Seminario de Sistemas Dinámicos

El Seminario de Sistemas Dinámicos de Santiago es el encuentro semanal de matemáticas con mayor tradición en el país pues se realiza ininterrumpidamente desde la década del '80. Se realiza alternadamente en alguna de las instituciones de Santiago donde hay miembros del grupo de Sistemas Dinámicos. Participan así las universidades de Chile, de Santiago, Andrés Bello y Católica de Chile.

 

2011-06-05
Huaibin Li. Pontificia Universidad Católica de Chile
The thermodynamic formalism of rational maps with backward contracting property
Sala 2 (Víctor Ochsenius) Facultad de Matemáticas - 16.30 hrs
Abstract:
In this talk, we shall discuss the thermodynamic formalism of rational maps satisfying backward contracting property.
For a rational map $f$ of degree at least two which is expanding away from critical points and is backward contracting, we prove that every H{o}lder continuous potential $varphi$ on the Julia set is contracting. It follows known results due to Denker, Urbanski, Przytycki and Haydn that there exists a unique equilibruim state of $f$ for the potential $varphi$ and this measure is exponentially mixing. (This is joint work with Juan Rivera-Letelier).


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2011-05-29
Manuel Cabezas. Pontificia Universidad Católica de Chile
Una aplicación probabilística del teorema ergódico multidimensional
Sala 2 (Víctor Ochsenius) 16:30 Hrs.
Abstract:
Primero mostraremos como el teorema ergódico puede ser visto como una generalización de la ley de los números grandes. El teorema ergódico multidimensional es una generalización natural del teorema ergódico que consiste en reemplazar la acción de $mathbb{Z}$ por la acción de $mathbb{Z}^d$ ($d>1$). Luego presentaremos un tipo de marcha aleatoria en $mathbb{Z}$ llamada randomly trapped random walk. Los límites en escala (scaling limits) de estos procesos pueden ser de varios tipos diferentes. Uno de los posibles límites en escala es el movimiento Browniano (MB). Veremos que para demostrar convergencia al MB es suficiente establecer la ley de los números grandes para un arreglo bidimensional de variables aleatorias. Finalmente mostraremos que el teorema ergó
2011-05-26
Ricardo Borquez. Pontificia Universidad Católica de Chile
Pricing Contingent Claims in a Discrete-Time Economy with Innite Horizon
Sala 2 Víctor Ochsenius)- Facultad de Matemáticas - 12:00 Hrs.
Abstract:

Contingent Claims (or derivatives) are fundamental on the developing of modern nance. Seminal on this area has been the work of Merton (1973) and Black and Scholes (1973) on european options, signifying for these authors to be rewarded with the maximal prize in economics. A general theory on pricing derivatives is today available from the work of Harrison and Kreps (1979) and their Fundamental Theorem of Asset Pricing for contingent claims. It relates the existence of an equivalent martingale measure for a price of a contingent claim process to the economic concept of no-arbitrage. Although the theorem was established for a continuous-time economy on a restricted class of derivatives, the result has been extended to many contexts. For the case of a discrete-time economy with in nite horizon, it has been shown else where that the theorem fails for an arbitrary price process and the notion of no-arbitrage has to be replaced with the more stringent concept of "no free lunch with bounded risk". This result precludes most applications in time-series to be aligned with the theory. My presentation shows the state of the art of an exploratory work on a restricted solution for a class of derivatives with strictly stationary payos, which seems to render a proof for the theorem which is closer to its original form. The class of processes considered here are exible enough to be useful on a number of time-series applications.

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2011-05-22
Juan Rivera-Letelier. Pontificia Universidad Católica de Chile
Rigidez cohomológica y entropía topológica de funciones racionales ultramétricas
Sala 2 (Víctro Ochsenius), Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Demostraremos unas encarnaciones ultrametricas de dos resultados de rigidez en la dinamica de polinomios complejos, ambos relacionados con la medida de equidistribucion. El primero es un resultado de sobre la medida de los puntos biaccesibles conjeturado por Hubbard y demostrado independientemente por Smirnov y Zdunik. El segundo es un resultado celebre de rigidez cohomologica de Zdunik. Usaremos estos resultados para dar una descripcion completa de la entropia topologica de una funcion racional ultrametrica con exponente de Lyapunov mayor o igual a cero. Ademas demostraremos que el caso restante, de las funciones racionales con exponente de Lyapunov extrictamente negativo, corresponde precisamente al caso en que la medida
2011-05-02
Alvaro Coronel. Pontificia Universidad Católica de Chile
Los conjuntos de Delone linealmente repetitivos son rectificables
Sala 2 (Víctor Ochsenius) Facultad de Matemáticas PUC - 16:30 Hrs.
Abstract:
En este seminario mostraremos que, para todo entero d>0, cada conjunto de Delone linealmente repetitivo en el espacio euclidiano de dimensión d esta en biyección a Z^d por un homeomorfismo bi-Lipschitz. También mostraremos que cuando el conjunto de Delone proviene de una sustitución primitiva y se satisface una condición sobre los valores
propios de la matriz de sustitución entonces existe un homeomorfismo
con desplazamiento acotado del conjunto de Delone a un reescalamiento
de Z^d. Está condición incluye las sustituciones de tipo Pisot pero es
mucho más grande. (este es un trabajo conjunto con José Aliste-Prieto y Jean-Marc Gambaudo)

2011-04-25
José Manuel Gutierrez. Universidad de la Rioja, España
El método de Newton: pasado, presente y futuro.
CENI-USACH sala de conferencias A
Abstract:
En esta charla analizaremos el método de Newton para resolver ecuaciones no lineales desde varios puntos de vista. Por una parte mostraremos una breve evolución histórica desde sus controvertidos comienzos (¿Newton-Raphson-Simpson?) hasta las teorías clásicas de Kantorovich o Smale. Por último haremos una incursión en su comportamiento dinámico, destacando el hecho de que todavía quedan muchos interrogantes por resolver.

2011-04-18
Godofredo Iommi. Pontificia Universidad Católica de Chile
Almost-additive thermodynamic formalism for countable Markov shifts
Sala 2 (Víctor Ochsenuis) Facultad de Matemáticas - 16:30 Hrs.
Abstract:
We introduce a definition of pressure for almost-additive sequences of potentials defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices. (This is joint work with Yuki
Yayama).
2011-04-11
María Isabel Cortez. Usach
Clases de equivalencia orbital topológica de los sistemas minimales de Cantor unicamente ergódicos
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
Abstract:
Resumen: Se dice que dos sistemas dinámicos topológicos (X, T) e (Y,S) están en la misma clase de equivalencia orbital (topológica), si existe un homeomorfismo F de X en Y que envía las órbitas de (X,T) en las órbitas de (Y,S). En esta charla mostraremos que todo sistema minimal de Cantor unicamente ergódico es orbitalmente equivalente a la extensión natural de una escala de numeración proveniente de una función unimodal. Los últimos resultados de Giordano, Matui, Putnam y Skau permiten extender este resultado a las acciones minimales unicamente ergódicas de Z^d sobre el Cantor. Este es un trabajo en conjunto con J. Rivera-Letelier.
2011-04-04
Jan Kiwi. Pontificia Universidad Católica de Chile
Un teorema de Montel no-arquimedeano
Sala 2 (Víctor Ochsenius) - 16:30 Hrs. Facultad de Matemáticas - UC
Abstract:
Presentaremos una versión del teorema de Montel en el contexto no-arquimedeano. También presentaremos una posible definición de familias normales y aplicaciones. Este es un trabajo conjunto con Charles Favre y Eugenio Trucco
2011-03-21
Antoine Gournay. Universidad de Neuchâtel
A dynamical approach to Von Neumann dimension
Sala 2 Facultad de Matemáticas PUC
Abstract:
Let G be an amenable group (e.g. Z^n) and V a linear subspace of finite dimension. Gromov showed that the von Neumann dimension of (linear) subspaces of l^2(G;V) invariant under the natural action of G can be obtained as some growth exponent for a dynamical (pseudo-)distance. This point of view (reminiscent of metric entropy) does not require the Hilbertian structure, unlike the usual definition of von Neumann dimension. Thus one can associate to Y, a linear subspace of l^p(G;V) a positive real number, dim_p Y (which coincides with von Neumann dimension when p=2). By showing some properties of this number it is possible to conclude there are no G-equivariant map from l^p(G;V) into l^p(G; V´) when dim V > dim V´.

2011-03-07
Jairo Bochi. Puc-Rio de Janeiro
Uniform Hyperbolicity and Gap Opening
CENI-USACH (sala por confirmar
Abstract:
The study of products of random matrices naturally leads to the concept of a linear cocycle over a dynamical system endowed with an invariant probability measure. The asymptotic growth of these matrix products is described in terms of Lyapunov exponents by Oseledets theorem.

The simplest interesting case is given by matrices in SL(2,R). In that case, the cocycle is called uniformly hyperbolic if the matrix products grow uniformly exponentially with length. This is equivalent to the existence of uniformly expanding and contracting subbundles a la Smale.

In this talk I will consider SL(2,R)-cocycles over strictly ergodic dynamics (that is, uniquely ergodic dynamics with an invariant measure of full support).

I will explain when such a cocycle can be C^0-approximated by a uniformly hyperbolic
2011-01-10
Yves de Cornulier. Université Paris Sud, Orsay
Contaje de Clases de Conjugacion en Grupos Lineales
Sala 2 (víctor Ochsenius) - Facultad de Matemáticas - PUC
2011-01-03
Weixiao Shen. National University of Singapur
On stochastic stability of non-uniformly expanding interval maps
Sala 2 (Víctor Ochsenius) Fac. de Matemáticas PUC 16:230 hrs.
Abstract:
We shall discuss stochastic stability with respect to additive nosie for regular intervals that satisfies a summability condition. This is based on a careful analysis on the derivative growth along pesudo-orbits which has potential applications to other problems related to perturbation of non-uniformly expanding interval maps.

2010-12-10
Yann Bugeaud. Université de Strasbourg
On the decimal expansion of algebraic numbers
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas PUC - 16:30 hrs.
Abstract:
Classical ways to represent a real number are by its continued fraction
expansion or by its expansion in some integer base. It is commonly
expected that algebraic irrational numbers behave, in many respects, like almost all numbers. For instance, their decimal expansion should contain every finite block of digits from ${0, ldots , 9}$. We are very far away from establishing such a strong assertion. However, there has been some recent progress, that will be surveyed in the lecture.

2010-11-23
Raphael Krikorian. École Polytechnique, France
Lyapunov exponents of quasi-periodic cocycles non-homotopic to the identity: regularity and rigidity
Sala de Reuniones, Decanato Facultad de Ciencia, USACH
Abstract:
Resumen: "We prove that Lyapunov exponents of quasi-periodic SL(2,R)-valued cocycles non-homotopic to the identity display some remarkable features such as smooth regularity with respect to a parameter or rigidity: a (monotonic) quasi-periodic cocycle with zero Lyapunov exponent is smoothly conjugated to a cocyle of rotations. This is a joint work with Artur Avila."
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2010-11-22
Alexis Ballier. Cmm-Universidad de Chile
Computing (or not) quasi-periodicity of tilings
Sala de Seminarios 7mo piso, CMM-U. de Chile - 17:00
Abstract:
We know that, in any dimension, any subshift contains a configuration that is uniformly recurrent (or quasi-periodic): a configuration for which any pattern that appears must appear in every sufficiently large part of the configuration. One can then define the recurrence function q of an uniformly recurrent configuration as a function from N into itself such that q(n) is the smallest integer in which any pattern of size q(n) contains every pattern of size n.

This function is a way to measure the regularity of an uniformly recurrent configuration and, to some extent, of a subshift since it must contain at least one such configuration. Since even subshifts of finite type (SFTs) hold many uncomputable properties, one can expect this function to inherit some of these
2010-11-15
Samuel Senti. Ufrj Brasil
Lifting measures to inducing schemes
Sala 2 (Víctor Ochsenius) Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Inducing schemes are used in dynamical systems in order to find a symbolic coding of a subset of the phase space. The induced map - defined on each element of the inducing scheme as an iterate of the original map on - is conjugated to the full symbolic shift on a countable alphabet. Although certain ergodic properties of the full shift carry over to original map, there is no one to one correspondence between invariant probabilities of the shift and those of the original map. The liftability problem consists in studying which measures have shift invariant counterparts. We study the problem for piecewise continuous maps of compact metric spaces. Time permitting we will present some ideas towards dealing with the higher dimensional cases
2010-11-08
Karl-Olof Lindahl. Linnaeus University, Sweden
The Siegel Center Problem in Ultrametric Dynamics
Sala 2 (Víctor Ochsenius), Facultad de Matemáticas PUC - 16.30 hrs.
Abstract:
We consider the `Siegel center problem´ (i.e. local linearization of power series about indifferent fixed points) for ultrametric fields. It is known since a paper of Herman and Yoccoz in 1981 that Siegel´s linearization theorem for the complex field case is true also for ultrametric fields. They also showed that there exist examples in fields of prime characteristics where the Siegel, nor the weaker Brjuno condition is satisfied, imposing a problem of small divisors. However, until recently, it was still an open question whether these small divisors yield divergence of the corresponding conjugacy or not. In fact, it has been shown that the same multiplier may yield divergence of the conjugacy for some analytic functions and convergence for others. In other words,
2010-10-28
Katrin Gelfert. Universidade Federal de Rio de Janeiro
Thermodynamics for geodesic flows
Sala 2 (Víctor Ochsenius) Facultad de Matemáticas - 16.30 hrs
Abstract:
Abstract: I will talk about some properties of geodesic flows of a compact surface in the case that the flow is non-uniformly hyperbolic.

2010-10-25
Juan Rivera Letelier. Pontificia Universidad Católica de Chile
Potenciales Estocásticos de Aplicaciones del Intervalo
Sala 2 (Víctor Ochsenius) - Facultad de Matemáticas - 16:30 Hrs.