arXiv subjects

Sonia Pinto-de-Carvalho

Publications and source records attributed to Sonia Pinto-de-Carvalho.

5 recordsLinked to original sources

arXiv2024

On the role of the surface geometry in convex billiards

This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties that have long been known for billiards on the plane are established. We prove the twist property and investigate conditions on the billiard for the existence and non existence of rotational invariant curves.

math.DS↗

arXiv2016

Periodic Orbits of Oval Billiards on Surfaces of Constant Curvature

In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show, in this new version, that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the Euclidean and the hyperbolic planes. For the proof of the density, the techniques we use for the Euclidean and hyperbolic cases, do not work for the spherical case, due to a constraint (the perimeter of the polygonal trajectory being a multiple of π). We finish this paper studying the stability of these nondegenerate orbits.

math.DS↗

arXiv2014

Billiard dynamics near an invariant horizontal circle

Sufficiently differentiable oval billiards always have invariant rotational curves, but there are only two types of ovals with an invariant horizontal circle in its phase-space: the constant width ovals and some very special symmetric curves. In this work we study the dynamics near the horizontal circle for the billiard map of those two types of ovals, and show that the horizontal circle is approached, from both sides, by other invariant rotational curves. We will also describe some dynamical consequences.

math.DS↗

arXiv2011

Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

math.DS↗

arXiv2011

Limit Sets of Convex non Elastic Billiards

Inspired by the work of Pujals and Sambarino on dominated splitting, we present billiards with a modified reflection law which constitute simple examples of dynamical systems with limit sets with dominated splitting and where the dynamics is a rational or irrational rotation.

math.DS↗