# 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

2018-12-18
16:30hrs.
Alexander Bufetov. Cnrs
DETERMINANTAL POINT PROCESSES
Sala 2
Abstract:
Determinantal point processes arise in a wide range of problems.
How does the determinantal property behave under conditioning?

The talk will first address this question for specific examples such as the sine-process, where one can explicitly write the analogue of the Gibbs condition in our situation.
We will then consider the general case, where, in joint work with Yanqi Qiu and Alexander Shamov, proof is given of the Lyons-Peres conjecture on completeness of random kernels.

The talk is  based on the preprint arXiv:1605.01400
as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexander Shamov.
2018-12-03
14:30hrs.
Felipe Riquelme. Puc-V
Ground states at zero temperature in negative curvature
Sala CS-101
Abstract:
Let $X$ be the unit tangent bundle of a complete negatively curved Riemannian manifold and let $(g_t):X\to X$ be its associated geodesic flow. After the work of R. Bowen and D. Ruelle, it is well know that, if $X$ is compact, then any H\"older-continuous potential $F:X\to\mathbb{R}$ admits an unique equilibrium measure. Moreover, there is a fair enough description of some properties of the pressure map $t\mapsto P(tF)$ such as its regularity and its asymptotic behavior. For non-compact situations, the existence of equilibrium measures has been successfully studied over the last years. Moreover, regularity properties of the pressure map have been established in recent works by G. Iommi, F. Riquelme and A. Velozo.

In this talk we will be interested on the study of ground states at zero temperature for positive H\"older-continuous potentials. More precisely, for $F:X\to\mathbb{R}$ a positive potential going to 0 through infinity, we will study the asymptotic behavior of the equilibrium state $m_{tF}$ for the potential $tF$ as $t\to+\infty$. Indeed, we will show precise constructions of potentials having convergence/divergence to ergodic/non-ergodic ground states. This is a joint work with Anibal Velozo.
2018-12-03
15:45hrs.
Jairo Bochi. PUC
Emergence
Sala CS-101
Abstract:
I will talk about ongoing work with Pierre Berger.

Topological entropy is a way of quantifying the complexity of a dynamical system. It involves counting how many segments of orbit of some length $t$ can be distinguished up to some fine resolution $\epsilon$. If we are allowed to disregard a set of orbits of small measure, then we are led to the concept of metric entropy. Now suppose we don't care \emph{when} a piece of orbit visits a certain region of the space, but only \emph{how often}. Pursuing this idea, we are led to fundamentally new ways of quantifying dynamical complexity. This program was initiated by Berger a couple of years ago.

The first new concept that I'll explain is \emph{topological emergence} of a dynamical system: the bigger it is, the more different statistical behaviors are allowed by the system. We will explain how topological emergence is bounded from above in terms of the dimension of the ambient space. I'll also present examples of dynamical systems where this bound is essentially attained.

Then we'll come to another key concept: \emph{metric emergence} of a dynamical system with respect to a reference measure. Roughly speaking, it quantifies how far from ergodic our system is. (To draw a comparison, topological emergence quantifies how far from uniquely ergodic the system is.) KAM theory reveals that non-ergodicity is somewhat typical among conservative dynamical systems, and metric emergence provides a way of measuring the complexity of the KAM picture. I'll present examples and questions.
2018-12-03
16:50hrs.
Natalia Jurga. University of Surrey
Rigorous estimates on the top Lyapunov exponent for random matrix products
Sala CS-101
Abstract:
We study the Lyapunov exponent of random matrix products of positive $2 \times 2$ matrices and describe an efficient algorithm for its computation, which is based on the Fredholm theory of determinants of trace-class linear operators. Moreover, we obtain rigorous bounds on the error term in terms of two constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the average amount of projective contraction of the positive cone under the action of the matrices. This is joint work with Ian Morris from the University of Surrey.
2018-11-29
16:30hrs.
Gangloff Silvere. Ens de Lyon, Francia
Computing the entropy of multidimensional subshifts of finite type
CMM (Sala B08, Beauchef 851, Torre Poniente)
Abstract:
Multidimensional subshifts of finite type are discrete dynamical systems as a set of colorings of an infinite regular grid with elements of a finite set A together with the shift action. The set of colorings is defined by forbidding a finite set of patterns all over the grid (also called local rules). The most simple and most considered grids of this type are Z2 and more generally Zd for d = 1. In this case, one can consider a coloring as a bi-dimensional and infinite word on the alphabet A.

They are notably involved in statistical physics in the study of so-called lattice models. These models are often simple to describe: for instance, the hard square model is defined on alphabet A = {0, 1} and by forbidding two 1 to appear on horizontally or vertically adjacent positions of the lattice. However, these models and their physical constants, such as the entropy are difficult to apprehend with general methods, and involve specific properties of the considered model.

Although it is known that it is possible to compute the entropy of one-dimensional version of these models by computing the greatest eigenvalue of a matrix which derives from the description of the subshift, this is not possible for multidimensional subshifts. This is the consequence of a result by M. Hochman and T. Meyerovitch in 2010, which states that the possibles values of the entropy for multidimensional subshifts of finite type are the  Π1-computable numbers, including in particular non-computable numbers.

The method developed for this purpose originates in the work of R. Berger and R. Robinson. It has been developed further in order to characterize other dynamical aspects of SFT with computability conditions, with similar constructions. It consists of the implementation of Turing machines in hierarchical structures that emerge from the local rules.

However, models studied in statistical physics obey to strong dynamical constraints and there is still hope to include them into a sub-class of subshifts of finite type for which the entropy is uniformly computable (this means that there is an algorithm which can provide arbitrarily precise approximations of the entropy, provided the precision and the local rules of the subshift). An example of a constraint defining a class where this is verified is the block gluing: this was proved by R. Pavlov and M. Schraudner. This property means that two square blocks can be viewed in any relative positions in some element of the subshift provided that the distance between the two blocks is sufficiently large, with minimal distance not depending on the size of the blocks) is a computable real number. Although they provided a construction to realize some class numbers as the entropy of block gluing SFT, they did not prove a characterization, and this problem seems difficult. However, it could be possible to find broader class for which the entropy is still computable.

A strategy to understand the limit between the general regime where Hochman and Meyerovitch's result holds and this restricted block gluing class is to quantify this property. This means imposing that two patterns can be glued in any two positions in a configuration of the subshift, provided that the distance is great enough, where the minimal distance is a linear function of the size of these patterns.

In a work with Mathieu Sablik, we made a step towards the limit, proving that the result of Hochman and Meyerovitch is robust under the linear version of this property (where the minimal distance function is O(n) where n is the size of the two square blocks). The aim of this talk would be, after a presentation of the problem, to give an insight on the obstacles to this property in the initial construction of Hochman and Meyerovitch, using a construction slightly simpler to present, and on the methods used to overcome the obstacles. These methods involve, in particular, a modification of the Turing machine model and an operator on subshifts that acts by distortion.
2018-11-26
16:30hrs.
Samuel Petite. Université de Picardie Jules Verne, Francia
Restrictions on the group of automorphisms preserving a minimal subshift
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
A subshift is a closed shift invariant set of sequences over a finite alphabet. An automorphism is an homeomorphism of the space commuting with the shift map. The set of automorphisms is a countable group generally hard to describe. We will present in this talk a survey of various restrictions on these groups for zero entropy minimal subshifts.

2018-11-19
16:30hrs.
Arnaldo Nogueira. Université D' Aix-Marseille, Francia
On the action of the semigroup of non singular integral matrices on R^n
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
Hidden Gibbs measures on shift spaces over countable alphabet
Sala 2
Abstract:
We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets. This is a joint work with Godofredo Iommi and Camilo  Lacalle.
2018-11-05
15:45-16:45hrs.
Rhiannon Dougall. University of Bristol
Critical exponents for normal subgroups via a twisted Bowen-Margulis current and ergodicity
Auditorio Bralic
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.
2018-11-05
14:30-15:30hrs.
Optimal lower bounds for multiple recurrence
Auditorio Bralic
2018-11-05
16:50-17:50hrs.
Mario Ponce. Puc-Chile
A geometric approach to the cohomological equation for cocycles of isometries
Auditorio Bralic
Abstract:
We present some geometrical tools in order to obtain solutions to cohomological equations that arise in the reducibility problem of cocycles by isometries of negatively curved metric spaces. The main ingredient is the relation between the solution to the corresponding equation of reducibility for the boundary action and the solution in the metric space. This is a joint work with A. Moraga.
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.
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
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.

2018-08-20
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.
2018-08-13
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.