We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural requirements. In the case of Reeb flows, this leads to simple expressions (purely in terms of the mean indices) for the mean Euler characteristics. These are invariants of the underlying contact structure which are capable of distinguishing some contact structures that are homotopic but not diffeomorphic.
We characterise boundary shaped disc like neighbourhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. We prove uniqueness of homotopy and diffeomorphism type of such contact manifolds assuming non-existence of short periodic Reeb orbits.
We use Lerman's contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed 3-manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the 2-disc constructed by Fayad-Katok embed into the Reeb flow of a dynamically convex contact form on the 3-sphere.
Let $φ$ be any flow on $T^n$ obtained as the suspension of a diffeomorphism of $T^{n-1}$ and let $\mathcal A$ be any compact invariant set of $φ$. We realize $(\mathcal A, φ|_{\mathcal A})$ up to reparametrization as an invariant set of the Reeb flow of a contact form on $\mathbb R^{2n+1}$ equal to the standard contact form outside a compact set and defining the standard contact structure on all of $\mathbb R^{2n+1}$. This generalizes the construction of Geiges, Röttgen and Zehmisch.
This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the gener…
A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods …
We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.
We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non-contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a non-degenerate (or even isolated and homologically non-trivial) periodic orbit with non-zero homology class and the set of one-periodic orbits in that class is finite.
We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that such a symplectomorphism necessarily has infinitely many periodic orbits whenever it has one and all periodic points are non-degenerate.
We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.
We prove that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed, oriented surface admits a sequence of equidistributed periodic orbits. This is a quantitative refinement of the recently established generic density theorem for area-preserving surface diffeomorphisms. The proof has two ingredients. The first is a "Weyl law" for PFH spectral invariants, which was used to prove the generic density theorem. The second is a variational argument inspired by the work of Marques-Neves-Song and Irie on equidistribution results for minimal hypersurfaces and three-dimensional Reeb flows, r…
We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…
We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have simple periodic orbits of arbitrarily large period when it has fixed points which are not necessary from a homological perspective.
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.
We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.
We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $α,$ whence its restriction to the subgroup of $α$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously de…
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…
The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underl…
