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

2019-05-06
17:00-18:00hrs.
María Isabel Cortez. Usach
Algebraic Invariant of Minimal Group Actions on The Cantor Set: Topological Full Group and Group of Automorphism
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
2019-04-22
16:30--17:30hrs.
Mónica Moreno Rocha. Cimat
On the dynamics of elliptic functions of the form P+b
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
The dynamical system obtained by iteration of the Weierstrass P function over real square lattices can be characterized by the behavior of its single free critical orbit. In contrast, as soon as P is “perturbed” by the addition of a complex parameter b, the elliptic function P+b exhibits at least two free critical orbits, which complicates the study of its dynamics and connectedness locus. This talk I will present some of the results and open questions regarding the rich structures found in dynamical and parameter plane of P+b when b is restricted to a complex line and P is defined over real square lattices. This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.
2019-04-22
15:30-16:20hrs.
Cristobal Rivas. Usach
Sobre el grupo de Higman
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
Les contaré sobre el grupo de Higman. Porqué no tiene cocientes finitos y porqué no admite representaciones lineales. Si aún hay tiempo, diré algunas palabras sobre sus representaciones en grupos de difeomorfismos y homeomorfismos.
2019-04-15
15:30--16:20hrs.
Ryo Moore. PUC
Nonconventional Coboundaries
Sala 1, PUC, Facultad de Matemáticas
2019-04-15
16:30--17:30hrs.
Thomas Jordan. University of Bristol
Multifractal analysis for self-affine systems
Sala 1, PUC, Facultad de Matemáticas
Abstract:
joint work with Balazs Barany, Antti Kaenmaki and Michal Rams. If you consider a uniformly expanding Markov map on the interval and a continuous function. You can consider level sets of point for which the Birkhoff average is some fixed point. A typical problem I need multifractal analysis is to look at the dimension of these level sets. We will show how this can be done using the topological pressure and then how results can be obtained in the setting of certain self-affine sets in two dimensions using the sub-additive pressure and approximation by dominated subsystems.
2019-04-08
15:30--16:20hrs.
Italo Cipriano. PUC
Typical determinant of random matrices
Sala 1, PUC, Facultad de Matemáticas
Abstract:
Let $M_n$ be an $n\times n$ matrix with random entries in $\{+1,-1\}.$   In this talk I will discuss properties of the determinant of these matrices when $n$ tends to infinity. In particular, what is the probability that the determinant is zero? What is the typical value of the determinant? The talk is based on a result in [Tao and Vu, On random +-1 matrices, singularity and determinant. Random Structures and Algorithms, 2005 ].
2019-04-08
16:30--17:30hrs.
Gabriela Alexandra Estevez Jacinto. Universidade de São Paulo
Renormalization of multicritical circle maps
Sala 1, PUC, Facultad de Matemáticas
Abstract:
We study $C^3$ orientation preserving circle homeomorphisms with irrational rotation number and non-flat critical points. By Yoccoz, two of these maps with same irrational rotation are topologically conjugate. In this talk, we define the Renormalization operator of this kind of maps and assuming some properties of this operator we prove that the conjugacy is a $C^{1+\alpha}$ diffeomorphism. This result is valid for a total Lebesgue measure set of irrational rotation numbers. This is a joint work with Pablo Guarino (Universidade Federal Fluminense, Brazil).
2019-04-02
16:00-17:00hrs.
Roberto Markarian . Universidad de la República
Billares Caóticos. Resultados y métodos
PUC, Sala 1
Abstract:
La teoría matemática de billares estudia modelos simples de dinámicas en que hay choques de partículas y con las fronteras de un recipiente. Luego de indicar algunas motivaciones se expondrán propiedades ergódicas y estadísticas sobre las que se han hecho avances sustantivos en los últimos decenios, siguiendo muchos de los lineamientos abiertos, en particular, por la escuela de Ya. G. Sinai
2019-04-01
15:30-16:20hrs.
Sebastián Donoso. Cmm, Universidad de Chile
On subsets with no arithmetic progressions
CMM
Abstract:
For $N\in \mathbb{N}$, let $\nu(N)$ be the maximal cardinality of a subset of \{1,\ldots,N\} that contains no arithmetic progression of length 3. Finding upper and lower bounds for $\nu(N)$ has been a challenging problem for decades. In this talk I will survey this problem and present a proof of a theorem by Behrend in the 40's, that gave a surprising lower bound to $\nu(N)$.
2019-04-01
16:30-17:30hrs.
Angel Pardo. Cmm, Universidad de Chile
Counting problem on infinite periodic billiards and translation surfaces
CMM
Abstract:
The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.

Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.

In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces. The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.

I will first present some quite precise results that are only valid for the wind-tree model (and some natural generalizations) and then, a general result which is valid for a.e. infinite periodic translation surfaces that uses completely different techniques: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups.
2019-03-25
16:30 - 17:30hrs.
Michele Triestino. U. Bourgogne
Cantor dynamics and simple left-orderable groups
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central)
Abstract:
I will present a construction of simple groups of homeomorphisms of the real line.

Given a homeomorphism of a Cantor set $\sigma: X \to X$, consider the suspension $Y=X \times [0,1] / (x,1) \sim (\sigma(x),0)$, and look at the group $H_0(Y)$ of homeomorphisms of $Y$, isotopic to the identity. If $\sigma$ is minimal, then $H_0(Y)$ is simple [Aliste-Prieto - Petite], and I will describe countable subgroups $T(Y)$ which are also simple. These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, $T(Y)$ is finitely generated.

Joint work with Nicolás Matte Bon.
2019-03-18
16:30-17:30hrs.
Yiwei Zhang. Center for Mathematical Sciences, Huazhong University of Science and Technology, China
Understanding physical mixing processes via transfer operator approach
Sala 1, PUC, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, Macul, La Florida
Abstract:
Industrial and chemical mixing processes of various kinds occur throughout nature and are vital in many technological applications.In the context of discrete dynamical systems, the transfer operator approach has been shown as a powerful tools from both theoretic and numerical viewpoint.

In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introductionon the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has "cutting and shuffling'' behaviour (i.e., composing with a permutation).

If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.
2019-03-14
16:30 -- 17:30hrs.
Neil Dobbs. University of Geneva
Tba
Sala 5
2019-03-12
16:30 -- 18:00hrs.
Hamal Hubbard. Cornell University
Construccion de Aplicaciones Pseudo-Anosov
Sala 1
2019-03-11
16:30 -- 18:00hrs.
Hamal Hubbard. Cornell University
Construccion de Aplicaciones Pseudo-Anosov
Sala 1
2019-01-14
16:30hrs.
Alberto Pinto. Faculty of Sciences, University of Porto
Piecewice chaotic maps
Sala 2
Abstract:
We will consider the class of Cr unidimensional piecewise maps with a transitive attractor. These maps can have simultaneously discontinuities, criticalities and singularities. We will show that topological chaos is equivalent to metric chaos. We recall that this result is known in several classes strictly contained in the general class that we are presenting.
2019-01-10
16:30hrs.
Alberto Pinto. Faculty of Sciences, University of Porto
Pseudo-Anosov diffeomorphisms
Sala 2
Abstract:
We will introduce pseudo Cr smooth structures on surfaces that will have the following property; the Pseudo-Anosov diffeomorphisms are uniformly Cr hyperbolic. We will conjecture that the Bochi-Mane theorem will extend to such pesudo C1 smooth structures recovering the duality of the result for all surfaces.
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
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.
2018-12-03
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.