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.

Su coordinador es Cristóbal Rivas; cristobal.rivas@usach.cl

2018-08-13
16:30hrs.
Alejandro Kocsard. Instituto de Matemática e Estatística Universidade Federal Fluminense
Desvíos Rotacionales Para Mapas del Toro y Aplicaciones.
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
El número de rotación de Poincaré es sin duda alguna el invariante más importante en el estudio dinámico de homeomorfismos del círculo (que preservan orientación). En general, estos sistemas exhiben lo que llamamos "desvíos rotacionales uniformemente acotados", es decir, cualquier órbita de un homeomorfismo de este tipo siempre se mantiene a distancia uniformemente acotada de la órbita de la rotación rígida correspondiente. Esta importante propiedad tiene implicaciones profundas en dinámica unidimensional.

En dimensiones superiores, en analogía con la teoría de Poincaré del círculo, es posible definir el "conjunto de rotación" de homeomorfismos del d-toro homotópicos a la identidad, que a diferencia del caso unidimensional, en general no se reduce a un punto.

En esta charla discutiremos varias consecuencias de la acotación uniforme de los desvíos rotacionales en dimensiones superiores, enfocándonos fundamentalmente en homeomorfismos sin puntos periódicos en dimensión 2. También presentaremos algunos resultados recientes que relacionan la geometría del conjunto de rotación con la acotación a priori de los desvíos rotacionales. 
2018-07-23
16:30hrs.
Daniel Coronel. Unab
Sensitive Dependence of Geometric Gibbs Measures At Positive Temperature
Av. República 399, edificio R5 sala 101, UNAB
Abstract:
In this talk we give the main ideas of the construction of  the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states diverge at positive temperature. These are the first examples of divergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel.
2018-07-09
17:30hrs.
Umberto Hryniewicz. Universidade Federal Do Rio de Janeiro
Morse Theory for The Action Functional and a Poincare-Birkhoff Theorem for Flows
Sala 2, PUC
Abstract:
The goal of this talk is twofold. Firstly I would like to explain how pseudo-holomorphic curves can be used to study Morse theory of the action functional from classical mechanics. Then I will move to applications, focusing on a generalization of the Poincare-Birkhoff theorem for Reeb flows on the three-sphere.
2018-07-09
16:30hrs.
Anibal Velozo. Princeton
Analogies Between The Geodesic Flow on a Negatively Curved Manifold and Countable Markov Shifts
Sala 2, PUC
Abstract:
By the work of Bowen and Ratner it is known that the geodesic flow on a compact negatively curved manifold can be modeled as a suspension flow over a subshift of finite type. Unfortunately, a symbolic representation is not available if the manifold is non-compact. In this talk I will briefly explain some recent developments on the study of the thermodynamic formalism of the geodesic flow on non-compact negatively curved manifolds. Surprisingly some of the methods used to understand the geodesic flow have consequences to the theory of countable Markov shifts. I will explain such consequences, as well as some open problems.
2018-07-05
16:30hrs.
Andreas Koutsogiannis. The Ohio State University
Norm and Pointwise Convergence of Averages of Multiple Ergodic Averages and Applications
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann).
Abstract:
Via the study of multiple ergodic averages for a single transformation, Furstenberg, in 1977, was able to provide an ergodic theoretical proof of Szemerédi's theorem, i.e., every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions. We will present some recent developments in the area for more general averages, e.g., for multiple commuting transformations with iterates along specific classes of integer valued sequences. We will also get numerous applications of the aforementioned study to number theory, as we will present the corresponding results along prime (and shifted prime) numbers, topological dynamics and combinatorics. Finally, we will present a result to the most general, and far more difficult case of pointwise convergence along special sublinear functions. This is part of independent, as well as joint work with D. Karageorgos (norm case); and S. Donoso and W. Sun (pointwise case).
2018-05-31
16:30hrs.
Pierre Arnoux. Université Aix-Marseille
Multidimensional Continued Fractions and Symbolic Dynamics for Toral Translations
CMM
Abstract:
We give a dynamical, symbolic and geometric interpretation to multi-dimensional continued fractions algorithms. For some strongly convergent algorithms, the construction gives symbolic dynamics of sublinear complexity for almost all toral translations; it can be used to obtain a symbolic model of the diagonal flow on lattices in $\mathbb R^3$.
2018-05-28
16:30hrs.
Felipe Riquelme. Pontificia Universidad Católica de Valparaíso
Entropías Intermedias y Temperatura Nula en Curvatura Negativa
sala 2
Abstract:
Un problema bastante general en teoría ergódica consiste en estudiar al conjunto de entropías de un sistema dinámico respecto a sus medidas ergódicas. Katok conjeturó que dicho conjunto contiene al intervalo $[0,h_{top}(f))$ en el caso de difeomorfismos suaves en variedades compactas. Si bien la conjetura permanece abierta, muchos avances se han logrado a la fecha. Se conoce, por ejemplo, que el flujo geodésico en variedades compactas a curvatura negativa verifica esta propiedad. La demostración de esto último recae en la realización del flujo geodésico como un flujo de suspensión sobre un shift de Markov de tipo finito.

En esta charla mostraremos que la tesis de la conjetura sigue siendo válida para el flujo geodésico sin la hipótesis de compacidad. Ante la ausencia de una realización simbólica genérica, las herramientas de la demostración serán puramente geométricas. Estas consisten en gran parte en el estudio del formalismo termodinámico del sistema, particularmente en los estados a temperatura nula. Este trabajo es un trabajo en curso junto a Anibal Velozo.
2018-05-22
15:30hrs.
Constantine Medynets. Mathematics Department At The United States Naval Academy
Invariant Random Subgroups of Full Groups of Bratteli Diagrams
Sala de Seminarios Depto de Matemáticas USACH.
Abstract:
In the talk, we will classify the ergodic invariant random subgroups (IRS) of simple AF full groups. AF full groups arise as the transformation groups of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram.  AF full groups are complete (algebraic) invariants for the isomorphism of Bratteli diagrams. Given a simple AF full group $G$, we will prove that every ergodic IRS of $G$ arises as the stabilizer distribution of a diagonal action on $X^n$ for some $n$, where $X$ is the path-space of the Bratteli diagram associated to $G$. This is joint work with Artem Dudko.
2018-05-14
16:30hrs.
Eduardo Jorquera. Pontificia Universidad Católica de Valparaíso
Common Fixed Points of Set-Valued Mappings in Hyperconvex Metric Spaces
Sala Seminarios, DMCC, USACH
Abstract:
In this work, we establish several common fixed point theorems for families of set-valued mappings defined in hyperconvex metric spaces. Then we give several applications of our results. This is a joint work with M. Balaj and M. A. Khamsi. Eduardo.
2018-03-12
16:30hrs.
Thomas Fernique. Paris Xiii, Lipn
Local Rules for Planar Tilings
Sala von Neumann, CMM
Abstract:
The cut and project method is one of the prominent method to define quasiperiodic tilings. In order to model quasicrystals, where energetic interactions are only short range, it is important to know which of these tilings can be characterized by local configurations (in dynamical terms: which of these tiling spaces are of finite type or sofic). In this talk we shall review known results, in particular those obtained these last years with Nicolas Bedaride and Mathieu Sablik. 
2018-01-22
16:30hrs.
Marcelo Sobottka. Ufsc
Curtis-Hedlund-Lyndon Theorem for Ultragraph Shift Spaces
Sala von Neumann, CMM
Abstract:
In this work we characterize the class of continuous shift commuting maps between ultragraph shift spaces, proving a Curtis-Hedlund-Lyndon type theorem. Then we use it to characterize continuous, shift commuting, length preserving maps in terms of generalized sliding block codes.

This is a joint work with Prof. Daniel Gonçalves  (UFSC, Brazil) 
2018-01-15
16:30hrs.
Piere Berger. Paris Xiii
The Herman Positive Entropy Conjecture
Sala 1, PUC
Abstract:
We show that any sympletic $C^\infty$-diffeomorphism which displays a non-hyperbolic periodic point can be $C^\infty$-approximated by a symplectic one whose metric entropy is positive.

The proof involves developments of the concept of universal dynamics, some developments of Katok, Przytycki, Arroyo-Pujals examples, and new techniques to ensure that up to the composition with a map of the form
$(x,y)\mapsto (x,y+\phi(x))$ with $\phi$ small, some heteroclinic links can be restored (i.e. the stable and unstable manifolds of some hyperbolic periodic points coincide).

Joint work with Dimitry Turaev.
2017-12-18
16:30hrs.
Jiagang Yang. U. F. Fluminense
A Rigidity Property of 3 Dimensional Partially Hyperbolic Anosov Diffeomorphism
Sala 1, PUC
Abstract:
This is a joint work with Abel Rios Bravo and Radu Saghin.

It is well known that, on 3 dimensional torus, every Anosov diffeomorphism $f$ is conjugate to a linear Anosov diffeomorphism $A$ by a homeomorphism $h$.
We are going to show that, if $f$ is $C^2$, volume preserving and partially hyperbolic, and the Lyapunov exponents of $f$ are the same with the Lyapunov exponents of $A$, then $h$ is indeed smoooth.
2017-12-11
16:30hrs.
Felipe García. Universidad Autónoma de San Luis Potosí
Entropía Topológica y Parejas Asintóticas en Acciones de Grupo
Sala von Neumann, CMM
Abstract:
En esta charla hablaremos de la relación entre entropía topologica y las parejas asíntoticas ( (x,y) es una pareja asíntótica si para g grande gx y gy están arbitrariamente cercanos). 
En particular nos enfocaremos en acciones expansivas y con sombreo (en ingles shadowing o pseudo orbit tracing).

Veremos que si el grupo es promediable las parejas asíntoticas son densas en las parejas de entropía, pero que esto no es necesariamente cierto si el grupo no es promediable. 
También estudiaremos la relación con condiciones suficientes para tener entropía positiva completa  en shifts de tipo finito estudiadas por Pavlov y daremos un ejemplo para contestar negativamente una de sus preguntas.

(trabajo en conjunto con Sebastían Barbieri)
2017-12-04
16:30hrs.
Pablo Shmerkin. Universidad Torcuato Di Tella
Intersecciones de Conjuntos de Cantor
Sala 1 Fac. Mates. PUC
2017-11-27
*Horario especial* 17:00hrs.
Lev Birbrair. Federal University of Ceara, Fortaleza, Brazil
Focal Decomposition of Peixoto and Related Problems in Geometry, Number Theory and Combinatorics
Sala Neumann, CMM
Abstract:
Focal Decomposition is a Geometric Object, created by Peixoto in oder to study qualitative properties of Differential Equations of Second Order. We are going to discuss some topologiacal and arithmetical
structures related to the Focal Decompositions.
2017-11-20
16:30hrs.
Hongming Nie. Indiana University
Rescaling Limits for Newton?s Method
Sala 1
Abstract:
In this talk, I will give a complete description of the rescaling limits for holomorphic families of degree $d\ge 3$ Newton’s method. The main ingredients are Berkovich dynamics and weak limits of measures of maximal entropy.
 
2017-11-13
16:00hrs.
Mathieu Hoyrup. Inria Loria
Computability in Ergodic Theory
Edificio R-5 (Av. Republica 399), SALA 202, UNAB
Abstract:
A common task when analysing or simulating dynamical systems is to design algorithms that compute quantities or characteristics associated to a given system: entropy, attractor, invariant measures, etc. We study this general problem in a theoretical way, requiring the algorithms to correctly compute the objects at any precision. In this approach one often reaches the logical limitations of the computer, leading to negative results: such algorithms sometimes do not exist. I will present a few such results and give a flavour of the technics involved, often based on toplogical considerations.
2017-11-13
17:00hrs.
Lorenzo Sadun. University of Texas Austin
Tiling Spaces and Their Homeomorphisms
Edificio R-5 (Av. Republica 399), SALA 202, UNAB
Abstract:
Tiling spaces, and the dynamics induced by translation, have connections to many areas of mathematics. One dimensional tiling spaces generalize symbolic dynamics. Substitution tiling spaces (in one or higher dimensions) model expanding attractors. The dynamical properties of higher dimensional tiling spaces describe the diffraction properties of physical quasicrystals. In this talk, I'll review the dynamics and topology of tiling spaces and then present some new results on understanding homeomorphisms between tilings spaces. To wit: under some mild assumptions, every homeomorphism of tiling spaces is the composition of three maps: a self-map homotopic to the identity, a "shape change" that preserves the combinatorics of the tilings but distorts the shapes and sizes of the tiles, and a local relabeling.
2017-10-02
16:30hrs.
Nicolas Bédaride. Université Aix-Marseille
Thermodynamic Formalism and Substitutions
Sala von Neumann, CMM
Abstract:
We give sufficient conditions on a uniquely ergodic subshift K (into the full D-shift) such that an explicit family of potentials have freezing phase transition with ground state supported onto K. Then, we exhibit a class of substitutions, which contains the Thue-Morse substitution, such that the associated attractor K satisfies the previous conditions.