Ergodic properties of triangle partitions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Explosion of smoothness from a point to everywhere for conjugacies between diffeomorphisms on surfaces

For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria and ours for one-dimensional expanding dynamics.

La teoría de Morales-Ramis y el algoritmo de Kovacic

La teor\'\ı a de Morales–Ramis es la teor\'\ı a de Galois en el contextode los sistemas din\'amicos y relaciona dos tipos diferentes de integrabilidad:integrabilidad en el sentido de Liouville de un sistema hamiltonianoe integrabilidad en el sentido de la teor\'\ı a de Galois diferencial deuna ecuaci\'on diferencial. En este art\'\i culo se presentan algunas aplicacionesde la teor\'\i a de Morales–Ramis en problemas de no integrabilidadde sistemas hamiltonianos cuya ecuaci\'on variacional normal a lo largode una curva integral particular es una ecuaci\'on diferencial lineal desegundo orden con coeficientes funciones racionales. La integrabilidadde la ecuaci\'on variacional normal es analizada mediante el algoritmode Kovacic.

Unified formalism for non-autonomous mechanical systems

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

On the configuration of Herman rings of meromorphic functions

We prove some results concerning the possible configuration s of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period a nd give a sufficient condition for a configuration to be realizable.

Sistemes lineals: una aproximació a algun dels seus conceptes i problemes rellevants

L'objectiu que m'he proposat al preparar aquesta exposicióha estat el de mostrar. per una banda, com alguns delsproblemes bàsics de la teoria de Sistemes Dinàmics Linealstenen una resposta senzilla en termes de l' Álgebra lineal i,d'altra, com alguns problemes importants d'aquesta teorias'entronquen amb conceptes i tècniques bàsiques de la geometriai la topologia

Modelos dinámicos en finanzas

El problema de la modelización dinámica enfinanzas tiene mucho que ver con el tipo deproblema que se pretende estudiar. Es preciso teneren cuenta el subyacente así como las magnitudesque se pretende estimar para elegir el modeloadecuado.-

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences

Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features.

Notas del primer seminario de integrabilidad de la Universitat Politècnica de Catalunya

Estas notas corresponden a las exposiciones presentadas en el \emph{Primer Seminario de Integrabilidad}, dentro de lo que se denomina \emph{Aula de Sistemas Din\'amicos}. Durante este evento se realizaron seis conferencias, todas presentadas por miembros del grupo de Sistemas Din\'amicos de la UPC. El programa desarrollado fue el siguiente:\\\begin{center}AULA DE SISTEMAS DIN\'AMICOS\end{center}\begin{center}\texttt{http://www.ma1.upc.es/recerca/seminaris/aulasd-cat.html}\end{center}\begin{center}SEMINARIO DE INTEGRABILIDAD\end{center}\begin{center}Martes 29 y Mi\'ercoles 30 de marzo de 2005\\Facultad de Matem\'aticas y Estad\'{\i}stica, UPC\\Aula: Seminario 1\end{center}\bigskip\begin{center}PROGRAMA Y RES\'UMENES\end{center}{\bf Martes 29 de marzo}\begin{itemize}\item15:30. Juan J. Morales-Ruiz. \emph{El problema de laintegrabilidad en Sistemas Din\'amicos}\medskip {\bf Resumen.} En esta presentaci\'on se pretende dar unaidea de conjunto, pero sin entrar en detalles, sobre las diversasnociones de integrabilidad, asociadas a nombres de matem\'aticostan ilustres como Liouville, Galois-Picard-Vessiot, Lie, Darboux,Kowalevskaya, Painlev\'e, Poincar\'e, Kolchin, Lax, etc. Adem\'astambi\'en mencionaremos la revoluci\'on que supuso en los a\~nossesenta del siglo pasado el descubrimiento de Gardner, Green,Kruskal y Miura sobre un nuevo m\'etodo para resolver en algunoscasos determinadas ecuaciones en derivadas parciales. \medskip\item16:00. David G\'omez-Ullate. \emph{Superintegrabilidad, pares deLax y modelos de $N-$cuerpos en el plano}\medskip{\bf Resumen.} Introduciremos algunas t\'ecnicas cl\'asicas paraconstruir modelos de N-cuerpos integrables, como los pares de Laxo la din\'amica de los ceros de un polinomio. Revisaremos lanoci\'on de integrabilidad Liouville y superintegrabilidad, ydiscutiremos un nuevo m\'etodo debido a F. Calogero para contruirmodelos de N-cuerpos en el plano con muchas \'orbitasperi\'odicas. La exposici\'on se acompa\~nar\'a de animaciones delmovimiento de los cuerpos, y se plantear\'an algunos problemasabiertos.\medskip\item17:00. Pausa\medskip\item17:30. Yuri Fedorov. \emph{An\'alisis de Kovalevskaya--Painlev\'ey Sistemas Algebraicamente Integrables}\medskip{\bf Resumen.} Muchos sistemas integrables poseen una propiedadremarcable: todas sus soluciones son funciones meromorfas deltiempo como una variable compleja. Tal comportamiento, que serefiere como propiedad de Kovalevskaya-Painleve (KP) y que se usafrecuentemente como una ensayo de integrabilidad, no es accidentaly tiene unas ra\'{\i}ces geom\'etricas profundas. En esta charladescribiremos una clase de tales sistemas (conocidos como lossistemas algebraicamente integrables) y subrayaremos suspropiedades geom\'etricas principales que permiten predecir laestructura de las soluciones complejas y adem\'as encontrarlasexpl\'{\i}citamente. Eso lo ilustraremos con algunos sistemas dela mec\'anica cl\'asica. Tambi\'en mencionaremos unasgeneralizaciones \'utiles de la noci\'on de integrabilidadalgebraica y de la propiedad KP.\end{itemize}\medskip{\bf Mi\'ercoles 30 de marzo}\begin{itemize}\item 15:30. Rafael Ram\'{\i}rez-Ros. \emph{El m\'etodo de Poincar\'e}\medskip{\bf Resumen.} Dado un sistema Hamiltoniano aut\'onomo cercano acompletamente integrable Poincar\'e prob\'o que, en general, noexiste ninguna integral primera adicional uniforme en elpar\'ametro de perturbaci\'on salvo el propio Hamiltoniano.Esbozaremos las ideas principales del m\'etodo de prueba ycomentaremos algunas extensiones y generalizaciones.\newpage\item16:30. Chara Pantazi. \emph{El M\'etodo de Darboux}\medskip{\bf Resumen.} Darboux, en 1878, present\'o su m\'etodo paraconstruir integrales primeras de campos vectoriales polinomialesutilizando sus curvas invariantes algebraicas. En estaexposici\'on presentaremos algunas extensiones del m\'etodocl\'asico de Darboux y tambi\'en algunas aplicaciones.\medskip\item17:30. Pausa\medskip\item18:00. Juan J. Morales-Ruiz. \emph{M\'etodos recientes paradetectar la no integrabilidad}\medskip{\bf Resumen.} En 1982 Ziglin utiliza la estructura de laecuaci\'on en variaciones de Poincar\'e (sobre una curva integralparticular) como una herramienta fundamental para detectar la nointegrabilidad de un sistema Hamiltoniano. En esta charla sepretende dar una idea de esta aproximaci\'on a la nointegrabilidad, junto con t\'ecnicas m\'as recientes queinvolucran la teor\'{\i}a de Galois de ecuaciones diferencialeslineales, haciendo \'enfasis en los ejemplos m\'as que en lateor\'{\i}a general. Ilustraremos estos m\'etodos con resultadossobre la no integrabilidad de algunos problemas de $N$ cuerpos enMec\'anica Celeste.\end{itemize}

On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

Non-commutative symbolic coding

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

Stability criteria for the contextual emergence of macrostates in neural networks

More than thirty years ago, Amari and colleagues proposed a statistical framework for identifying structurally stable macrostates of neural networks from observations of their microstates. We compare their stochastic stability criterion with a deterministic stability criterion based on the ergodic theory of dynamical systems, recently proposed for the scheme of contextual emergence and applied to particular inter-level relations in neuroscience. Stochastic and deterministic stability criteria for macrostates rely on macro-level contexts, which make them sensitive to differences between different macro-levels.

Transitivity of codimension-one Anosov actions of R(k) on closed manifolds

We consider Anosov actions of R(k), k >= 2, on a closed connected orientable manifold M, of codimension one, i.e. such that the unstable foliation associated to some element of R(k) has dimension one. We prove that if the ambient manifold has dimension greater than k + 2, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension-one Anosov flows.

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.