Abstract
We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results.
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References
P. Biran and K. Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel Journal of Mathematics 127 (2002), 221–244.
P. Biran, L. Polterovich and D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Mathematical Journal 119 (2003), 65–118.
K. Cieliebak, Subcritical Stein manifolds are split, preprint, math.DG/0204351.
K. Cieliebak, A. Floer and H. Hofer, Symplectic homology. II. A general construction, Mathematische Zeitschrift 218 (1995), 103–122.
K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology. II. Stability of the action spectrum, Mathematische Zeitschrift 223 (1996), 27–45.
K. Cieliebak, V. Ginzburg and E. Kerman, Symplectic homology and periodic orbits near symplectic submanifolds, Commentarii Mathematici Helvetici 79 (2004), 554–581.
G. Contreras, The Palais-Smale condition for contact type energy levels for convex lagrangian systems, math.DS/0304238.
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé’s critical values, Geometric and Functional Analysis 8 (1998), 788–809.
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, The Palais-Smale condition and Mañé’s critical values, Annales de l’Institut Henri Poincaré 1 (2000), 655–684.
G. Contreras, L. Macarini and G. P. Paternain, Periodic orbits for exact magnetic flows on surfaces, International Mathematics Research Notices (2004), 361–387.
P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric, Annales Scientifiques de l’École Normale Supérieure 27 (1994), 611–660.
Ya. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, International Journal of Mathematics 1 (1990), 29–46.
Ya. Eliashberg, Symplectic geometry of plurisubharmonic functions, With notes by Miguel Abreu. NATO Advanced Science Institutes Ser. C Math. Phys. Sci., 488, Gauge theory and symplectic geometry (Montreal, 1995), Kluwer Acad. Publ., Dordrecht, 1997, pp. 49–67.
Ya. Eliashberg and M. Gromov, Convex symplectic manifolds, in Several Complex Variables and Complex Geometry, Proceedings, Summer Research Institute, Santa Cruz, 1989, Part 2, (E. Bedford et al., eds.), Proceedings of Symposia in Pure Mathematics 52, American Mathematical Society, Providence, RI, 1991, pp. 135–162.
A. Floer, A relative Morse index for the symplectic action, Communications on Pure and Applied Mathematics 41 (1988), 393–407.
A. Floer, The unregularized gradient flow of the symplectic action, Communications on Pure and Applied Mathematics 41 (1988), 775–813.
A. Floer, Morse theory for Lagrangian intersections, Journal of Differential Geometry 28 (1988), 513–547.
A. Floer, Witten’s complex and infinite-dimensional Morse theory, Journal of Differential Geometry 30 (1989), 207–221.
A. Floer and H. Hofer, Symplectic homology. I Open sets in ℂ n, Mathematische Zeitschrift 215 (1994), 37–88.
A. Floer, H. Hofer and D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Mathematical Journal 80 (1995), 251–292.
U. Frauenfelder, Floer homology of symplectic quotients and the Arnold-Givental conjecture, Diss. ETH No. 14981. Zürich 2003. Available at http://e-collection.ethbib.ethz.ch/ecol-pool/diss/fulltext/eth14981.pdf
U. Frauenfelder, V. Ginzburg and F. Schlenk, Energy capacity inequalities via an action selector, in Geometry, spectral theory, groups, and dynamics, Contemporary Mathematics 387, American Mathematical Society, Providence, RI, 2005, pp. 129–152.
U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, math.SG/0303282.
U. Frauenfelder and F. Schlenk, Slow entropy and symplectomorphisms of cotangent bundles, math.SG/0404017.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983.
V. Ginzburg, On closed trajectories of a charge in a magnetic field, An application of symplectic geometry. Contact and symplectic geometry (Cambridge, 1994), Publications of the Newton Institute 8, Cambridge University Press, Cambridge, 1996, pp. 131–148.
V. Ginzburg, Hamiltonian dynamical systems without periodic orbits, Northern California Symplectic Geometry Seminar, American Mathematical Society Translations Ser. 2, 196, American Mathematical Society, Providence, RI, 1999, pp. 35–48.
V. Ginzburg, The Weinstein conjecture and theorems of nearby and almost existence, The breadth of symplectic and Poisson geometry, Progress in Mathematics 232, Birkhäuser Boston, Boston, MA, 2005, pp. 139–172.
V. Ginzburg and B. Gürel, Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Mathematical Journal 123 (2004), 1–47.
V. Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimensions greater than two, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemporary Mathematics 246, American Mathematical Society, Providence, RI, 1999, pp. 113–121.
V. Ginzburg and E. Kerman, Periodic orbits of Hamiltonian flows near symplectic extrema, Pacific Journal of Mathematics 206 (2002), 69–91.
D. Gromoll and J. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bulletin of the American Mathematical Society 77 (1971), 545–552.
M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 (1985), 307–347.
D. Hermann, Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary, Duke Mathematical Journal 103 (2000), 335–374.
H. Hofer and C. Viterbo, The Weinstein conjecture in cotangent bundles and related results, Annali della Scuola Normale Superiore di Pisa Cl. Sci. IV 15 (1988), 411–445.
H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Inventiones Mathematicae 90 (1987), 1–9.
H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 405–427.
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, Basel, 1994.
E. Kerman, Periodic orbits of Hamiltonian flows near symplectic critical submanifolds, International Mathematics Research Notices, 1999, pp. 953–969.
V. V. Kozlov, Calculus of variations in the large and classical mechanics, Russian Mathematical Surveys 40 (1985), 37–71.
F. Laudenbach and J.-C. Sikorav, Hamiltonian disjunction and limits of Lagrangian submanifolds, International Mathematics Research Notices, 1994, pp. 161–168.
F. Lalonde and D. McDuff, The geometry of symplectic energy, Annals of Mathematics 141 (1995), 349–371.
F. Lalonde and L. Polterovich, Symplectic diffeomorphisms as isometries of Hofer’s norm, Topology 36 (1997), 711–727.
G. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu Journal of Mathematics 52 (1998), 331–351 and 54 (2000), 181–182.
L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Communications in Contemporary Mathematics 6 (2004), 913–945.
L. Macarini and F. Schlenk, A refinement of the Hofer-Zehnder theorem on the existence of closed characteristics near a hypersurface, The Bulletin of the London Mathematical Society 37 (2005), 297–300.
D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Clarendon Press, New York, 1995, 425 pp.
D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, AMS Colloquium Publications 52. American Mathematical Society, Providence, RI, 2004.
S. P. Novikov, The Hamiltonian formalism and a many-valued analogue of Morse theory, Russian Mathematical Surveys 37 (1982), 1–56.
Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Communications in Analysis and Geometry 7 (1999), 1–54.
Y.-G. Oh, Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian Journal of Mathematics 6 (2002), 579–624.
Y.-G. Oh, Spectral invariants,analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Mathematical Journal 130 (2005), 199–295.
O. Osuna, Periodic orbits of weakly exact magnetic flows, 2005, preprint.
G. Paternain, Magnetic rigidity of horocycle flows, Pacific Journal of Mathematics 225 (2006), 301–323.
G. Paternain and M. Paternain, Critical values of autonomous Lagrangian systems, Commentarii Mathematici Helvetici 72 (1997), 481–499.
G. Paternain, L. Polterovich and K. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Moscow Mathematical Journal 3 (2003), 593–619, 745.
S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton Inst. 8, Cambridge University Press, Cambridge, 1996, pp. 171–200.
L. Polterovich, An obstacle to non-Lagrangian intersections, in The Floer Memorial Volume, Progr. Math. 133, Birkhäuser, Basel, 1995, pp. 575–586.
L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math 1998, Extra Vol. II, pp. 401–410.
L. Polterovich, Growth of maps, distortion in groups and symplectic geometry, Inventiones Mathematicae 150 (2002), 655–686.
D. Salamon, Lectures on Floer homology, Symplectic geometry and topology (Park City, UT, 1997), IAS/Park City Math. Ser. 7, American Mathematical Society, Providence, RI, 1999, pp. 143–229.
D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Communications on Pure and Applied Mathematics 45 (1992), 1303–1360.
F. Schlenk, Applications of Hofer’s geometry to Hamiltonian dynamics, Commentarii Mathematici Helvetici 81 (2006), 105–121.
M. Schwarz, Morse homology, Progress in Mathematics 111, Birkhäuser Verlag, Basel, 1993.
M. Schwarz, Cohomology operations from S 1-cobordisms in Floer homology, Ph. D. thesis, ETH Zürich, Diss. ETH No. 11182, 1995. http://www.math.unileipzig.de/-schwarz/diss.pdf.
M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific Journal of Mathematics 193 (2000), 419–461.
C. Viterbo, Symplectic topology as the geometry of generating functions, Mathematische Annalen 292 (1992), 685–710.
C. Viterbo, Functors and computations in Floer homology with applications. I, Geometric and Functional Analysis 9 (1999), 985–1033.
S. Yau, On the fundamental group of compact manifolds of non-positive curvature, Annals of Mathematics 93 (1971), 579–585.
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Partially supported by the Swiss National Foundation.
Supported by the Swiss National Foundation and the von Roll Research Foundation.
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Frauenfelder, U., Schlenk, F. Hamiltonian dynamics on convex symplectic manifolds. Isr. J. Math. 159, 1–56 (2007). https://doi.org/10.1007/s11856-007-0037-3
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DOI: https://doi.org/10.1007/s11856-007-0037-3