scholarly journals Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles

2006 ◽  
Vol 14 (4) ◽  
pp. 617-630 ◽  
Author(s):  
César J. Niche ◽  
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Flows ◽  
Cotangent Bundles

scholarly journals The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

2016 ◽  
Vol 08 (03) ◽  
pp. 545-570 ◽  
Author(s):  
Luca Asselle ◽  
Gabriele Benedetti
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Critical Value ◽  
Closed Manifold ◽  
Hamiltonian Flow ◽  
Cotangent Bundles ◽  
Almost All

Let [Formula: see text] be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian [Formula: see text] and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if [Formula: see text] is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of [Formula: see text].

scholarly journals On periodic orbits in cotangent bundles of non-compact manifolds

2016 ◽  
Vol 14 (4) ◽  
pp. 1145-1173 ◽  
Author(s):  
J.B. van den Berg ◽  
F. Pasquotto ◽  
T. Rot ◽  
R.C.A.M. Vandervorst
Keyword(s):  
Periodic Orbits ◽  
Compact Manifolds ◽  
Cotangent Bundles

scholarly journals Periodic orbits of Hamiltonian flows near symplectic extrema

2002 ◽  
Vol 206 (1) ◽  
pp. 69-91 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Ely Kerman
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Flows

scholarly journals Controlled Reeb dynamics — Three lectures not in Cala Gonone

2019 ◽  
Vol 6 (1) ◽  
pp. 118-137
Author(s):  
Hansjörg Geiges
Keyword(s):  
Periodic Orbits ◽  
Group Actions ◽  
Main Theme ◽  
Contact Topology ◽  
Hamiltonian Flows ◽  
Weinstein Conjecture ◽  
Reeb Flows ◽  
And Topology

AbstractThese are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby–Wang bundles, that might be useful for other applications in contact topology.

Symplectic Banach–Mazur distances between subsets of ℂn

2020 ◽  
pp. 1-56
Author(s):  
Michael Usher
Keyword(s):  
Finite Dimension ◽  
Convex Bodies ◽  
The Other ◽  
Symplectic Homology ◽  
Hamiltonian Flows ◽  
Dimension Formula ◽  
Isometric Embeddings ◽  
Cotangent Bundles ◽  
Definition Of ◽  
General Method

Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of [Formula: see text] and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach–Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in [Formula: see text] admits quasi-isometric embeddings of [Formula: see text] for every finite dimension [Formula: see text]. Our constructions are obtained from a general method of constructing [Formula: see text]-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given [Formula: see text]-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from [J. Gutt and M. Usher, Symplectically knotted codimension-zero embeddings between domains in [Formula: see text], Duke Math. J. 168 (2019) 2299–2363].

scholarly journals Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles

2004 ◽  
Vol 123 (1) ◽  
pp. 1-47 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel
Keyword(s):  
Periodic Orbits ◽  
Cotangent Bundles
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