On the Hofer-Zehnder conjecture
Egor Shelukhin
Abstract
We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.
Topics & Concepts
Contractible spaceMathematicsDiffeomorphismConjectureSymplectic geometryHomology (biology)Pure mathematicsHamiltonian (control theory)Monotone polygonSymplectic manifoldCombinatoricsGeometryBiochemistryMathematical optimizationGeneChemistryTopological and Geometric Data AnalysisHomotopy and Cohomology in Algebraic TopologyGeometric and Algebraic Topology