On the Hofer girth of the sphere of great circles

An oriented equator of [Formula: see text] is the image of an oriented embedding [Formula: see text] such that it divides [Formula: see text] into two equal area halves. Following Chekanov, we define the Hofer distance between two oriented equators as the infimal Hofer norm of a Hamiltonian diffeomorphism taking one to another. Consider [Formula: see text] the space of oriented equators. We define the Hofer girth of an embedding [Formula: see text] as the infimum of the Hofer diameter of [Formula: see text], where [Formula: see text] is homotopic to [Formula: see text]. There is a natural embedding [Formula: see text], sending a point on the sphere to the positively oriented great circle perpendicular to it. In this paper, we provide an upper bound on the Hofer girth of [Formula: see text].

Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

THE g-AREAS AND THE COMMUTATOR LENGTH

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham (M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham (M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k+-area, which measures the "distance", in a certain sense, to the subspace [Formula: see text] of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with [Formula: see text].

TOTALLY NON-COISOTROPIC DISPLACEMENT AND ITS APPLICATIONS TO HAMILTONIAN DYNAMICS

In this paper, we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic manifold. Essential to the proofs is a displacement principle for such submanifolds. Namely, we show that a topologically displaceable nowhere coisotropic submanifold is also displaceable by a Hamiltonian diffeomorphism, partially extending the well-known non-Lagrangian displacement property.