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Homogenization of a spectral equation with drift in linear transport

Guillaume Bal — 2001

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two...

Dislocation measure of the fragmentation of a general Lévy tree

Guillaume Voisin — 2011

ESAIM: Probability and Statistics

Given a general critical or sub-critical branching mechanism and its associated Lévy continuum random tree, we consider a pruning procedure on this tree using a Poisson snake. It defines a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. This work generalizes the work made for a Brownian tree [R. Abraham and L. Serlet, 7 (2002) 1–15] and for a tree without Brownian part [R. Abraham and J.-F. Delmas, 141 (2008) 113–154].

Mesures invariantes pour les fractions rationnelles géométriquement finies

Guillaume Havard — 1999

Fundamenta Mathematicae

Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if p ( T ) + 1 p ( T ) δ > 2 . Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.

Length of continued fractions in principal quadratic fields

Guillaume Grisel — 1998

Acta Arithmetica

Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l ( ( √ d ) 2 n + 1 ) be the length of the continued fraction expansion of ( √ d ) 2 n + 1 . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that C ₁ ( √ d ) 2 n + 1 ≥ l ( ( √ d ) 2 n + 1 ) ≥ C ₂ ( √ d ) 2 n + 1 for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

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