On the number of periodic orbits in billiards with flat points
A billiard is a map that describes the motion of a ball without mass in a closed region on the plane such that the collisions with the boundary are elastic. The region where the ball moves is the billiard table. In this thesis, we present the convex billiards (the boundary of the billiard table is a...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/360853 |
| Acceso en línea: | https://hdl.handle.net/2117/360853 |
| Access Level: | acceso abierto |
| Palabra clave: | Differentiable dynamical systems Ergodic theory Convex billiards Twist maps Are-preserving maps Generating function Action functional Invariant curves Caustics Periodic orbits Symmetry Sistemes dinàmics diferenciables Teoria ergòdica Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications Classificació AMS::28 Measure and integration::28D Measure-theoretic ergodic theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
| Sumario: | A billiard is a map that describes the motion of a ball without mass in a closed region on the plane such that the collisions with the boundary are elastic. The region where the ball moves is the billiard table. In this thesis, we present the convex billiards (the boundary of the billiard table is a convex Jordan curve of class $C^2$) and some of their properties. In particular, we will study caustics which are curves that often appear in the billiard problem and they are related with rotational invariant curves (RIC). Lazutkin and Douady proved that convex billiards have caustics if all points of the boundary have curvature strictly positive and the boundary has 6 continuous derivatives. Guktin and Katok, under the hypothesis of Lazutkin and Douady, give estimations of the size of the regions free of caustics contained inside the billiard table. Mather proves the non existence of caustics if there is a flat point in the boundary and Hubacher proves the non existence of caustics close to the boundary if the second derivative of the boundary is not continuous. Finally we do a numerical study about symmetric periodic orbits with odd period and we expose a conjecture that relates the number of symmetric periodic orbits with its period. |
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