Leafwise fixed points for C⁰-small Hamiltonian flows

2014/08/20 by Ziltener, Fabian
#53D05 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1408.4578

Abstract

Consider a closed coisotropic submanifold N of a symplectic manifold (M,ω) and a Hamiltonian diffeomorphism ϕ on M. The main result of this article states that ϕ has at least the cup-length of N many leafwise fixed points w.r.t. N, provided that it is the time-1-map of a global Hamiltonian flow whose restriction to N stays C⁰-close to the inclusion N→ M. If (ϕ,N) is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of N. The nondegeneracy condition is generically satisfied. This appears to be the first leafwise fixed point result in which neither ϕ|_N is assumed to be C¹-close to the inclusion N→ M, nor N to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the C⁰-condition on ϕ cannot be replaced by the assumption that ϕ is Hofer-small.

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