951 research outputs found
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
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
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
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 -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
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