Related papers: Action-Index Relations for Perfect Hamiltonian Dif…

Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and…

The quantum measurement axiom dictates that physical observables and in particular the Hamiltonian must be diagonalizable and have a real spectrum. For a time-independent Hamiltonian (with a discrete spectrum) these conditions ensure the…

Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a…

A time-dependent completely integrable Hamiltonian system is quantized with respect to time-dependent action-angle variables near an instantly compact regular invariant manifold. Its Hamiltonian depends only on action variables, and has a…

The action--Maslov homomorphism $I\co\pi_1(\text{Ham}(X,\omega))\to\R$ is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this…

In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds. Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphisms has infinitely many periodic…

We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the…

In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold $(M,\omega)$ canonically relates the action spectra of different normalized Hamiltonians on {\it arbitrary}…

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\omega) \supeq…

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least…

On an open, connected symplectic manifold $(M,\omega)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fr\'echet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this…

We consider semi-free Hamiltonian $S^1$-manifolds of dimension six and establish when the equivariant cohomology and data on the fixed point set determine the isomorphism type. Gonzales listed conditions under which the isomorphism type of…

In a non-compact context the first natural step in the search for periodic orbits of a hamiltonian flow is to detect bounded ones. In this paper we show that, in a non-compact setting, certain algebraic topological constraints imposed to a…

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will…

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact…

Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas…

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few…

In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of…

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of ${\mathbb C}{\mathbb P}^n$ with the minimal possible number of periodic points (equal to $n+1$ by Arnold's conjecture), called here Hamiltonian pseudo-rotations.…