On the Fixed Points of a Hamiltonian Diffeomorphism in Presence of Fundamental Group

Kaoru Ono () and Andrei Pajitnov ()
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Kaoru Ono: Kyoto University, Research Institute for Mathematical Sciences
Andrei Pajitnov: Faculté des Sciences, Université de Nantes, Laboratoire Mathématiques Jean Leray UMR 6629

A chapter in Essays in Mathematics and its Applications, 2016, pp 199-228 from Springer

Abstract: Abstract Let M be a weakly monotone symplectic manifold and H be a time-dependent 1-periodic Hamiltonian; we assume that the 1-periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M and derive from it new lower bounds for the number of 1-periodic orbits. Using these invariants we prove in particular that if π 1(M) is finite and solvable or simple, then the number of 1-periodic orbits is not less than the minimal number of generators of π 1(M). For a general closed symplectic manifold with infinite fundamental group, we show the existence of 1-periodic orbit of Conley–Zehnder index 1 − n for any non-degenerate 1-periodic Hamiltonian system.

Keywords: Modulus Space; Chain Complex; Symplectic Manifold; Floer Homology; Principal Ideal Domain (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-31338-2_10

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DOI: 10.1007/978-3-319-31338-2_10

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