Growth and complexity in contact dynamics, contact topology and billiards
Project responsable Félix Schlenk
Team member Lucas Dahinden
Abstract We propose three projects on properties of Reeb flows and contact isotopies: 1. Lower bounds for topological entropy, that are known for a while for geodesic flows and now also for Reeb flows on spherizations, hold for all positive contact isotopies on spherizations. In a similar direction, the full Bott-Samelson theorem on the topology of manifolds all of whose geodesics are closed holds on spherizations for which there is a fiber sphere that can be moved to itself by a positive contact isotopy. 2. Positive topological entropy of all Reeb flows on a given contact manifold is known only for the very special class of spherizations and for a few contact 3-manifolds. This project of M. Alves aims to show positive entropy of all Reeb flows on large classes of contact manifolds, such as the boundaries of plumbings of cotangent bundles and connected sums thereof. 3. General billiards is concerned with billiards on Riemannian manifolds with boundaries, subject to general force fields. In this situation, we study the length of the shortest periodic billiard orbit, lower bounds on the number of such orbits, and the complexity of the billiard map. The main tools are variants of Floer homology and symplectic field theory: Rabinowitz-Floer homology, periodic and strip Legendrian contact homology, and wrapped Fukaya category.
Keywords Contact dynamics and topology
Type of project Fundamental research project
Research area Mathématiques
Method of financing FNS - Encouragement de projets (Div. I-III)
Status Completed
Start of project 1-10-2015
End of project 31-7-2018
Overall budget 403'367.00
Contact Félix Schlenk