A symplectic proof of a theorem of Franks

Brian Collier, Ely Kerman, Benjamin M. Reiniger, Bolor Turmunkh, Andrew Zimmer

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1969-1984
Number of pages16
JournalCompositio Mathematica
Issue number6
StatePublished - Nov 2012


  • Floer homology
  • Hamiltonian flows
  • periodic orbits

ASJC Scopus subject areas

  • Algebra and Number Theory


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