Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

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 C 0 -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 C 1 -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 C 0condition on ϕ cannot be replaced by the assumption that ϕ is Hofer-small.

Fragility and persistence of leafwise intersections

“…Note that in Theorem 1.1, the convergence to zero is much stronger than the convergence in Hofer's norm and not obviously related to the C 0 -convergence of the maps ϕ F k . (Apparently, the sequence ϕ F k we constructed does not C 0 -converge; by [Zi14], it cannot C 0 -converge to i d.)…”

“…It readily follows from the proof that M can be chosen to be diffeomorphic and arbitrarily C 0 -close to the round sphere S 2n−1 , and the Hamiltonians F k can also be chosen to be supported in an arbitrarily small neighborhood of S 2n−1 . To be more precise, for any δ > 0, we can ensure that M is the image of an embedding which is δ-close to the standard embedding S 2n−1 → R 2n and that for all k the Hamiltonians F k are supported in the δ-neighborhood of S 2n−1 and ϕ F k is also δ-close to i d. As has been pointed out above, ϕ F k cannot C 0 -converge to i d for a fixed M due to the results from [Zi14].…”

“…Recently, Moser's theorem was strengthened by Ziltener in [Zi14], where it was shown that leafwise intersections must exist for any closed coisotropic submanifold M whenever ϕ is the time-one map of a Hamiltonian isotopy ϕ t which is C 0 -close to i d or, more generally, when ϕ t (M ) stays C 0 -close to M . Furthermore, in this case one still has the Lusternik-Schnirelmann (cup-length) and Morse type multiplicity results.…”

Note on coisotropic Floer homology and leafwise fixed points

“…Find conditions under which Fix(ψ, N ) is non-empty and find lower bounds on its cardinality.This generalizes the problems of showing that a given Hamiltonian diffeomorphism has a fixed point and that a given Lagrangian submanifold intersects its image under a Hamiltonian diffeomorphism. References for solutions to the general problem are provided in [20,22].Example (translated points). As explained in [19, p. 97], translated points of the time-1-map of a contact isotopy starting at the identity are leafwise fixed points of the Hamiltonian lift of this map to the symplectization.…”

“…This generalizes the problems of showing that a given Hamiltonian diffeomorphism has a fixed point and that a given Lagrangian submanifold intersects its image under a Hamiltonian diffeomorphism. References for solutions to the general problem are provided in [20,22].…”