Outer Symplectic Billiard Map

• Outer symplectic billiard map is a generalized symplectic correspondence defined by a midpoint condition and chord orthogonality, connecting submanifold geometry with dynamical systems.
• It extends classical outer billiard maps to higher dimensions by incorporating convex curves, Lagrangian submanifolds, and hypersurfaces, revealing rich periodic orbit behavior.
• The dynamics leverage variational principles, symplectic invariants, and volume preservation to establish integrability and rigidity phenomena in both finite and infinite phase spaces.

The outer symplectic billiard map is a family of symplectic correspondences and (partially defined) maps associated with submanifolds in symplectic vector spaces, generalizing the two-dimensional outer billiard map and encompassing crucial connections to convex geometry, higher-dimensional symplectic geometry, and dynamical systems. In this setting, the outer symplectic billiard map is defined such that two points are dual-related if the midpoint lies on the submanifold, and the chord joining them is symplectically orthogonal to the tangent space at that midpoint (Albers et al., 12 Sep 2024). Various special cases (convex curves, convex hypersurfaces, Lagrangian submanifolds) correspond to previously studied outer billiard or dual billiard phenomena, while the most general setting yields rich and unexplored dynamics on both finite and infinite phase spaces.

1. Fundamental Definition and General Structure

Let $V = \mathbb{R}^{2d}$ with standard symplectic form $\omega = \sum_{i=1}^d dx_i \wedge dy_i$, and let $M \subset V$ be an immersed submanifold. The outer symplectic billiard correspondence is the relation on $V$ such that points $z, z' \in V$ are related if

• the midpoint $Q = \frac{z + z'}{2}$ lies on $M$, and
• the vector $z' - z$ belongs to the symplectic orthogonal complement $T_Q^{\omega} M$, i.e.,

$$\omega(z' - z, \xi) = 0 \quad \forall\, \xi \in T_Q M$$

For immersed closed submanifolds, this relation is typically multi-valued and only partially defined (i.e., the chord, midpoint, and orthogonality conditions may admit several or zero solutions for $z'$ given $z$).

This construction specializes:

• To the classical planar outer billiard for $d=1$, $M$ a strictly convex curve and $\omega$ the standard area form.
• To “Lagrangian outer billiards” when $M$ is a Lagrangian submanifold (Fuchs et al., 2015).
• To higher-dimensional convex hypersurfaces, with the characteristic direction of $\ker \omega|_{T_Q M}$ dictating the correspondence (Albers et al., 12 Sep 2024).

2. Symplectic Properties and Invariant Structures

The outer symplectic billiard correspondence is (in the restricted setting to an open dense subset) a symplectic correspondence. In the planar case, when the correspondence is a map, it is area-preserving; in higher dimensions, when $M$ is Lagrangian, it is a symplectomorphism (locally) (Fuchs et al., 2015, Albers et al., 12 Sep 2024). For hypersurfaces and general submanifolds, the correspondence is a symplectic relation (its graph is a Lagrangian submanifold of $V \times V$ with $\omega \oplus (-\omega)$).

If $M$ is a Lagrangian submanifold in $\mathbb{R}^{2n}$ and locally given as the graph $L = \{(q, p = \nabla F(q))\}$ for some generating function $F(q)$, the outer symplectic billiard correspondence is locally a symplectomorphism. In coordinates, the relation takes $(x, y)$ and $(\bar{x}, \bar{y})$ such that $$\left(\frac{x + \bar{x}}{2}, \frac{y + \bar{y}}{2}\right) \in L$$ and $\bar{y} - y \in \nu_q(L)$, where $\nu_q(L)$ is the conormal bundle at $q$ (Fuchs et al., 2015).

The existence of symplectic invariants and the preservation of symplectic volume are fundamental to the paper of such systems, underpinning their dynamical richness.

3. Variational Formulation and Existence of Periodic Orbits

The orbits and periodic points of the outer symplectic billiard correspondence are characterized using a variational principle. For any odd $n \geq 3$, define

$$F(Q_1, \ldots, Q_n) = 2 \sum_{1 \leq i < j \leq n} (-1)^{i+j-1} \omega(Q_i, Q_j)$$

on $M^n$. The critical points of $F$ correspond to $n$-periodic billiard configurations (where every consecutive chord obeys the midpoint and orthogonality constraints) (Albers et al., 12 Sep 2024). For an immersed closed $M$, the existence of such critical points is guaranteed for all odd $n$.

Furthermore, for any pair of transverse affine Lagrangian subspaces and any $n \geq 1$, assuming that $M$ satisfies a certain "largeness" condition (dimension at least $d$), there exist at least two nondegenerate $n$-reflection orbits from one Lagrangian to another (Albers et al., 12 Sep 2024).

For Lagrangian $M$ with a cubic generating function ($F$ homogeneous cubic), the system is completely integrable in the Liouville sense; the coordinates

$G_i(Q, P) = P_i - \partial_{Q_i} F(Q)$

are $n$ independent Poisson-commuting integrals, invariant under the billiard correspondence (Albers et al., 12 Sep 2024).

4. Special Cases: Curves, Lagrangian Manifolds, and Absence of Periodic Orbits

For $M$ a smooth, closed, symplectically convex curve (i.e., $\omega(\gamma', \gamma'') > 0$), the map is well-defined and locally of multiplicity 2 on a neighborhood outside a “wall” (a singular hypersurface in the ambient space) (Albers et al., 12 Sep 2024). However, not every period is realized: for instance, the Chebyshev curve $\gamma(t) = (\cos t, \sin t, \cos 2t, \sin 2t)$ in $\mathbb{R}^4$ admits no nondegenerate 4-periodic outer symplectic billiard orbits; this is due to constraints on the possible midpoints and associated trigonometric polynomial root counts.

When $M$ is Lagrangian and given by a cubic generating function, periodic and connecting orbits are abundant, the correspondence is as regular as the cubic structure allows, and the system is integrable (Albers et al., 12 Sep 2024).

5. Generalizations and Dynamics in Geometric Settings

Analogous constructions extend to curved geometric contexts:

• In three-dimensional space forms (Euclidean, spherical, hyperbolic), the outer billiard on the space of oriented geodesics—equipped with a Kähler structure via the Killing form—yields a symplectic (or Poisson, in the Euclidean case) correspondence (Godoy et al., 2021).
• For hypersurfaces in the complex hyperbolic plane $\mathbb{C}H^2$, the characteristic rays of the restricted symplectic form provide a double geodesic foliation of the exterior, on which the outer billiard map is both a diffeomorphism and a symplectomorphism (Godoy et al., 10 Mar 2025).
• In the context of tangent ray foliations, the outer billiard map is constructed using bifoliating vector fields whose eigenvalue properties control the regularity and volume preservation of the correspondence (Godoy et al., 2022).

6. Rigidity, Integrability, and Invariant Hypersurfaces

The existence of rotationally invariant periodic orbits (e.g., a family of 4-periodic orbits on a hypersurface) imposes strong restrictions on $M$.

• In dimension two, the only $C^1$-smooth strictly convex planar curves admitting a one-parameter family of 4-periodic orbits under the outer billiard map are boundaries of unit balls of Radon norms; in higher dimensions, symplectically self-polar convex bodies are forced (Berezovik et al., 21 Jan 2025).
• For planar symplectic billiards (using the area generating function), total integrability (foliation by invariant curves), or rational integrability (invariant curves of $q$-periodic orbits for each $q \geq 3$), is rigid: in a strong $C^\infty$ topology, only ellipses admit such integrable dynamics (Bialy, 2023, Tsodikovich, 15 Jan 2025, Baracco et al., 2023).
• For higher-dimensional self-polar domains, the outer symplectic billiard map admits invariant hypersurfaces comprised entirely of, for example, 4-periodic orbits, providing genuine examples of invariant sets outside classical ellipsoids (Berezovik et al., 21 Jan 2025).

7. Asymptotics and Dynamics at Infinity

For points far from the given domain $M$, the second iterate $T^2$ of the outer symplectic billiard map is approximated by the time-2 flow of a Hamiltonian system, where the Hamiltonian $H$ is a 1-homogeneous function whose level set defines the symplectic polar dual of the symmetrization $M \ominus M$ (Albers et al., 21 Aug 2025). The error in this approximation decays as $c/|x|$ for $|x| \rightarrow \infty$. Escaping orbits have distances from the origin growing no faster than $\sqrt{k}$ in $k$ iterates, and periodic orbits of period $k$ are confined to be within an explicit radius depending on $k$ and $M$, assuring that periodic behavior is localized near the defining domain.

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The outer symplectic billiard map thus unifies rich branches of symplectic geometry, variational mechanics, and dynamical systems, providing a framework for exploring the interplay of local geometric structure, global symplectic invariance, the existence (or obstruction) of periodic orbits, and rigidity phenomena across classical and higher-dimensional settings (Fuchs et al., 2015, Godoy et al., 2021, Godoy et al., 2022, Baracco et al., 2023, Bialy, 2023, Albers et al., 12 Sep 2024, Tsodikovich, 15 Jan 2025, Berezovik et al., 21 Jan 2025, Godoy et al., 10 Mar 2025, Albers et al., 21 Aug 2025).