Publicaciones del CMAT

Publicaciones de los docentes del Centro de Matemática

On weak KAM theory for N-body problems

We consider N-body problems with (1/r)^2κ potential where κ ∈ (0, 1), including the Newtonian case (κ = 1/2). Given R > 0 and T > 0, we find a uniform upper bound for the minimal action of paths binding in time T any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding in free time two configurations). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax-Oleinik semigroup and we show that they are global viscosity solutions of the corresponding Hamilton-Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Translation invariance of weak KAM solutions of the Newtonian N-body problem

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c ≥ 0, of the classical N -body problem in an Euclidean space E of dimension k ≥ 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

Recurrence of non-resonant homeomorphisms on the torus

 

Wild Milnor attractors accumulated by lower dimensional dynamics

 

Translation invariance of weak KAM solutions of the Newtonian N-body problem

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c ≥ 0, of the classical N -body problem in an Euclidean space E of dimension k ≥ 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

Generic bi-Lyapunov stable homoclinic classes

En este trabajo se estudian clases homoclinicas que verifican una propiedad mas debil que tener interior no vacio. Probamos que para difeomorfismos genericos estas deben tener interior no vacio y en algunos casos mostramos que deben tener interior no vacio e incluso que deben ser toda la variedad en otros casos. Algunos resultados son mas generales y se aplican a clases homoclinicas llamadas Quasi-Atractores.

A review of some recent results on Random Polynomials over R and over C

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere.

Geometría y álgebra lineal 1

 

Stochastic Perturbations and Smooth Condition Numbers

En este artículo definimos un nuevo número de condición adaptado a perturbaciones uniformes en todas las direcciones en un contexto general de mapas entre variedades Riemannianas. Las definiciones y teoremas se pueden aplicar a una gran cantidad de ejemplos. Estudiaremos la relación con el número de condición clásico, y estudiaremos algunos ejemplos importantes.

A note about the average number of real roots of a Bernstein polynomial system

We prove that the real roots of normal random homogeneous polynomial systems with n+1 variables and given degrees are, in some sense, equidistributed in the projective space P(R^{n+1}). From this fact we compute the average number of real roots of normal random polynomial systems given in the Bernstein basis.

Random systems of polynomial equations. The expected number of roots under smooth analysis.

We consider random systems of equations over the reals, with m equations and m unknowns P_i(t) + X_i(t) = 0, t 2 Rm, i = 1...m, where the P_i's are non-random polynomials having degrees d_i's (the "signal") and the X_i's (the "noise") are indepen- dent real-valued Gaussian centered random polynomial fields defined on R^m, with a probability law satisfying some invariance properties. For each i, P_ i and X_ i have degree d_ i. The problem is the behavior of the number of roots for large m. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e. P_i identically zero for all i = 1...m), tends to zero geometrically fast as m tends to in¯nity. In particular, this means that the behavior of this expected value, is governed by the noise part.

Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials

We prove that points in the sphere associated with roots of random polynomials via the stereographic projection, are surprisignly well-suited with respect to the minimal logarithmic energy on the sphere. That is, roots of random polynomials provide a fairly good approximation to Elliptic Fekete points.

Local product structure for expansive homeomorphisms

En este trabajo se estudia la estructura de homeomorfismos expansivos en variedades asumiendo hipotesis sobre la existencia de puntos periodicos y su estructura local. Se obtiene un resultado bastante completo en el caso que haya un punto periodico topologicamente hiperbolico de codimension uno. Los trabajos generalizan trabajos anteriores de Vieitez y estan tambien motivados por un importante trabajo de Lewowicz.

Local implications of almost global stability

En este trabajo se estudian propiedades locales que surgen de la estabilidad casi global. Las tecnicas tienen que ver con las variedades centrales junto con la existencia de densidades. Se presentan tambien ejemplos que cierran algunas preguntas cuya respuesta no era conocida en la literatura.

Codimension one generic homoclinic classes with interior

En este trabajo se estudia una conjetura de Abdenur, Bonatti y Diaz acerca de la existencia de clases homoclinicas para difeomorfismos genericos con interior no vacio. Se resuelven algunos casos particulares y se da un resultado general sobre su estructura en caso de admitir descomposicion dominada de codimension 1.

Compact coalgebras, quantum groups and the positive antipode.

We consider compact ◦-coalgebras and Hopf algebras. In the case of a ◦–Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness –up to conjugation by an automorphism of Hopf algebras– of the compact involution. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it –as well as the unitary antipode and Nakayama automorphism– in order to enhance our understanding of the antipode itself.

Globally minimizing parabolic motions in the Newtonian N-body problem

We consider th N-body problem with the Newtonian potential 1/r. We prove that for every initial configuration x and for every minimal normalized central configuration u, there exists a collision-free parabolic solution starting at x and asymptotic to u. This solution is a minimizer in every time interval. The proof exploits the variational structure of the problem, and it consist in finding a convergent subsequence in a family of minimizing trajectories. The hardest part is to show that this solution is parabolic and asymptotic to u.

On the free time minimizers of the Newtonian N-body problem

The Hamiltonian formulation of the Newtonian N-body problem assures that motions are characterized by the local minimization property of the Lagrangian action. In this paper we study the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. A simple example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central configuration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice of y in E^N , there should be at least one free time minimizer x on [0,+∞) which satisfies x(0)=y. We prove that such motions are completely parabolic and asymptotic to some central configuration. This means that the velocity of each body goes to zero as t goes to +∞, and that the normalized configuration u(t)=I(x(t))^{-1/2}.x(t) converges to a central configuration. We show that the moment of inertia grows like c.t^{4/3} where the constant c>0 depends on the limit configuration. Moreover, we prove that the parabolic homothetic motion by the limit configuration must be also a free time minimizer. Using Marchal's theorem, which states that minimizers avoid collisions we can deduce as a corollary that there are no complete free time minimizers, i.e. defined on (-∞,+∞).

Princeton Lecture Notes in Analysis I. Fourier Analysis: an Introduction.

 

Radford's formula for biFrobenius algebras and applications