Hypersemitoric Systems: Invariants & Bifurcations

• Hypersemitoric systems are integrable Hamiltonian systems on four-dimensional symplectic manifolds with an effective S¹-action and controlled nondegenerate and parabolic singularities.
• They generalize toric and semitoric systems by relaxing singularity restrictions to include hyperbolic blocks and mildly degenerate (parabolic) points, leading to intricate bifurcation phenomena.
• Classification employs affine and combinatorial invariants that capture unique geometric and topological features, inspiring new research directions in integrable systems theory.

Hypersemitoric systems are integrable Hamiltonian systems on four-dimensional symplectic manifolds that generalize both toric and semitoric systems by permitting mildly degenerate singularities, especially parabolic (cuspidal) points, alongside standard nondegenerate singularities and enforcing an underlying effective Hamiltonian S¹-action generated by a proper function. Recent developments demonstrate that every compact Hamiltonian S¹-space can be extended to a hypersemitoric system, and motivate the paper of new invariants and classification schemes that capture the geometric and topological intricacies introduced by these additional singularities and bifurcation phenomena. The theory is actively expanding toward a comprehensive symplectic and topological classification, drawing on foundational ideas from toric and semitoric systems and extending them in the presence of more complex singular behavior.

1. Definition and Structural Properties

A hypersemitoric system is defined on a compact, connected four-dimensional symplectic manifold $(M, \omega)$ by a pair of smooth functions $(J, H)$ where $J$ is proper and its Hamiltonian flow generates an effective S¹-action. All singularities of the momentum map $F = (J, H)$ are either nondegenerate (in the sense of Williamson, i.e., elliptic–elliptic, elliptic–regular, hyperbolic–regular, hyperbolic–elliptic, focus–focus) or mildly degenerate parabolic points that satisfy specific local regularity conditions.

A parabolic singularity is characterized (see (Hohloch et al., 2021)) by the restriction of the second component to the level set of the first, i.e., for $(f_1, f_2) = g \circ F$ with $g$ a local diffeomorphism, define $\tilde{f}_2 := f_2|_{f_1^{-1}(f_1(p))}$, then $p$ is parabolic if:

• $p$ is a critical point of $\tilde{f}_2$,
• the Hessian $d^2\tilde{f}_2(p)$ has rank one,
• there exists $v$ in the kernel of $d^2\tilde{f}_2(p)$ such that $v^3(\tilde{f}_2) \neq 0$.

No hyperbolic–hyperbolic singularities can occur in hypersemitoric systems with S¹-symmetry, but hyperbolic–regular and hyperbolic–elliptic singularities are permitted (Hohloch et al., 2021).

This generalization is motivated by the need to extend semitoric classification to physically relevant systems exhibiting singularities not treated by previous theories, including rigid body dynamics, coupled angular momentum systems, and the quadratic spherical pendulum (Henriksen et al., 2023).

2. Relation to Toric and Semitoric Systems

Hypersemitoric systems extend toric and semitoric systems by relaxing restrictions on singularity types and the connectedness of fibers.

• Toric systems admit only nondegenerate elliptic singularities and are completely classified by Delzant polytopes; all symplectic invariants arise from the momentum map image and its affine structure (Efstathiou et al., 26 Nov 2024).
• Semitoric systems allow nondegenerate focus–focus points (arising via supercritical Hamiltonian–Hopf bifurcations), but exclude hyperbolic blocks and degenerate singularities (Alonso et al., 2019, Palmer et al., 2019). Their classification uses invariants including the Taylor series invariant, polygon invariant, height invariant, and twisting index.
• Hypersemitoric systems further admit hyperbolic blocks and parabolic degeneracies (often born in subcritical Hamiltonian–Hopf bifurcations), producing extra topological phenomena (flaps, pleats, swallowtails) and possibly disconnected fibers (Henriksen et al., 2023, Efstathiou et al., 26 Nov 2024).

A key result (attributed to Hohloch and Palmer (Hohloch et al., 2021, Palmer, 6 Oct 2025)) is that every compact Hamiltonian S¹-space (classified by Karshon graphs) can be lifted to a hypersemitoric system, even when no semitoric lifting is possible.

3. Singularities and Bifurcation Phenomena

Hypersemitoric systems feature a rich spectrum of singularities:

• Nondegenerate types: elliptic–regular, elliptic–elliptic, focus–focus, hyperbolic–regular, hyperbolic–elliptic.
• Parabolic singularities: characterized by a local normal form $q_1 = x^2 - y^3 + \lambda y,\ q_2 = \lambda$, representing a “cusp” (Henriksen et al., 2023).
• Bifurcation phenomena:
  • Supercritical Hamiltonian–Hopf bifurcations (semitoric): yield focus–focus points.
  • Subcritical Hamiltonian–Hopf bifurcations (hypersemitoric): yield flaps/pleats and allow parabolic points; the momentum map image may acquire branchings (regions bounded by curves of regular and elliptic/parabolic values, sometimes with disconnected fibers) (Gullentops et al., 2023).

The presence of S¹-symmetry (properness and effectiveness of J) constrains the singularity types and allows effective quotienting to paper fiber components via decorated graphs or bouquets (Gullentops et al., 2023).

4. Classification and Affine Invariant

The classification of hypersemitoric systems requires refinement beyond the polygon invariants of toric and semitoric settings:

• Affine invariant: Introduced in (Efstathiou et al., 26 Nov 2024), generalizing the Delzant and semitoric polytope invariants. The construction uses “straightened” action–angle coordinates and vertical cuts along critical values so that each connected fiber component is categorized by piecewise-affine jump relations. For vertical coordinates, if $f: F(M) \setminus (\text{cuts}) \to \mathbb{R}^2$, then

$$\lim_{x \to c^-} df(x, y) = M_c \cdot \lim_{x \to c^+} df(x, y) \quad \text{with} \quad M_c = \begin{bmatrix} 1 & 0 \ k(c) & 1 \end{bmatrix},$$

where $k(c)$ depends on the degeneracies and isotropy data. The resulting invariant is a rational convex set often with holes or indentations reflecting disconnected fibers (e.g., flaps/pleats).

• Combinatorial invariants: Quotienting hyperbolic–regular leaves by S¹-action yields decorated graphs, encoding crossings, marked points, and component information (“generalized bouquets”) (Gullentops et al., 2023). The unfolded bifurcation diagram $t: U \to F(M)$ with $|t^{-1}(r)|$ tracking fiber component count records data omitted from momentum map images.

A plausible implication is that a complete classification scheme will combine the affine invariants with bouquet-type combinatorial objects, mirroring the roles of moment polytopes and molecule invariants in Fomenko–Zieschang theory (Gullentops et al., 2023, Henriksen et al., 2023).

5. Examples and Explicit System Construction

Several explicit constructions illustrate the breadth of hypersemitoric systems:

• Quadratic spherical pendulum: Exhibits singular fibers outside the toric/semitoric classes (Henriksen et al., 2023).
• Euler and Lagrange tops: For generic Casimir values, dynamics involve hyperbolic/parabolic singularities (Henriksen et al., 2023).
• Coupled angular momenta and spin oscillator systems: By adjusting Hamiltonian parameters, transitions between semitoric and hypersemitoric regimes occur; novel bifurcations produce flaps and $k$-stacked tori (with $k$ up to 13 in certain families) (Gullentops et al., 2023).
• CP² and Hirzebruch surfaces: One-parameter families can interpolate between toric, semitoric, and hypersemitoric types, with explicit affine invariants computed and plotted (Floch et al., 2023, Efstathiou et al., 26 Nov 2024).

The general construction method is to start from a well-understood toric or semitoric system and perturb the Hamiltonian, explicitly introducing bifurcations (nodal trades, Hamiltonian–Hopf bifurcations) to create focus–focus, parabolic, and hyperbolic singularities in a controlled manner (Floch et al., 2023). For any Hamiltonian S¹-space, such extensions exist and can be engineered algorithmically.

6. Topological and Symplectic Implications

Hypersemitoric systems provide building blocks for integrable surgeries and the construction of almost-toric fibrations, generalizing semitoric “faithful” subsystems (Hohloch et al., 2017). The existence of flaps and disconnected fibers prompts the use of layer-wise decomposition of the momentum map image and integrable system glueing techniques.

A key foundational result is the nonexistence of closed loops of hyperbolic–regular points in the image (the images are always intervals), ensuring controlled topological behavior (Hohloch et al., 2021). Blow-up and blow-down operations can be tracked through changes in Karshon graph data and the corresponding affine invariants (Hohloch et al., 2021, Floch et al., 2023).

From the perspective of spectral theory, these invariants are expected to reflect in quantum/spectral properties, with implications for quantum inverse spectral theory in systems with degenerate or exotic singularities (Hohloch et al., 2017).

7. Research Directions and Open Problems

Current challenges and active areas of research include:

• Completing the symplectic classification of hypersemitoric systems, including the design of complete invariants incorporating flaps, bouquets, and unfolded bifurcation diagrams (Efstathiou et al., 26 Nov 2024, Gullentops et al., 2023).
• Developing effective techniques for lifting arbitrary Hamiltonian S¹-spaces to hypersemitoric systems, and determining the minimal number and type of permissible singular degeneracies in such extensions (Hohloch et al., 2021, Palmer, 6 Oct 2025).
• Extending classification methods to higher-dimensional complexity-one systems and understanding connections to Lagrangian topology, mirror symmetry, and tropical geometry (Hohloch et al., 2017).
• Clarifying the role of parabolic singularities (genericity, stability under perturbation) and their impact on fiber connectivity and monodromy (Gullentops et al., 2023, Henriksen et al., 2023).
• Investigating quantum analogs and the relationship between classical affine invariants and semiclassical spectral data (Hohloch et al., 2017).

A plausible implication is that the affine invariant, once fully developed in conjunction with combinatorial (bouquet) data, will serve as a complete symplectic invariant for hypersemitoric systems, paralleling the roles of the Delzant and semitoric polygon invariants (Efstathiou et al., 26 Nov 2024).

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In summary, hypersemitoric systems serve as a comprehensive framework for four-dimensional integrable Hamiltonian systems with S¹-symmetry and controlled singularities. Their paper advances the classification of integrable systems, facilitates explicit constructions, and extends the reach of symplectic invariants and topological techniques into settings featuring exotic fiber phenomena, bifurcation structures, and mild degeneracies. The field is actively evolving, with foundational theorems in place and ongoing work addressing invariants, classification, and applications in mathematics and mathematical physics.