arXiv subjects

Ana Chavez Caliz

Publications and source records attributed to Ana Chavez Caliz.

4 recordsLinked to original sources

arXiv2026

Outer Contact Billiards

We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation. We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits.

math.DS↗

arXiv2026

Symplectic billiards as Minkowski billiards

We establish a connection between Minkowski billiards and symplectic billiards, two classes of dynamical systems that have been studied largely independently. We show that the Minkowski billiard map can be described in symplectic terms via reduction from the canonical symplectic structure on $V \times V^*$, and that symplectic billiards can be viewed as a ``square root'' of a symplectic version of Minkowski billiards. As an application, we recover several known results on symplectic billiards from the more general Minkowski setting, and extend some of them to higher dimensions and to periodic orbits of even period. In particular, we prove the existence of at least $(r-1)(n-1)$ $2r$-periodic symplectic billiard orbits in dimension $2n$.

math.DS↗

arXiv2025

Outer symplectic billiards

A submanifold of the standard symplectic space determines a partially defined, multi-valued symplectic map, the outer symplectic billiard correspondence. Two points are in this correspondence if the midpoint of the segment connecting them is on the submanifold, and this segment is symplectically orthogonal to the tangent space of the submanifold at its midpoint. This is a far-reaching generalization of the outer billiard map in the plane; the particular cases, when the submanifold is a closed convex hypersurface or a Lagrangian submanifold, were considered earlier. Using a variational approach, we establish the existence of odd-periodic orbits of the outer symplectic billiard correspondence. On the other hand, we give examples of curves in 4-space which do not admit 4-periodic orbits at all. If the submanifold satisfies certain conditions (which are always satisfied if its dimension is at least half of the ambient dimension) we prove the existence of two $n$-reflection orbits connecting two transverse affine Lagrangian subspaces for every $n\geq1$. In addition, for every immersed closed submanifold, the number of single outer symplectic billiard ``shots" from one affine Lagrangian subspace to another is no less than the number of critical points of a smooth function on this submanifold. We study, in detail, the behavior of this correspondence when the submanifold is a curve or a Lagrangian submanifold. For Lagrangian submanifolds in 4-dimensional space we present a criterion for the outer symplectic billiard correspondence to be an actual map. We show, in every dimension, that if a Lagrangian submanifold has a cubic generating function, then the outer symplectic billiard correspondence is completely integrable in the Liouville sense.

math.SG↗

arXiv2025

Outer symplectic billiard map at infinity

We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely, denote by $N$ the symplectic polar dual of the symmetrization $M\ominus M$ of $M$. If we write $N$ as the unit level set of a 1-homogeneous function $H$, then the difference between $T^2$ and the time-2-Hamiltonian flow of $H$ applied to a point $x$ is smaller than $c/|x|$ for some constant $c$ depending only on $M$. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows not faster than $\sqrt{k}$ in the number of iterations. Finally, we prove that a $k$-periodic orbit needs to be close, in terms of $k$, to $M$.

math.SG↗