Antoine Gournay

On a Holder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on lp(Γ). We introduce a modification of mean dimension which is an obstruction for lp(Γ) and lq(Γ) to be Holder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A remark on the connectedness of spheres in Cayley graphs

The aim of this note is to prove an elementary yet useful property of finitely presented groups. This property is called “connected spheres” in Blachère’s work [1] (where he shows that the Heisenberg group has this property). Filimonov & Kleptsyn [3] use this remark to get some nice results on certain groups of diffeomorphisms of the circle. Recall that, for a finitely generated group Γ and S ⊂ Γ a finite set such that s ∈ S =⇒ s−1 ∈ S, the Cayley graph is the graph whose vertices are the elements of Γ and where g, h ∈ G are connected by an edge whenever there exists s ∈ S such that gs = h. This 1-complex is central to the study of Γ as a geometric object. A very rough property of Cayley graphs is the number of ends. Let Bn be the ball of radius n with centre at the identity element. This is defined to be the number of infinite connected components in the complement of Bn as n → ∞. Hopf [5] showed that a Cayley graph may have only 0 (finite group), 1, 2 , or ∞ many ends. Stallings [7] described the case of groups with 2 ends (virtually-Z) and ∞ many ends (certain amalgamated products and HNN-extensions). Thus, it turns out “most” groups have 1 end. The subject matter here is the number of “important” connected components in the spheres of thickness r. The term “important” needs to be added because the complement of Bn may have many finite connected components (and only the infinite one is of interest here). The aim is to show that when the group is finitely presented, there exists r (independent of n) such that these spheres are always connected. The complement of a set A will be denoted Ac.

An isoperimetric constant for signed graphs

A sign is introduced in the usual Laplacian on graphs and the corresponding analogue of the isoperimetric constant for this Laplacian is presented, i.e. a geometric quantity which enables to bound from above and below the first eigenvalue. The introduction of the sign in the Laplacian is motivated by the study of

TIME DELAY AND CALABI INVARIANT IN CLASSICAL SCATTERING THEORY

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A formula relating sojourn times to the time of arrival in Hamiltonian dynamics

-lifts of graphs and of the combinatorial Laplacian in higher degree.

The Liouville property and Hilbertian compression

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H0, 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 Buslaev and Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincare scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincare ball.

A dynamical approach to von Neumann dimension

We consider on a manifold M equipped with a Poisson bracket { ?, ?} a Hamiltonian H with complete flow and a family ? ? (?1, ?, ?d) of abstract position observables satisfying the condition {{?j, H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of ? to the time of arrival of 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, central force systems, the Poincar? ball model, the wave equation, the nonlinear Schr?dinger equation, the Korteweg?de Vries equation and quantum Hamiltonians defined via expectation values.