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

By V. Ginzburg, Basak Z. Gurel. 2003

58 citations recorded by the source corpus.

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum. The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a non-trivial contractible one-periodic orbit, when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb4be1888fa11c59d9824072b2bc24cbf554cf79

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Citations in the corpus

Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, methodology): “In fact, it suffices to carry out the calculation for M D R , which is done in [26], for then, by Example 5.”
The Palais-Smale condition on contact type energy levels for convex Lagrangian systems (2003, unknown): “9 of [11].”
Quantitative symplectic geometry (2005, unknown): “Since then, lots of new capacities have been defined [16, 30, 32, 44, 49, 59, 60, 90, 99] and they were further studied in [1, 2, 8, 9, 17, 26, 21, 28, 31, 35, 37, 38, 41, 42, 43, 46, 48, 50, 52, 56, 57, 58, 61, 62, 63, 64, 65, 66, 68, 74, 75, 76, 88, 89, 91, 92, 94, 97, 98].”
Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, methodology): “We use a version of symplectic homology associated to negative compactly supported Hamiltonians as in [3; 10; 25], see Section 4 below for the definition and conventions about the grading.”
The Conley Conjecture (2006, methodology): “an is displaceable is less important and, in some cases, not required at all. In fact, what appears to matter is that the set where the maximum is attained is relatively small (e.g., symplectic as in [GG] or displaceable as in [LMc] or just isolated as in [Gi, KL]). It is one of these methods, combi”
Contact Homology, Capacity and Non-Squeezing in R^2n x S^1 via Generating Functions (2009, methodology): “rshaped domains in R2n× S1. This is a special instance of the Symplectic Field Theory, and is related to a version of the ﬁltered symplectic homology of domains in R2n as used in [BPS03], [CGK04] and [GG04]. We will present here a proof of the same results using generating functions instead of holom”
Applications of Hofer's geometry to Hamiltonian dynamics (2003, background): “Following [27], [28] and [38], [53], [17] we define for each subset A of (V , ω) the π1-sensitive Hofer–Zehnder capacity”