Document Zbl 1208.53089

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Math.</a> <a href="/?q=in%3A273472" title="Articles in this Issue">96, No. 1, 85-100 (2011)</a>. </div> <div class="abstract">Summary: We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5851489">Cited in <strong>11</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A53D40" title="MSC2020">53D40</a> </td> <td class="space"> Symplectic aspects of Floer homology and cohomology </td> </tr><tr> <td class="mono"> 37J45 </td> <td class="space"> Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70H12" title="MSC2020">70H12</a> </td> <td class="space"> Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AConley+conjecture">Conley conjecture</a>; <a href="/?q=ut%3Aperiodic+orbits">periodic orbits</a>; <a href="/?q=ut%3AFloer+homology">Floer homology</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1208.53089" data-ciurl="/ci/05851489" data-biburl="/bibtex/05851489.bib" data-amsurl="/amsrefs/05851489.bib" data-xmlurl="/xml/05851489.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05851489.pdf" title="Zbl 1208.53089 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00013-010-0208-z" aria-label="DOI for “The Conley conjecture for the cotangent bundle”" title="10.1007/s00013-010-0208-z">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1006.0372"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/nc.svg" alt="Creative Commons NC license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abbondandolo A., Schwarz M.: On the Floer homology of cotangent bundles. Comm. Pure Appl. Math. 59, 254–316 (2006) · <a href="/1084.53074" class="nowrap">Zbl 1084.53074</a> · <a href="https://doi.org/10.1002/cpa.20090" class="nowrap">doi:10.1002/cpa.20090</a></td> </tr><tr> <td>[2]</td> <td class="space">Biran P., Polterovich L., Salamon D.: Propagation in Hamiltonian dynamics and relative symplectic homology. Duke Math. J. 119, 65–118 (2003) · <a href="/1034.53089" class="nowrap">Zbl 1034.53089</a> · <a href="https://doi.org/10.1215/S0012-7094-03-11913-4" class="nowrap">doi:10.1215/S0012-7094-03-11913-4</a></td> </tr><tr> <td>[3]</td> <td class="space">Conley C.C., Lecture at the University of Wisconsin, April 6, 1984</td> </tr><tr> <td>[4]</td> <td class="space">Conley C.C., Zehnder E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 37, 207–253 (1984) · <a href="/0559.58019" class="nowrap">Zbl 0559.58019</a> · <a href="https://doi.org/10.1002/cpa.3160370204" class="nowrap">doi:10.1002/cpa.3160370204</a></td> </tr><tr> <td>[5]</td> <td class="space">Franks J., Handel M.: Periodic points of Hamiltonian surface diffeomorphisms. Geom. Topol. 7, 713–756 (2003) · <a href="/1034.37028" class="nowrap">Zbl 1034.37028</a> · <a href="https://doi.org/10.2140/gt.2003.7.713" class="nowrap">doi:10.2140/gt.2003.7.713</a></td> </tr><tr> <td>[6]</td> <td class="space">V. L. Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, in: Contact and Symplectic Geometry, Ed.: C. B. Thomas, INI Publications, Cambridge University Press, Cambridge (1996), 131–148. · <a href="/0873.58034" class="nowrap">Zbl 0873.58034</a></td> </tr><tr> <td>[7]</td> <td class="space">Ginzburg V.L.: The Conley Conjecture. Ann. of Math. 172, 1127–1180 (2010) · <a href="/1228.53098" class="nowrap">Zbl 1228.53098</a> · <a href="https://doi.org/10.4007/annals.2010.172.1129" class="nowrap">doi:10.4007/annals.2010.172.1129</a></td> </tr><tr> <td>[8]</td> <td class="space">Ginzburg V.L.: Gürel B. Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles. Duke Math. J. 123, 1–47 (2004) · <a href="/1066.53138" class="nowrap">Zbl 1066.53138</a></td> </tr><tr> <td>[9]</td> <td class="space">Ginzburg V.L., Gürel B.Z.: Action and index spectra and periodic orbits in Hamiltonian dynamics. Geom. Topol. 13, 2745–2805 (2009) · <a href="/1172.53052" class="nowrap">Zbl 1172.53052</a> · <a href="https://doi.org/10.2140/gt.2009.13.2745" class="nowrap">doi:10.2140/gt.2009.13.2745</a></td> </tr><tr> <td>[10]</td> <td class="space">Ginzburg V.L., Gürel B.Z.: Local Floer homology and the action gap. J. Sympl. 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Long, Index Theory for Symplectic Path with Applications, Progress in Math. 207, Birkhäuser Verlag, Basel, 2002.</td> </tr><tr> <td>[15]</td> <td class="space">Lu G.: The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems. J. Funct. Anal. 256, 2967–3034 (2009) · <a href="/1184.53080" class="nowrap">Zbl 1184.53080</a> · <a href="https://doi.org/10.1016/j.jfa.2009.01.001" class="nowrap">doi:10.1016/j.jfa.2009.01.001</a></td> </tr><tr> <td>[16]</td> <td class="space">M. Mazzucchelli, The Lagrangian Conley Conjecture, to appear in Comment. Math. Helv. · <a href="/1209.37067" class="nowrap">Zbl 1209.37067</a></td> </tr><tr> <td>[17]</td> <td class="space">D. A. Salamon, Lectures on Floer homology, in: Symplectic Geometry and Topology, Eds.: Y. Eliashberg and L. Traynor, IAS/Park City Mathematics series, 7 (1999), pp. 143–230. · <a href="/1031.53118" class="nowrap">Zbl 1031.53118</a></td> </tr><tr> <td>[18]</td> <td class="space">Salamon D., Weber J.: Floer homology and the heat flow. Geom. Funct. Anal. 16, 1050–1138 (2006) · <a href="/1118.53056" class="nowrap">Zbl 1118.53056</a> · <a href="https://doi.org/10.1007/s00039-006-0577-4" class="nowrap">doi:10.1007/s00039-006-0577-4</a></td> </tr><tr> <td>[19]</td> <td class="space">Salamon D., Zehnder E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45, 1303–1360 (1992) · <a href="/0766.58023" class="nowrap">Zbl 0766.58023</a> · <a href="https://doi.org/10.1002/cpa.3160451004" class="nowrap">doi:10.1002/cpa.3160451004</a></td> </tr><tr> <td>[20]</td> <td class="space">P. Seidel, A biased view of symplectic cohomology, Lecture notes from Current Developments in Mathematics, 2006, 211–253, Int. Press, Somerville, MA, 2008.</td> </tr><tr> <td>[21]</td> <td class="space">Usher M.: Floer homology in disk bundles and symplectically twisted geodesic flows. J. Mod. Dyn. 3, 61–101 (2009) · <a href="/1186.53099" class="nowrap">Zbl 1186.53099</a> · <a href="https://doi.org/10.3934/jmd.2009.3.61" class="nowrap">doi:10.3934/jmd.2009.3.61</a></td> </tr><tr> <td>[22]</td> <td class="space">C. Viterbo, Functors and Computations in Floer homology and applications, Part II, Preprint 1996 (revised 2003), available from http://www.math.polytechnique.fr/cmat/viterbo/Prepublications.html .</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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