Introducing symplectic billiards

In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure

Periodic bounce orbits of prescribed energy

We prove the existence of periodic bounce orbits of prescribed energy on an open bounded domain in Euclidean space. We derive explicit bounds on the period and the number of bounce points.Comment: 18 pages, 3 figures; v2: improved main result, removed typo

Spectral Invariants in Rabinowitz Floer homology and Global Hamiltonian perturbations

Spectral invariant were introduced in Hamiltonian Floer homology by Viterbo, Oh, and Schwarz. We extend this concept to Rabinowitz Floer homology. As an application we derive new quantitative existence results for leaf-wise intersections. The importance of spectral invariants for the presented application is that spectral invariants allow us to derive existence of critical points of the Rabinowitz action functional even in degenerate situations where the functional is not Morse.Comment: 29 page

Cup-length estimates for leaf-wise intersections

We prove that on a restricted contact type hypersurface the number of leaf-wise intersections is bounded from below by a certain cup-length.Comment: 13 pages, 4 figures; v2: minor modification

Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.Comment: 46 pages, 2 figures; v2: major revisio

On the Weinstein conjecture in higher dimensions

The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.Comment: 5 page

The space of linear anti-symplectic involutions is a homogenous space

In this note we prove that the space of linear anti-symplectic involutions is the homogenous space Gl(n,\R)\Sp(n). This result is motivated by the study of symmetric periodic orbits in the restricted 3-body problem.Comment: 5 page

Cuplength estimates in Morse cohomology

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms, and solutions to a Dirac-type equation.Comment: 25 pages, 1 figure, appeared online in Journal of Topology and Analysi