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A symplectic proof of a theorem of Franks

Part of: Hamiltonian and Lagrangian mechanics Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry

Published online by Cambridge University Press:  11 October 2012

Brian Collier ,
Ely Kerman ,
Benjamin M. Reiniger ,
Bolor Turmunkh  and
Andrew Zimmer
Brian Collier
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: collier3@illinois.edu)
Ely Kerman
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: ekerman@illinois.edu)
Benjamin M. Reiniger
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: reinige1@illinois.edu)
Bolor Turmunkh
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: turmunk1@illinois.edu)
Andrew Zimmer
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: aazimmer@umich.edu)
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Abstract

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A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area-preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we re-prove Franks’ theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorphisms.

Keywords

periodic orbitsHamiltonian flowsFloer homology

MSC classification

Secondary: 53D40: Floer homology and cohomology, symplectic aspects 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 70H12: Periodic and almost periodic solutions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1969 - 1984
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AK70]Anosov, D. V. and Katok, A. B., New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč 23 (1970), 3–36 (in Russian).Google Scholar
[Arn89]Arnold, V. I., Mathematical methods of classical mechanics (Springer, New York, 1989).Google Scholar
[BH12]Bramham, B. and Hofer, H., First steps towards a symplectic dynamics, Surv. Differ. Geom. 17 (2012), 127–178.Google Scholar
[BH01]Burghelea, D. and Haller, S., Non-contractible periodic trajectories of symplectic vector fields, Floer cohomology and symplectic torsion, Preprint (2001), arXiv:math/0104013.Google Scholar
[CZ84]Conley, C. C. and Zehnder, E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207–253.Google Scholar
[Cot09]Cotton-Clay, A., Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol. 13 (2009), 2619–2674.CrossRefGoogle Scholar
[Cot10]Cotton-Clay, A., A sharp bound on fixed points of surface symplectomorphisms in each mapping class, Preprint (2010), arXiv:1009.0760.Google Scholar
[DS94]Dostoglou, S. and Salamon, D. A., Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994), 581–640.Google Scholar
[FH77]Fathi, A. and Herman, M. R., Existence de difféomorphismes minimaux, Astérisque 49 (1977), 37–59.Google Scholar
[FK04]Fayad, B. and Katok, A. B., Constructions in elliptic dynamics, Ergod. Th. & Dynam. Sys. 24 (2004), 1477–1520.CrossRefGoogle Scholar
[Flo88]Floer, A., Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513–547.Google Scholar
[Fra92]Franks, J., Geodesics on S 2 and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), 403–418.CrossRefGoogle Scholar
[Fra96]Franks, J., Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math. 2 (1996), 1–19.Google Scholar
[FH03]Franks, J. and Handel, M., Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol. 7 (2003), 713–756.CrossRefGoogle Scholar
[Gin10]Ginzburg, V. L., The Conley conjecture, Ann. of Math. (2) 172 (2010), 1127–1180.Google Scholar
[GG09a]Ginzburg, V. L. and Gürel, B. Z., Action and index spectra and periodic orbits in Hamiltonian dynamics, Geom. Topol. 13 (2009), 2745–2805.Google Scholar
[GG09b]Ginzburg, V. L. and Gürel, B. Z., On the generic existence of periodic orbits in Hamiltonian dynamics, J. Mod. Dyn. 4 (2009), 595–610.Google Scholar
[GG10]Ginzburg, V. L. and Gürel, B. Z., Local Floer homology and the action gap, J. Symplectic Geom. 8 (2010), 323–357.Google Scholar
[GK10]Ginzburg, V. L. and Kerman, E., Homological resonances for Hamiltonian diffeomorphisms and Reeb flows, Int. Math. Res. Not. (2010), 53–68; doi:10.1093/imrn/rnp120.Google Scholar
[Hei12]Hein, D., The Conley conjecture for irrational symplectic manifolds, J. Symplectic Geom. 10 (2012), 183–202.CrossRefGoogle Scholar
[Hin09]Hingston, N., Subharmonic solutions of Hamiltonian equations on tori, Ann. of Math. (2) 170 (2009), 529–560.Google Scholar
[HZ94]Hofer, H. and Zehnder, E., Symplectic invariants and Hamiltonian dynamics (Birkhäuser, Basel, 1994).Google Scholar
[LO95]Lê, H. V. and Ono, K., Symplectic fixed points, the Calabi invariant and Novikov homology, Topology 34 (1995), 155–176.Google Scholar
[Le06a]Le Calvez, P., Periodic orbits of Hamiltonian homeomorphisms of surfaces, Duke Math. J. 133 (2006), 125–184.CrossRefGoogle Scholar
[Le06b]Le Calvez, P., From Brouwer theory to the study of homeomorphisms of surfaces, in Proc. int. congress of mathematicians, Madrid, Spain, 2006, vol. III (European Mathematical Society, Zürich, 2006), 77–98.Google Scholar
[Lon02]Long, Y., Index theory for symplectic paths with applications, Progress in Mathematics, vol. 207 (Birkhäuser, Basel, 2002).Google Scholar
[MS95]McDuff, D. and Salamon, D., Introduction to symplectic topology (Oxford University Press, New York, 1995).Google Scholar
[MS04]McDuff, D. and Salamon, D., J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
[SZ92]Salamon, D. and Zehnder, E., Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.Google Scholar
[Sei97a]Seidel, P, Floer homology and the symplectic isotopy problem, PhD thesis, University of Oxford (1997).Google Scholar
[Sei97b]Seidel, P., π 1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046–1095.CrossRefGoogle Scholar
[Sei02a]Seidel, P., Symplectic Floer homology and the mapping class group, Pacific J. Math. 206 (2002), 219–229.Google Scholar
[Sei02b]Seidel, P., Braids and symplectic four-manifolds with abelian fundamental group, Turkish J. Math. 26 (2002), 93–100.Google Scholar
[vdBGVW09]van den Berg, J.-B., Ghrist, R., Vandervorst, R. and Wojcik, W., Braid Floer homology, Preprint (2009), arXiv:0910.0647.Google Scholar