arXiv subjects

Antoine Gournay

Publications and source records attributed to Antoine Gournay.

26 records · Page 2Linked to original sources

arXiv2012

Further properties of $\ell^p$ dimension

This article establishes more properties of the $\ell^p$ dimension introduced in a previous article. Given an amenable group $Γ$ acting by translation on $\ell^p(Γ)$, this is just a number, associated to the (usually infinite dimensional) subspaces $Y$ of $\ell^p(G)$ which are invariant under the action of $G$, satisfying dimension-like properties. As a consequence, for $p\in [1,2]$, if $Y$ is a closed non-trivial $Γ$-invariant subspace of $\ell^p(Γ;V)$ and let $Y_n$ is an increasing sequence of closed $Γ$-invariant subspace such that $\srl{\cup Y_n} = \ell^p(Γ;V)$, then there exist a $k$ such that $Y_k \cap Y \neq \{0\}$.

math.FA↗

arXiv2012

A Runge approximation theorem for pseudo-holomorphic maps

The Runge approximation theorem for holomorphic maps (U -> C) is a fundamental result in complex analysis. The aim of this article is to prove such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a compact (almost-)complex manifold M under certain assumptions. Though the setting is definitively that of pseudo-holomorphic maps it also covers some complex varieties.

math.SG↗

arXiv2012

A dynamical approach to von Neumann dimension

Let G be an amenable group and V be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of l^2(G;V) (with respect to G) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $Γ$-invariant linear subspaces Y of l^p(G;V) a real positive number dim_{l^p} Y (which is the von Neumann dimension when p=2). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective G-equivariant linear map of finite-type from l^p(G;V) -> l^p(G; V') if dim V > dim V'. A generalization of the Ornstein-Weiss lemma is developed along the way.

math.FA↗

arXiv2011

Time delay and Calabi invariant in classical scattering theory

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincaré ball.

math-ph↗

arXiv2011

A formula relating localisation observables to the variation of energy in Hamiltonian dynamics

We consider on a symplectic manifold M with Poisson bracket {,} an Hamiltonian H with complete flow and a family Phi=(Phi_1,...,Phi_d) of observables satisfying the condition {{Phi_j,H},H}=0 for each j. Under these assumptions, we prove a new formula relating the time evolution of localisation observables defined in terms of Phi to the variation of energy along classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, the simple pendulum, central force systems, the Poincare ball model, covering manifolds, the wave equation, the nonlinear Schroedinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.

math-ph↗

arXiv2010

Complex surfaces and interpolation on pseudo-holomorphic cylinders

Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. In the present paper, a similar interpolation theorem for pseudo-holomorphic maps from the cylinder S to an almost-complex manifold (M,J) is proved. Properties of the space of pseudo-holomorphic maps from S to (M,J) are derived, in particular a lower bound for the size of this space (in terms of mean dimension). When M is a moduli space of curves, this gives a construction of non-compact complex surfaces. The methods involve an infinite number of surgeries. On the way, a refinement of the gluing process for two pseudo-holomorphic curves is obtained, establishing the geometric behavior of two glued pseudo-holomorphic curves.

math.DG↗

arXiv2010

On a Hölder covariant version of mean dimension

Let G be a infinite countable group which acts naturally on l^p(G). We introduce a modification of mean dimension which is an obstruction for l^p(G) and l^q(G) to be Hölder conjugates.

math.MG↗

arXiv2008

Width of l^p balls

We say a map f:X \to Y is an ε-embedding if it is continuous and the diameter of the fibres is less than ε. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). It is the smallest real number such that there exists an ε-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. Following Gromov, we take the slightly different point of view by looking at the smallest dimension n for which there exists a ε-embedding to a polyhedron of dimension n. While bounds are obtained using Hadamard matrices, the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n+1 points not contained in a hemisphere (obtained by methods very close to those of Ivanov and Pichugov). We are also able to give a complete description in dimension 3 for 1 \leq p \leq 2.

math.MG↗