolated intersection, the base neighborhood is trivialized via the normal bundle of one curve, a parallel curve family is chosen avoiding the other curves (using a pigeonhole argument over uncountably many angular parameters), and the displacement is implemented by pulling back the Hamiltonian along a compactly supported family of diffeomorphisms of $\dim Q \geq 3$4, interpolated against the original $\dim Q \geq 3$5 by a cutoff. Non-intersection is then $\dim Q \geq 3$6-open among the finitely many non-degenerate orbits, so the property is $\dim Q \geq 3$7-open and $\dim Q \geq 3$8-dense locally, and a Baire argument over time horizons and a compact exhaustion yields global residuality.

The technical core is step (2), i.e., Theorem on discrete intersections, proved via jets of submanifolds. A Hamiltonian $\dim Q \geq 3$9 induces a section $Ham_s$0 assigning to each point its short flow line as a marked one-dimensional submanifold; this section depends only on the hypersurface $Ham_s$1, not on the defining function, since the characteristic line field $Ham_s$2 determines the conformal class of $Ham_s$3. Two projected flow lines with a $Ham_s$4-jet disagreement at an intersection point cannot accumulate intersections nearby (a Taylor estimate); hence discreteness follows if the composite multisection $Ham_s$5 hits the diagonal of $Ham_s$6 only over the diagonal of $Ham_s$7. The key transversality statement is:

• For a $Ham_s$8-residual set of Hamiltonians ($Ham_s$9, $Q$0), the antipodal set and the off-diagonal isopodal set (points whose projected flow-line $Q$1-jets agree, with matching or opposite orientation) are submanifolds of $Q$2 of dimension $Q$3.

The dimension count is decisive: for $Q$4 and $Q$5, or $Q$6 and $Q$7, the dimension is negative, so the off-diagonal homopodal set is empty — which is precisely the hypothesis needed for discreteness. The terminology extends the classical notion of antipodes in convex Finsler geometry (the unique point with oppositely oriented parallel momentum) to arbitrary orders of jet agreement, distinguishing isopodes from antipodes by orientation.

Transversality is achieved by exploiting the local structure of perturbations. The authors cover $Q$8 by radial charts, in which the level set is described as the graph of a positive function $Q$9 over $\dim Q \geq 3$0, and the characteristic flow becomes the Reeb flow of $\dim Q \geq 3$1 for a local contact form $\dim Q \geq 3$2 — equivalently, the autonomous contact Hamiltonian flow of $\dim Q \geq 3$3. They prove that the jet map $\dim Q \geq 3$4 sending the jet of a positive contact Hamiltonian to the $\dim Q \geq 3$5-jet of its flow line through the base point is surjective and submersive. The proof constructs, for any given jet of a path transverse to the contact distribution, an autonomous contact Hamiltonian realizing it, using the isotropic isotopy extension theorem together with a deliberate choice of one free component of the prescribed 1-jet (along the Reeb direction) that enforces autonomy via the Leibniz rule. Surjectivity plus submersivity means the preimage of the relevant multijet stratum $\dim Q \geq 3$6 is a fixed submanifold independent of the perturbation, and transversality of the graph of $\dim Q \geq 3$7 against $\dim Q \geq 3$8 follows from Mather's multijet transversality theorem in the modern formulation of Gootjes-Dreesbach.

Regularity considerations

An important remark establishes sharpness of the regularity assumptions. At $\dim Q \geq 3$9 level the theorem fails outright, since $Ham_s$0 perturbations can create closed orbits and chords at will. At $Ham_s$1 regularity it also fails: curvature bumps that are $Ham_s$2-small but $Ham_s$3-large can create conjugate points at a target fiber, causing a single chord to bifurcate into a controlled-by-nobody family of mutually intersecting chords, within any $Ham_s$4 neighborhood of a given metric. The $Ham_s$5 threshold in the main theorem is therefore essential, and the residual statements require correspondingly high differentiability ($Ham_s$6 with $Ham_s$7 up to 4 or 5).

Limitations and open questions

Several qualifications are stated plainly in the paper. The class $Ham_s$8 excludes vertical dynamics: submersivity of $Ham_s$9 guarantees $C^7$0, which is used throughout; Hamiltonians with fibers contained in $C^7$1 fall outside the framework. Compactness and properness are not assumed, so the flow need not be complete; the authors note that the main theorem is local in nature and can be extended via exhaustion by compact sets, but completeness itself is not recovered. The discreteness theorem gives no quantitative control — no lower bound on the spacing of intersections, and no continuity of the first intersection time. The example of the "heart"-shaped fiber shows that the first-order homopodal set can contain genuine non-manifold points at inflection points (where $C^7$2 is nontrivial); the transversality theorem handles inflection–non-inflection pairs by parametrization and claims generic manifoldness for inflection–inflection pairs off the diagonal, but the analysis of higher-order inflection phenomena is confined to what the dimension count requires.

Two specific questions remain open. First, the existence of simple closed geodesics on simply connected manifolds (in the sense of Adams–Hass–Scott) is untouched; the genericity machinery here does not address existence, only non-intersection once geodesics exist. Second, the residual statements are established in the $C^7$3 Baire space with specified finite regularity thresholds; whether the thresholds $C^7$4/$C^7$5 for discreteness, or the intermediate exponents in the transversality theorem, are optimal is not investigated.

Conclusion

The paper extends Rademacher's generic non-intersection theorems for geodesics to the full Hamiltonian category of submersive regular energy hypersurfaces in cotangent bundles, resolving his question about non-reversible Finsler metrics affirmatively. The method replaces Rademacher's implicit-function-theorem perturbation step with a multijet transversality argument on bundles of submanifold jets, powered by a surjective-submersive jet map from contact Hamiltonians to jets of their flow lines. Combined with filtered Floer-theoretic lower bounds, the results convert homological counts into counts of genuinely geometrically distinct chords, with linear or exponential growth rates dictated by loop-space topology.