Seminario de Sistemas Dinámicos

2013-08-26

Mahsa Allahbakshi. Cmm-U. de Chile
Properties of the class degree
sala 1 Facultad de Matemáticas - 16:00 Hrs.
Abstract:
Resumen:

Given a finite-to-one factor code $pi$ from a one dimensional shift of finite type $X$ onto a sofic shift $Y$ there is a well-known quantity assigned to $pi$ called the degree of the code and is defined to be the minimal number of preimages of points of $Y$. When $pi o Y$ is infinite-to-one, a notion analogous to the degree of a finite-to-one code, called the class degree, was defined recently. The class degree is the minimal number of transition equivalence classes over points of $Y$ where the definition of transition classes is motivated by communicating classes in Markov chains. We describe the structure of such transition classes and show that the class degree which is a generalization of the degree satisfies similar properties.
Joint work with S. Hong and U. Jung.

2013-08-26

Jorge Vargas. Universidad Nacional de Córdoba
Grupos de Lie
Facultad de Matemáticas - UC
Abstract:

Este cursillo, que tratará de cubrir las principales herramientas de Grupos de Lie, será el puntapié inicial de un seminario que estamos coordinando bajo la tutela del profesor Juan Rivera acerca de los Teoremas de Ratner ( http://terrytao.wordpress.com/2007/09/29/ratners-theorems/ ). El cursillo también está orientado a tratar de entender una demostración y algunas aplicaciones de estos teoremas (se avisará acerca de este seminario cuando sea pertinente).


Las sesiones siguientes del cursillo se realizarán en la facultad de Matemáticas UC :

Lunes 26 Agosto, Sala 1 a las 17:15 hrs.
Martes 27 Agosto, Sala 5 a las 15.30 hrs.
Jueves 29 Agosto, Sala 5 a las 17 hrs.
Viernes 30 Agosto, Sala 5 a las 10.00 hrs.

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2013-06-10

Gonzalo Castro. Usach
Aspectos analíticos de acciones de grupos nilpotentes sobre el intervalo
Depto. De Matemáticas USACH, tercer piso 15:15 Abstract:
Resumen:
Una pregunta interesante y lejos de estar satisfactoriamente resuelta es la siguiente:

¿Dado un grupo G finitamente generado, libre de torsión y nilpotente, cuál es el mejor alpha para el cual dicho grupo se incrusta como un subgrupo de difeomorfismo de clase C^{1+alpha}?

G. Castro, E. Jorquera y A. Navas demostraron que que el grupo N_{d} de matrices triangulares superiores de (d+1)d+1), con entradas entradas enteras y unos en su diagonal, se puede incrustar en el grupo de difeomorfismos C^{1+alpha}[0,1], para todo alpha< rac{n(n-1)}{2} (ver, http://arxiv.org/pdf/1108.5223.pdf ). Lo importante de lo anterior radica en que todo grupo que satisface las condiciones de la pregunta inicial se puede incrustar N_{d}, para algún d.

El método empleado en http://arxiv.org/pdf/1108.5223.pdf (esencialmente un método de conteo y caminatas aleatorias) para
verificar que no se podía obtener una mejor clase de diferenciablidad no parecía tener una fácil generalización a situaciones similares

2013-03-18

Matthieu Arfeux. Université Paul Sabatier, Toulouse, France
Deligne-Mumford Compactification and Dynamic on Trees of Spheres
Sala 1 de la Facultad de Matemáticas - 16:00 Hrs.

2013-03-17

Jean-Baptiste Aujogue. Usach
Embedding of Meyer sets into Model sets
Sala de Seminario Citecamp, USACH - 16:30 hRS.
Abstract:
Resumen:
In this talk I provide a presentation of highly structured point patterns of an Euclidean space R^d, the so-called model sets and the Meyer sets, as well as their associated dynamical systems. I moreover recall some basics on the maximal equicontinuous factor and proximality for general dynamical systems.
Then I will present the main steps of a construction which allows to embed any repetitive Meyer set into a canonically associated repetitive model set.
This will naturally leads to a characterization of repetitive model sets as the repetitive Meyer sets with almost automorphic associated dynamical system.

2013-03-11

Katrin Gelfert. Universidade Federal de Rio de Janeiro
Lyapunov exponents in non-hyperbolic dynamics
Sala 1 Facultad de Matematicas UC - 16:30 Hrs.
Abstract:
We study Lyapunov exponents for a family of partially hyperbolic and topologically transitive diffeomorphisms that are step skew-products over a horseshoe map. These maps are genuinely non-hyperbolic and the central Lyapunov spectrum contains negative and
positive values. We show that in a first model, besides one gap, this spectrum is complete. We also investigate how Lyapunov regular points with corresponding (central) exponents are distributed in the fibers. The principal ingredients of our proofs are minimality of the underlying iterated function system and shadowing- like arguments. In another model we study multiple phase transitions for the topological pressure of geometric potentials. We prove that for every k there is a diffeomorphism with a transitive set as abov

2013-02-21

Renaud Leplaideur. Université de Brest
Construction of explicit examples of diffeomorphisms at the border of the Uniformly hyperbolic ones and Kupka Smale
Sala 1 de la Facultad de Matemáticas UC - 16:30 Hrs.
Abstract:
Resumen: The goal is to construct diffeomorphisms that are not uniformly hyperbolic but such that every periodic point is hyperbolic and stable and unstable manifolds are mutually transverse. In the first part I will present some discussion on what hyperbolic means, and then present the main motivation: to get phase transition for the thermodynamic formalism.
Then, I will present several step of the construction, explaining what are the problems to solve and how we can do it.

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2013-01-14

Hiroki Takahasi. Tokyo University
Thermodynamic Formalism for The Henon Map At The First Bifurcation
Sala 1 - Facultad de Matemáticas - 16:30 Hrs.

2012-11-26

Yiwei Zhang. Pontificia Universidad Católica de Chile
On the mixing properties of piecewise expanding maps under composition with permutations (Joint work with Nigel Byott and Mark Holland)
Sala 1 Facultad de Matemáticas PUC
Abstract:
Resumen:

We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal
subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx mod 1$ for integers $m geq 2$is examined in detail. We give a combinatorial description of those permutations $sigma$ for which $sigma circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N o infty$. We then investigate the mixing rate of $sigma circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $sigma$ranges through all permutations; this can be made arbitrarily close to $1$ as $N o infty$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behavior than that obtained from the stretch-and-fold map.

2012-11-19

Pablo Guarino. Usp, Brasil
Rigidez geométrica de homeomorfismos críticos del círculo
Sala 1 de la Facultad de Matemáticas PUC - 16:30 hrs.
Abstract:
Resumen: Hablaremos de homeomorfismos del círculo que están en el borde de los difeomorfismos. Mas explícitamente, estudiaremos
homeomorfismos del círculo de clase C^3 que no son difeomorfismos, pues presentan un punto crítico (de grado impar). Nos concentraremos en el caso de número de rotación irracional de tipo limitado y mostraremos cómo se prueba la siguiente rigidez geométrica: dos homemorfismos críticos con igual número de rotación irracional de tipo limitado e igual grado en el punto crítico son conjugados por un difeomorfismo de clase C^{1+alpha}. Esto surgió como una conjetura a comienzos de los años ´80 a través de trabajos de Feigenbaum, Lanford, Rand, etc. Luego de muchas contribuciones para el caso real-analítico (de Faria-de Melo,
Yampolsky, Khanin-Teplinsky) he

2012-10-29

Rafael Tiedra. Pontificia Universidad Católica de Chile
Commutator methods for the spectral analysis of time changes of horocycle flows
Sala 1 de la Facultad de Matemáticas - 16:30 Hrs.
Abstract:
Resumen:

We prove that all time changes of the horocycle flow on compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators and the unique ergodicity of the horocycle flow.

2012-10-09

Fabio Tal. Universidade de Sao Paulo
Sublinear displacement for conservative
Auditorio del Departamento de Matemática USACH
Abstract:
Resumen: (j.w. with A. Koropecki) In this talk we will consider area preserving homeomorphisms of the 2-torus which are homotopic to the identity and whose rotation set (a concept analogous to the rotation number of a homeomorphisms of the circle) is either a single point or a nondegenerate line segment. Whenever the rotation set of an homeomorphisms consists of a single point $v$, it is called a pseudo-rotation, and these type of maps have been extensively studied in the case where $v$ has an irrational component. On the other hand not much is known when the rotation set of $f$ is reduced to a single point with rational coordinates.
We will show that there exists a pathological example of a smooth homeomorphisms whose rotation set is just the origin, meaning that there exists no point which

2012-10-08

Fabio Tal. Universidade de Sao Paulo
Essential dynamics for surface homeomorphisms
Auditorio del Departamento de Matemática USACH - 16:30 Hrs.
Abstract:
Resumen: (j.w. with A. Koropecki) We will discuss nonwandering homeomorphisms of closed surfaces of genus g which are homotopic to the identity, and we will try to determine when the dynamic of the homeomorphisms is intrinsic to the surface. This means, loosely speaking, that the dynamics cannot be seen as that of an homeomorphisms of a surface with strictly smaller genus.
The main result we obtain is that, if the set of fixed points of the homeomorphism $f$ homotopic to the identity is not essential, then any $f$ invariant open topological disk lifts to a bounded set in the universal covering. We will also discuss 2 consequences of this result, the first one being that the set of essential" points of the dynamic (those point whose orbit of any neighborhood is not homotopically trivial) is itself essential large.
For the specific case of the torus, we show that if the rotation set of $f$ has nonempty interior, then it is possible to partition the torus into 2 different regions, a chaotic one which is externally transitive and realizes all of the rotational dynamics, and an inessential one, consisting of bounded periodic topological disks.

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2012-10-01

Mickaël Crampon. Universidad de Santiago de Chile
El flujo geodésico de los convexos divisibles
Auditorio del Departamento de Matemática - USACH - 16:30 Hrs.
Abstract:
Un convexo divisible es un abierto convexo del espacio proyectivo que admite un cuociente compacto por un subgrupo discreto de transformaciones proyectivas. Cuando el grupo no tiene torsión, el cuociente es una variedad, que admite una métrica natural llamado métrica de Hilbert. De forma general, nos proponemos estudiar el flujo geodésico de esta métrica.
Yves Benoist mostro que mucho depende de si el convexo es estrictamente convexo o no. Si lo es, se puede decir muchas cosas sobre la dinámica del flujo geodesico y recordaré los resultados principales. Si no, no se sabe casi nada. Presentaré en este caso lo que se sabe de la geometria de estas variedades en dimension 3. Luego, introduciré varias preguntas respecto a las propiedades estadísticas del flujo geodésic

2012-09-03

Michal Szostakiewicz. Varsovia
Statistical properties of rational maps: introduction
Sala 1 Facultad de Matemáticas - PUC 15:30 a 16:30 Hrs.
Abstract:
I will explain some basic properties of rational maps on the Riemann Sphere from a dynamic point of view and present a list of known results concerning their statistical properties.

After short geometric introduction, I want to focus on describing Perron-Frobenius operator and using it to construct equilibrium states with Holder continous potential in this setting.

I will also connect convergence of this operator with the statistical properties of the map. I will mention the known results about this convergence and say something about techniques of proving them. If I have time, I will tell more about Young´s towers.

This talk will serve as an introduction to the mini-course with the same name.

2012-08-27

Martin Andersson. Uff-Niteroi
Comportamiento ergódico extraño en el mundo C^0 genérico
Auditorio del Depto. de Matemática USACH
Abstract:
Resumen:
Voy a exponer algunos temas de la teoría ergódica de sistemas C^0 genéricos. Lo más notable, en este contexto, es un resultado de la existencia q.t.p. de promedios de Birkhoff y la sorprendente falta de medidas físicas.
En colaboración con Flávio Abdenur.

2012-08-20

Jimena Royo-Letelier. Université de Versailles Saint-Quentin-En-Yvelines, France
Two-component Bose-Einstein Condensates
Sala 1 de la Facultad de Matemáticas PUC - 16:30
Abstract:
We deal with minimizers of the coupled Gross-Pitaevskii energy of a two-component Bose-Einstein condensate. We will show the links between our model and other mathematical problems : optimal partition problems and the Cahn-Hilliard model for phase transitions. These links appear in our model in the limit strongly repulsive limit, so the two components segregate. We show that in the weakly interacting regime, the minimal configuration of the segregated minimizers is two half-planes.

2012-05-28

Sandro Vaienti. Centre de Physique Théorique (Marseille)
Escape Rates Formulae and Metastablilty for Randomly perturbed maps
Sala 1, Facultad de Matemáticas - PUC 16:30 Hrs.
Abstract:
Resumen:
(Joint with W Bahsoun) We provide escape rates formulae for piecewise expanding interval maps with `random holes´. Then we obtain rigorous approximations of invariant densties of randomly perturbed metabstable interval maps. We show that our escape rates formulae can be used to approximate limits of invariant densities of randomly perturbed metastable systems.

2012-05-14

Radu Saghin. Puc-Valparaíso
Volume growth and entropy for partially hyperbolic diffeomorphisms
USACH (sala por confirmar)
Abstract:
Resumen: I will present an inequality between the metric entropy, Lyapunov exponents, and an invariant measuring the growth of the volume in the direction of the unstable foliation (the integrated volume growth of $W^u$) of a $C^1$ partially hyperbolic diffeomorphism. I will discuss some situations where this integrated volume growth is locally constant, and several applications.

2012-05-07

Daniel Coronel. Pontificia Universidad Católica de Chile
Low-temperature phase transitions in the quadratic family
Sala 403 del edificio R5 (República 399, 4to. piso), UNAB
Abstract:
We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function.
This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is
non-uniformly hyperbolic in a strong sense. Joint work with Juan Rivera-Letelier.