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;
Godofredo Iommi. Puc-Chile
Termodinámica de la Transformación de Jacobi-Perron
Sala 1, Fac Mates, PUC
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.
Arnaldo Nogueira. Inst. Mat. Marseille
Topological Dynamics Of Piecewise \lambda-Affine Maps Of The Interval
Sala J. Neumann CMM
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. 
Sebastian Donoso. Universidad O'higgins
Quantitative Multiple Recurrence For Two And Three Transformations.
Sala J. Neumann, CMM
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.
Mao Shinoda. Keio University
The Existence Of a Dense Subset Of Uncountably Maximized Continuous Functions
Sala 1, Fac. Mates, PUC
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.
Tanya Firsova. Kansas State University
Deformation Spaces Of Rational Functions
Sala 1, Fac. Mates, PUC
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.
Jiangang Yang. Uff
Continuity Of Lyapunov Exponents In The C0 Topology
Sala 1, Fac. Mates, PUC
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.
Luna Lomonaco. Usp
The Mandelbrot Set And Its Satellite Copies
Sala 1, Fac. Mates, PUC
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.
Mitsuhiro Shishikura. U. Kioto
Tropical Complex Dynamics
Sala 1, Fac. Mates, PUC
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.
Sebastian Herrero. Pontificia Universidad Católica de Chile
Distribución Asintótica de Puntos Hecke en C_P.
Sala 1, Facultad de Matematicas, PUC
Eduardo Oregón. Pontificia Universidad Católica de Chile
Propiedades de Conjuntos de Isometrías en Espacios Gromov Hiperbólicos
Sala seminarios (4to piso), Dpto. Mates, USACH.
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
Mike Todd. University Of St Andrews, Escocia
Stability Of Measures In Interval Dynamics
Sala 1, Fac. Mates, PUC.
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.
Eduardo Oregón
Propiedades de Conjuntos de Isometrías de Espacios Gromov-Hiperbólicos
Sala 3 de la Facultad de Matemáticas PUC a las 14:00 Hrs.
Arnaldo Nogueira. Institut de Mathématiques de Marseille , Francia
Piecewise Contraction Maps And Its Applications
Sala de Seminarios John Von Neumann CMM, séptimo piso a las 16:00 Hrs.
Glenn Merlet. Iml Marseille
Limit Theorems For Products Of Non-Negative And Tropical Random Matrices.
Sala Von Neumann, 7mo piso CMM
Carolina Canales. Orsay
Hipersuperficies Levi-Flat en Las Superficies Complejas
Sala Von Neumann, 7mo piso CMM.
Eduardo Oregón
Propiedades de Conjuntos de Isometrías de Espacios Gromov-Hiperbólicos
Sala 3 de la Facultad de Matemáticas PUC a las 14:00 Hrs.
Clark Butler. University Of Chicago
Continuity Of Measurable Invariant Conformal Structures
Sala 1 Facultad de Matemáticas a las 15:00 Hrs.
John Franks. Northwestern University.
The Fine Structure Of Entropy Zero Area Preserving Diffeomorphisms Of Surfaces
Sala Seminarios, Dpto. Matematicas y C.C, USACH a las 16:30 Hrs.
Alexis Moraga. P. Universidad Católica de Chile
El Teorema Ergódico de Karlsson- Ledrappier
Sala 1 - Facultad de Matemáticas a las 15:30 Hrs.
Vincent Delecroix. Cnrs-Cmm
Dinámica Genérica de Las Superficies de Translación
16.30 hrs, Sala John Von Neumann, 7mo piso CMM.