Related papers: C^0-rigidity of Poisson brackets
The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,\omega)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ \omega(A) > 0 \quad \text{and}…
In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its…
Let M be a bounded open plane domain. Let f be a continuous function on the closure of M, 3-times continuously differentiable in M, which vanish on the boundary. Polterovich and Sodin proved that the values of f cannot exceed the norm of…
We investigate rigidity phenomena associated to the stable norm and Mather's $\beta$-function for Riemannian geodesic flows on closed manifolds. Given two metrics $g_1$ and $g_2$, we compare these objects pointwise at individual homology…
We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + \lambda \mathcal{B}\right)dH_{\lambda}$ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets $\left(\mathcal{A} + \lambda \mathcal{B}\right)$ is…
Let G be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G] be its function algebra. C. Brouder and A. Frabetti introduced a…
We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This…
We construct Hamiltonian Floer complexes associated to continuous, and even lower semi-continuous, time dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps…
In this paper we introduce a new methodology for smooth rigidity of Anosov diffeomorphisms based on "matching functions." The main observation is that under certain bunching assumptions on the diffeomorphism the periodic cycle functionals…
Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the…
It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $\mathfrak{c}$ is a remainder in a soft compactification of $\mathbb{N}$. We also exhibit an example of a compact space of weight $\aleph_1$ --…
On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology. In this article, we generalize Schwarz's theorem…
One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling…
An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a Lie bracket on the space $\Omega^{1}(P)$ of all differential one-forms, under which…
We show that Banyaga's Hofer-like norm, a generalization of the Hofer norm coincides with the classical Hofer norm when restricted to Hamiltonian diffeomorphisms on compact symplectic manifolds. This result proves a conjecture of Banyaga…
Let Phi : M --> g^* be a proper moment map associated to an action of a compact connected Lie group, G, on a connected symplectic manifold, (M,\omega). A collective function is a pullback via \Phi of a smooth function on g^*. In this paper…
A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…
A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give…
We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated…
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…