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

2018-11-26
16:30hrs.
Michael Schraudner. CMM
Tba
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
2018-11-26
17:30hrs.
Samuel Petite. U de Picardie, Francia
Tba
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
2018-11-19
16:30hrs.
Sebastián Donoso. Universidad de O'higgins
Optimal lower bounds for multiple recurrence
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
2018-11-19
17:30hrs.
Fabien Durand. U. Picardie
Decidability of the isomorphism and the factorization for minimal substitution subshifts
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
Classification is a central problem in the study of dynamical systems, in particular for families of systems that arise in a wide range of topics. Hence it is important to have algorithms deciding wether a dynamical 
system have some given property.

Let us mention subshifts of finite type that appear, for example, in information theory, hyperbolic dynamics, $C^*$-algebra, statistical mechanics and thermodynamic formalism. The most important and longstanding open problem for this family originates in [Williams:1973] and is stated in [Boyle:2008] as follows : Classify subshifts of finite type up to topological isomorphism. In particular, give a procedure which decides when two non-negative 
integer matrices define topologically conjugate subshifts of finite type. 

Another well-known family of subshifts, that is also defined through matrices, with a wide range of interests is the family of substitution subshifts. These subshifts are concerned, for example, with automata theory, first order logic, combinatorics on words, quasicrystallography, fractal geometry, group theory and number theory. In this talk we will show that not only the existence of isomorphism between such subshifts is decidable but also the factorization.
2018-11-12
16:30hrs.
Yuki Yayama. Ubiobio
Tba
Sala 2
2018-11-05
16:30hrs.
Rhiannon Dougall. University of Bristol
Critical exponents for normal subgroups via a twisted Bowen-Margulis current and ergodicity
Sala 2
Abstract:
For a discrete group $\Gamma$ of isometries of a negatively curved space $X$, the critical exponent $\delta(\Gamma)$ measures the exponential growth rate of the orbit of a point. When $X$ is a manifold, this can be rephrased in terms the growth of periodic orbits for the geodesic flow in the $\Gamma$-quotient. We fix a group $\Gamma_0$ with good dynamical properties, and ask for $\Gamma < \Gamma_0$, when does $\delta(\Gamma)=\delta(\Gamma_0)$? We will motivate this problem, and discuss what is new: the construction of a twisted Bowen-Margulis current on the double-boundary, which highlights a feature of ergodicity, and extends the class for which the result is known. This is joint work with R. Coulon, B. Schapira and S. Tapie.
2018-10-29
16:30hrs.
Zoltan Buczolich. Department of Analysis, Eotvos Lorand University, Budapest, Hungary
Almost everywhere convergence of ergodic averages
Sala 2
Abstract:
In this talk I would like to discuss some of my results concerning almost everywhere convergence of non-conventional ergodic averages of $L^1$ functions.
 
These topics include:
• divergence of ergodic averages along the squares;
• convergence along some sequences of zero Banach density;
• convergence for arithmetic weights: the prime divisor functions $ω$ and $Ω$.
2018-10-22
16:30hrs.
Radu Saghin. Pucv
Exponentes de Lyapunov y rigidez para difeomorfismos hiperbólicos y parcialmente hiperbólicos
Sala 2
Abstract:
Voy a presentar unos resultados de rigidez en termino de exponentes de Lyapunov para difeomorfismos hiperbólicos y parcialmente hiperbólicos. Si un difeomorfismo hiperbólico (o parcialmente hiperbólico) es cerca a un automorfismo lineal (o un skew product sobre un automorfismo lineal), preserva el volumen, y tiene los mismos exponentes de Lyapunov (estables e inestable), entonces es suavemente conjugado al automorfismo lineal (o a un skew product sobre el automorfismo lineal). En el caso de difeomorfismos hiperbólicos el resultado puede ser visto como un análogo a la conjetura de rigidez de la entropía de Katok.
2018-10-01
16:30hrs.
Cristobal Rivas. Usach
Conos de Markov en grupos finitamente generados
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación (Las Sophoras nº 173, Santiago, Estación Central)
Abstract:
Un subconjunto de un grupo se dice cono si es cerrado bajo multiplicación y disjunto de su inverso. En esta charla nos interesamos en estudiar conos que pueden ser descritos por un lenguaje regular (i.e. por un automata). Veremos ejemplos, algunas obstrucciones geométricas generales y finalmente nos enfocaremos en el caso en que el grupo ambiente es hyperbólico.
2018-09-03
16:30hrs.
Çagri Sert. Eth Zürich
Random products of matrices and large deviations
Sala 2
Abstract:
We will start by surveying classical results of Furstenberg, Kesten, Guivarc’h, Le Page, Bougerol, Benoist-Quint and others, on random products of matrices such as a the non-commutative law of large numbers, properties of Lyapunov exponents, central limit theorem etc. In a second part, we will turn to large deviations and talk about the recent result on the existence of large deviation principle for random matrix products. Finally, we will make connections with the recently introduced notion of joint spectrum.
2018-08-27
16:30hrs.
Çagri Sert. Eth Zürich
The joint spectrum
Sala 2
Abstract:
We will define the notion of joint spectrum of a compact subset of $GLd(C)$ which is a multidimensional generalization of joint spectral radius. We will talk about its properties such as convexity, continuity (and discontinuity) and mention its realization and finiteness properties. Finally, we will make connections with random products of matrices. (Joint work with Emmanuel Breuillard).
 
2018-08-20
14:30 [Dynamical Day]hrs.
Mike Todd. University of St Andrews, United Kingdom
Phase transitions and limit laws.
Auditorio Bralic
Abstract:
The 'statistics' of a dynamical system is the collection of statistical limit laws it satisfies.  This starts with Birkhoff’s Ergodic Theorem, which is about averages of some observable along orbits: this is a pointwise result, for typical points for a given invariant measure.  Then we can look for forms of Central Limit Theorem, Large Deviations and so on: these are about how averages fluctuate, globally, with respect to the invariant measure.   In this talk, I’ll show how the form of the `pressure function´ for a dynamical system determines its statistical limit laws.  This is particularly interesting when the system has slow mixing properties, or, even more extreme, in the null recurrent case (where the relevant invariant measure is infinite). I’ll start by introducing these ideas for simple interval maps with nice Gibbs measures and then indicate how this generalizes. This is joint work with Henk Bruin and Dalia Terhesiu.
2018-08-20
15:30 [Dynamical Day]hrs.
Alejandro Kocsard. Universidade Federal Fluminense, Brazil
Cociclos sobre dinámicas hiperbólicas, exponentes de Lyapunov y aplicaciones
Auditorio Bralic
Abstract:
Las orbitas periódicas de los sistemas uniformemente hiperbólicos concentran gran parte de la información dinámica de los mismos. De esta forma, muchas veces es posible estudiar diversas propiedades de cociclos sobre estos sistemas (e.g. exponentes de Lyapunov) observando tan sólo lo que sucede sobre las órbitas periódicas.En esta charla discutiremos los alcances y limitaciones de este enfoque y  algunas aplicaciones.
 
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.