Daniel Massart

Université de Montpellier | UM1 · Institut Montpelliérain Alexander Grothendieck

This paper is a follow-up to a previous work where we considered a multi-patch model, each patch following a logistic law, the patches being coupled by symmetric migration terms. In this paper we drop the symmetry hypothesis. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic...

This paper is a follow-up to a previous work where we considered a multi-patch model, each patch following a logistic law, the patches being coupled by symmetric migration terms. In this paper we drop the symmetry hypothesis. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic...

We review the different notions of translation surfaces that are necessary to understand McMullen’s classification of \(\mathrm {GL}_2^+({\mathbb R})\)-orbit closures in genus two. We start by recalling the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending wi...

We study the function $$\mbox{KVol} : (X,\omega)\mapsto \mbox{Vol} (X,\omega) \sup_{\alpha,\beta} \frac{\mbox{Int} (\alpha,\beta)}{l_g (\alpha) l_g (\beta)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichm\"uller discs of the original Veech surface $(X_n,\omega_n)$ arising from right-angled tr...

We review the different notions about translation surfaces which are necessary to understand McMullen's classification of $GL_2^+(\mathbb{R})$-orbit closures in genus two. In Section 2 we recall the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending with Abeli...

WE consider a three-patch model with symmetric migration terms, where each patch follows a logistic law. We show numerically that the increase in number of patches from two to three gives a new behavior for the dynamics of the total equilibrium population as a function of the migration rate.

The paper considers a n-patch model with migration terms, where each patch follows a logistic law. In the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the n carrying capacities.

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n...

We study a volume related quantity $\mathrm{KVol}$ on the stratum ${\mathcal{H}(2)}$ of translation surfaces of genus $2$, with one conical point. We provide an explicit sequence $L(n, n)$ of surfaces such that $\mathrm{KVol}(L(n, n)) \rightarrow 2$ when n goes to infinity, $2$ being the conjectured infimum for $\mathrm{KVol}$ over ${\mathcal{H}(2)...

The setting is a square-tiled surface X. We study the quantity KVol, defined as the supremum over all pairs of closed curves, of their algebraic intersection divided by the product of their length, times the volume of X (so as to make it scaling-invariant). We give a hyperbolic-geometric construction to compute KVol in a family of Teichm\H{u}ller d...

The setting is a translation surface $X$ in the stratum H(2) of translation surfaces of genus 2, with one conical point. We study the quantity KVol, defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their length, times the volume of X. We provide an explicit sequence L(n, n) of surf...

We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their lengths, times the area of the surface. The surfaces we consider live in the stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with one conical point. We provide an explicit sequ...

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectrum into account rather than jus...

The paper considers a \begin{document}$ n $\end{document}-patch model with migration terms, where each patch follows a logistic law. First, we give some properties of the total equilibrium population. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might...

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in the first cohomology of $M$, which is a first step towards Ma\~n\'e's conjecture.

Given a closed, oriented surface M, the algebraic intersection of closed curves induces a symplectic form Int(.,.) on the first homology group of M. If M is equipped with a Riemannian metric g, the first homology group of M inherits a norm, called the stable norm. We study the norm of the bilinear form Int(.,.), with respect to the stable norm.

We prove Mañé's conjectures [9] in the context of codimension 1 Aubry-Mather theory. © 2011 Wiley Periodicals, Inc.

In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system. Comment: 14 pages, 1 figure

We review the author's results on Mather's $\beta$ function : non-strict convexity of $\beta$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $\beta$, correlation between the rationality of the homology class and the differentiability of $\beta$, equality of the Mather set and the...

On passe en revue les résultats de l'auteur sur la fonction $\beta$ de Mather.

We prove Mañé's conjectures in the context of codimension one Aubry-Mather theory

We consider Mañé’s conjectures and prove that the one he made in [1] is stronger than the one he made in [2]. Then we prove that the most straightforward approach to prove the strong conjecture doesn’t work in the C 4 topology. Key wordsLagrangian dynamics–minimizing measures

We prove that Ma\~n\'e's conjecture, as stated in {\em Lagrangian flows: the dynamics of globally minimizing orbits}, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141--153, contains another conjecture of Ma\~n\'e, stated in {\em Generic properties and problems of minimizing measures of Lagrangian systems} Nonlinearity 9 (1996) 273-310.

If the β-function of a time-periodic Lagrangian on a manifold M has a vertex at a k-irrational homology class h, then 2k<dim M, so if dim M=2, then h is rational.

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.

We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.

If the $\beta$-function of a time-periodic Lagrangian on a manifold $M$ has a vertex at a $k$-irrational homology class $h$, then $2k \leq \dim M$. Furthermore if $\dim M =2$ $h$ is rational.

We prove the existence of $C^{1}$ critical subsolutions of the Hamilton-Jacobi equation for a time-periodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system.

We study Lagrangian systems on a closed manifoldM. We link the differentiability of Mather’sβ-function with the topological complexity of the complement of the Aubry set. As a consequence, whenM is a closed, orientable surface, the differentiability of theβ-function at a given homology class is forced by the irrationality of the homology class. Thi...

Let (M, g) be a closed orientable surface, equipped with a smooth Finsler metric. The metric induces a norm on the real homology of M, called a stable norm. We show this norm is neither strictly convex, nor smooth.

Let (M.g) be a closed orientable surface, equipped with a Finsler metric. The metric induces a norm on the real homology of M, called stable norm. We show this norm is not strictly convex.

We study the stable norm on the first homology of a Riemannian manifold, with special focus on the case of a surface of genus greater than 1. We link the stable norm with Mather's beta function, then we prove that the stable norm is neither strictly convex, nor smooth. Then we compare the stable norm with the L2 norm.