# 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

2017-10-02
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.
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.
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.
2017-01-16
17:00hrs.
Luna Lomonaco. Usp
The Mandelbrot Set And Its Satellite Copies
Sala 1, Fac. Mates, PUC
Abstract:
For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family $P_c(z)=z^2+c$. The Mandelbrot set M  is the set of parameters c such that the filled Julia set of $P_c$ is connected. Douady and Hubbard, using renormalization, proved the existence of homeomorphic copies of M inside of M, which can be primitive (if, roughly speaking, they have a cusp) or satellite (if they don't). They conjectured that the primitive copies of M are quasiconformal homeomorphic to M, and that the satellite ones are quasiconformal homeomorphic to M outside any small neighbourhood of the root. These results are now theorems due to Lyubich. The satellite copies are not quasiconformal homeomorphic to M, but are they mutually quasiconformally homeomorphic? In a joint work with C. Petersen we prove that this question, which has been open for about 20 years, has in general a negative answer.
2017-01-16
16:00hrs.
Jiangang Yang. Uff
Continuity Of Lyapunov Exponents In The C0 Topology
Sala 1, Fac. Mates, PUC
Abstract:
This is a joint with Marcelo Viana. We prove that the Bochi-Mañé theorem is false, in general, for linear cocycles over non-invertible maps: there are $C_0$-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.
2016-09-12
16:30hrs.
Mitsuhiro Shishikura. U. Kioto
Tropical Complex Dynamics
Sala 1, Fac. Mates, PUC
Abstract:
Complex rational maps induce rich and interesting dynamics on the Riemann sphere.
The sphere is divided into two sets: the Fatou set where the dynamics is tame, and the Julia set where the dynamics is chaotic.
For a rational map with non-empty Fatou set, one can associate a piecewise linear map on a tree.  From this "tree map", on "toropicalized complex dynamics", we can derive
some information on whether certain type of dynamics can be realized, or at which degree such dynamics can be realized.  This tree map is supposed to describe the degeneration of rational maps under the limit of quasiconformal deformation, or the boundary of the moduli space. In this talk, we will discuss various problems related to the tropical complex dynamics.
2016-09-05
16:30hrs.
Sebastian Herrero. Pontificia Universidad Católica de Chile
Distribución Asintótica de Puntos Hecke en C_P.
Sala 1, Facultad de Matematicas, PUC
2016-08-29
16:30hrs.
Eduardo Oregón. Pontificia Universidad Católica de Chile
Propiedades de Conjuntos de Isometrías en Espacios Gromov Hiperbólicos
Sala seminarios (4to piso), Dpto. Mates, USACH.
Abstract:
En esta charla probaremos una desigualdad sobre isometrías en un espacio Gromov hiperbólico, que no requiere que el espacio sea propio o geodésico. Ésta tratará sobre el desplazamiento estable generalizado, una versión hiperbólica del radio espectral generalizado (o joint spectral radius), mostrándonos que los conjuntos de isometrías se comportan como conjuntos de matrices reales de 2x2. Además discutiremos consecuencias de la desigualdad, como la continuidad del desplazamiento estable generalizado y un análogo al teorema de Berger-Wang
2016-08-22
16:30hrs.
Mike Todd. University Of St Andrews, Escocia
Stability Of Measures In Interval Dynamics
Sala 1, Fac. Mates, PUC.
Abstract:
Given a family of interval maps, each map possessing a canonical measure (an invariant measure absolutely continuous w.r.t. Lebesgue - an acip), we have a weak form of stability if these measures change continuously through the family.  Even for uniformly hyperbolic dynamical systems this stability can fail.  I’ll give minimal conditions for a class of non-uniformly hyperbolic interval maps to satisfy this stability property.  This work forms part of a paper with Neil Dobbs, where more general thermodynamic properties are proved to be stable (entropy, pressure, equilibrium states), and I’ll give some indication of the general approach there.