31 research outputs found
A backward -Lemma for the forward heat flow
The inclination or -Lemma is a fundamental tool in finite
dimensional hyperbolic dynamics. In contrast to finite dimension, we consider
the forward semi-flow on the loop space of a closed Riemannian manifold
provided by the heat flow. The main result is a backward -Lemma for
the heat flow near a hyperbolic fixed point . There are the following
novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow
versus flow. Thirdly, suitable adaption provides a new proof in the finite
dimensional case. Fourthly and a priori most surprisingly, our -Lemma
moves the given disk transversal to the unstable manifold backward in time,
although there is no backward flow. As a first application we propose a new
method to calculate the Conley homotopy index of .Comment: 31 pages, 6 figures. Comments most welcome. v2: Theorem 1.2 and Lemma
2.1 slightly improved, corrected typos. v3: minor modifications. To appear in
{\it Math. Ann.
Three approaches towards Floer homology of cotangent bundles
Consider the cotangent bundle of a closed Riemannian manifold and an almost
complex structure close to the one induced by the Riemannian metric. For
Hamiltonians which grow for instance quadratically in the fibers outside of a
compact set, one can define Floer homology and show that it is naturally
isomorphic to singular homology of the free loop space. We review the three
isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and
Abbondandolo-Schwarz (2004).
The theory is illustrated by calculating Morse and Floer homology in case of
the euclidean n-torus. Applications include existence of noncontractible
periodic orbits of compactly supported Hamiltonians on open unit disc cotangent
bundles which are sufficiently large over the zero section.Comment: 30 pages, 6 figures. To appear in J. Symplectic Geom. (Stare Jablonki
conference issue
The shift map on Floer trajectory spaces
In this article we give a uniform proof why the shift map on Floer homology
trajectory spaces is scale smooth. This proof works for various Floer
homologies, periodic, Lagrangian, Hyperk\"ahler, elliptic or parabolic, and
uses Hilbert space valued Sobolev theory.Comment: 32 pages, 3 figure
Floer homology and the heat flow
We study the heat flow in the loop space of a closed Riemannian manifold
as an adiabatic limit of the Floer equations in the cotangent bundle. Our main
application is a proof that the Floer homology of the cotangent bundle, for the
Hamiltonian function kinetic plus potential energy, is naturally isomorphic to
the homology of the loop space.Comment: 83 pages, 1 figure. We introduce a class of abstract perturbations in
order to achieve transversality. The argument carries over to this clas
An almost existence theorem for non-contractible periodic orbits in cotangent bundles
Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper
function bounded from below. Suppose the sublevel set {H<d} contains the zero
section and \alpha is a non-trivial homotopy class of free loops in M. Then for
almost every s>=d the level set {H=s} carries a periodic orbit z of the
Hamiltonian system (T^*M,\omega_0,H) representing \alpha.
Examples show that the condition that {H<d} contains M is necessary and
almost existence cannot be improved to everywhere existence.Comment: 9 pages, 4 figures. v2: corrected typo