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

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:
(trabajo en conjunto con Sebastían Barbieri)
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.
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.
2017-09-11
15:30hrs.
Eduardo Garibaldi. Universidade Estadual de Campinas
An Alphabetical Approach To The Nivat?s Conjecture
Sala 1, PUC
Abstract:
Nivat’s conjecture claims that only periodic configurations on a two-dimensional integer lattice may satisfy a low complexity assumption. Since techniques used to address the Nivat’s conjecture usually relies on Morse-Hedlund Theorem, an improved version of this classical result may mean a new step towards a proof for the conjecture. In this talk, we discuss how, following methods highlighted by Cyr and Kra, an extension of the so far best known result to the Nivat’s conjecture may be derived from an alphabetical version of Morse-Hedlund Theorem. This a joint work with C. Colle.
2017-08-28
16:30hrs.
Rodolfo Gutierrez. Paris Vii
Clasificación de los Grupos de Rauzy?veech
Sala von Neumann, 7mo piso, CMM
Abstract:
El algoritmo de renormalización de Rauzy–Veech es un poderoso método para el estudio de la dinámica de las transformaciones de intercambios de intervalo y de los flujos de traslación. En efecto, se puede interpretar como una discretización del cociclo Kontsevich–Zorich, que es la parte no trivial de la derivada del flujo de Teichmüller. La acción en homología de este algoritmo es un subgrupo de Sp(2g, Z) y se conoce como un grupo de Rauzy–Veech. En 1999, Zorich conjeturó que estos grupos son densos en Sp(2g, R) para la topología de Zariski. Las consecuencias dinámicas de esta conjetura son múltiples: en particular, generaliza la demostración de Avila–Viana de la conjetura de Kontsevich–Zorich, la que postula la simplicidad del espectro de Lyapunov de casi toda transformación de intercambio de intervalos o flujo de traslación.
 
El trabajo pionero de Avila–Matheus–Yoccoz estableció esta conjetura para el caso hiperelíptico. Expandiendo sus técnicas, es posible encontrar una clasificación completa de los grupos de Rauzy–Veech, demostrando, en particular, la conjetura de Zorich. Por ejemplo, para los estratos conexos de superficies de género g ≥ 3 estos grupos resultan ser Sp(2g, Z).
2017-08-21
16:30hrs.
Angel Pardo. Université Aix-Marseille
Billares en Polígonos, Superficies Planas y Dinámica en Espacios de Moduli
Sala von Neumann, 7mo piso, CMM
2017-07-10
16:30hrs.
Wenbo Sun. Ohio State University
Equidistribution of Dilated Curves.
Sala von Neumann, 7mo piso, CMM
Abstract:

Consider a light source located in a polynomial room. It is a classic question whether the whole room is illuminated by the light. This question was recently settled by Leli?evre, Monteil and Weiss. In this talk, we study the variation on the illumination problem introduced by Chaika and Hubert in the context of closed curves on nilmanifolds. We give necessary and suffcient conditions for a nilmanifold being illuminated by a curve.

 
2017-06-19
16:30hrs.
Anibal Velozo. Princeton University
Entropy Theory for Geodesic Flows
Sala 1
Abstract:
The geodesic flow on negatively curved manifolds is one of the classical examples of  Anosov flows. In this talk I will review the main features of the ergodic theory for the geodesic flow on negatively curved manifolds, which are not necessarily compact. We will then focus on the study of the entropy map for such systems. In the non-compact setting a sequence of probability measures can lose mass. A novel feature of our result is that relates the escape of mass of a sequence of ergodic measures with their measure theoretic entropy.  As an application we obtain a criterion for the existence of measure of maximal entropy for geometrically finite manifolds due to Dalbo, Otal and Peigne. Our method has the advantage to also cover nonpositively curved manifolds with certain properties. Part of this talk is based on joint work with F. Riquelme.
2017-05-22
16:30hrs.
Jaqueline Siqueira. Puc-Rio
Equilibrium States of Partially Hyperbolic Horseshoes: Uniqueness and Statistical Properties.
Sala 1
Abstract:
We prove uniqueness of equilibrium states of partially hyperbolic horseshoes associated to Holder continuous potentials with small variation. (Joint  work with Isabel Rios). In order to derive some statistical properties for the unique equilibrium state  we define a projection map associated to the horseshoe and prove a spectral gap for its transfer operator acting on the space of Holder continuous observables. From this we deduce an exponential decay of correlations and a central limit theorem. Finally, we extend these results to the horseshoe. ( Joint work with Vanessa Ramos).
2017-05-15
16:30hrs.
Robin Tucker-Drob. Texas A&m University
Inner Amenable Groupoids and Compact Actions
Sala J. Neumann, CMM
Abstract:
We introduce the notion of inner amenability for discrete p.m.p. (=probability measure preserving) groupoids which generalizes the notion of inner amenability of groups. In the special case of of p.m.p. equivalence relations, this gives a new orbit equivalence invariant. We show that the orbit equivalence relation associated to any free compact action of an inner amenable group is itself inner amenable as a groupoid. Conversely, any group which freely generates an inner amenable p.m.p. equivalence relation must itself be inner amenable.
2017-05-08
16:30hrs.
Tuomas Sahlsten. University of Bristol
Quantum Ergodicity and Limit Multiplicities
Sala 1
Abstract:
We will give an introduction to the topic of “quantum ergodicity” and review the history and current challenges of the problem. The quantum ergodicity theorem states that on Riemannian surfaces with an ergodic geodesic flow, most eigenfunctions of the Laplacian equidistribute spatially in the large eigenvalue limit. In this talk, we will present an alternative equidistribution theorem for eigenfunctions where the eigenvalues stay bounded and we take instead sequences of compact hyperbolic surfaces that become large in, say, volume. Thus the result combines quantum ergodicity with the theory of limit multiplicities in spectral theory (after DeGeorge and Wallach).

The approach is motivated by the recent works of Anantharaman, Brooks, Le Masson, and Lindenstrauss on eigenvectors of the discrete Laplacian on regular graphs, and the holomorphic form analogues by Nelson, Pitale and Saha. In the dynamics side of the proof we require the exponential mixing structure of the geodesic flow on hyperbolic surfaces, in particular a quantitative mean ergodic theorem by Nevo.

This is a joint work with Etienne Le Masson (Bristol).
2017-04-24
16:30hrs.
Godofredo Iommi. Puc-Chile
Termodinámica de la Transformación de Jacobi-Perron
Sala 1, Fac Mates, PUC
Abstract:
El algoritmo de Jacobi-Perron provee aproximaciones simultáneas a dos números reales por racionales con denominadores comunes. En esta charla discutiré cómo una variante del formalismo termodinámico no aditivo (desarrollado conjuntamente con Yuki Yayama) permite estudiar la calidad de dichas aproximaciones. Este es parte de un trabajo en desarrollo realizado en conjunto con Jairo Bochi y Pablo Shmerkin.
2017-04-17
16hrs.
Arnaldo Nogueira. Inst. Mat. Marseille
Topological Dynamics of Piecewise \lambda-Affine Maps of The Interval
Sala J. Neumann CMM
Abstract:
Let 0 < a < 1, 0 ≤ b < 1 and I = [0,1). We call contracted rotation the interval map φa,b : x  I  ax+b mod1. Once a is fixed, we are interested in the dynamics of the one-parameter family φa,b, where b runs on the interval interval [0, 1). Any contracted rotation has a rotation number ρa,b which describes the asymptotic behavior of φa,b. In the first part of the talk, we analyze the numerical relation between the parameters a, b and ρa,b and discuss some applications of the map φa,b. Next, we introduce a generalization of contracted rotations. Let −1 < λ < 1 and f : [0, 1)  R be a piecewise λ-affine contraction, that is, there exist points 0 = c0 < c1 < ··· < cn−1 < cn = 1 and real numbers b1,...,bn such that f(x) = λx + bi for every x [ci−1,ci). We prove that, for Lebesgue almost every δ  R, the map fδ = f + δ (mod 1) is asymptotically periodic. More precisely, fδ has at most n + 1 periodic orbits and the ω-limit set of every x  [0, 1) is a periodic orbit. 
2017-04-10
16:00hrs.
Sebastian Donoso. Universidad O'higgins
Quantitative Multiple Recurrence for Two and Three Transformations.
Sala J. Neumann, CMM
Abstract:
In this talk I will provide some counter examples for quantitative multiple recurrence problems for systems with more than one transformation.  For instance, I will show that there exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $\ell < 4$ there exists $A\in \mathcal{X}$ such that \[ \mu(A\cap T_1^n A\cap T_2^n A) < \mu(A)^{\ell} \] 
for every $n \in \mathbb{N}$. 

The construction of such a system is based on the study of ``big'' subsets of $\mathbb{N}^2$ and $\mathbb{N}^3$  satisfying combinatorial properties.
 
This a joint work with Wenbo Sun.
2017-03-13
17:00hrs.
Tanya Firsova. Kansas State University
Deformation Spaces of Rational Functions
Sala 1, Fac. Mates, PUC
Abstract:
A celebrated Theorem of W.Thurston gives a topological condition when a postcritically finite branched cover can be realized by a rational map. A.Epstein, building on the work of Thurston, studied the spaces of maps constrained by certain postcritically finite relations. He defined deformation spaces for such maps that live in certain Teichmuller spaces. Epstein proved transversality results in holomorphic dynamics using deformation spaces. 

We will discuss how these deformation spaces relate to the ones studied by Mary Rees. We will also discuss topological properties of the Epstein's deformation spaces and give a sufficient condition that guarantees that a given deformation space is not contractible. This is a joint work with J. Kahn and N. Selinger.
2017-03-13
16:00hrs.
Mao Shinoda. Keio University
The Existence of a Dense Subset of Uncountably Maximized Continuous Functions
Sala 1, Fac. Mates, PUC
Abstract:
The main purpose of the ergodic optimization is to single out invariant measures which maximize the space average of a performance function on a dynamical system. We mainly consider a dynamical system defined by a continuous self-map on a compact metric space. There is a major conjecture that for ``many" performance functions there exist unique maximizing measures and the unique measures are supported by a single periodic orbit. Jenkinson shows that for a generic continuous function there exists unique maximizing measure. We prove, on the other hand, there exits a dense subset of continuous functions which have uncountably many ergodic maximizing measures. The main idea of our proof is the application of the Bishop Phelps theorem to the context of maximizing measures.