Abstract. We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C 0 -dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C 0 -Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces [25,26], we introduce a dynamical invariant, denoted by N , for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.Formal Spectral Invariants: Although the following definition makes sense on any symplectic manifold we will restrict our attention here to the case of a surface Σ which is either the plane R 2 or is closed and aspherical. Definition 3. A function c : C ∞ ([0, 1] × Σ) → R is a formal spectral invariant if it satisfies the following four axioms: 1. (Spectrality) c(H) ∈ spec(H) for all H ∈ C ∞ ([0, 1]×Σ), where spec(H), the spectrum of H, is the set of critical values of the Hamiltonian action, that is, the set of actions of fixed points of φ 1 H .

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold (M, ω) must have at least as many fixed points as the minimal number of critical points of a smooth function on M .It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher.More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point. arXiv:1609.09192v1 [math.SG]

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