LXIII Colloquium: Celebrating Rafael Labarca´s 60th Birthday
25-29 September 2017
USACH
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Rafael Labarca obtained his PhD at IMPA, Rio de Janeiro, under the supervision of Jacob Palis, in 1985. He has held positions at IMPA as a research assistant and at Universidad de Santiago de Chile, where he is a full professor. His mathematical work has been devoted mostly to the Bifurcation Theory of Dynamical Systems. His contributions have been deep and groundbreaking. He is one of the fundamental figures in the development of the Chilean Group of Dynamical Systems. He has organised seminars, conferences, and several international schools in the subject. Rafael Labarca has generously supported young researchers in the field of mathematics. Much of his time and effort has been devoted to develop outreach activities, such as the «Campeonato Escolar de Matemáticas» (CMAT) which is one of his most outstanding creations. Recently, he has developed the Project «Academias de Matemáticas», at public primary schools localised in some of the poorest areas of Santiago. Thereby, in the occasion of his 60th birthday, this Colloquium is devoted to celebrate not only his mathematical work but also his enthusiasm and generosity.
Godofredo Iommi
Andrés Navas
Mario Ponce
Bernardo San Martín
Carlos Vásquez
Horario
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Lunes
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Martes
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Miércoles
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Jueves
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Viernes
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9:00 a 09.30
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Acreditación e Inauguración
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9:30- 10:30
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Maria José Pacífico
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Curso 1 JONATHAN CONEJEROS
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Curso 2 FELIPE RIQUELME
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Curso 1 JONATHAN CONEJEROS
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Curso 2 FELIPE RIQUELME
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10:30-10:45
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Café
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Café
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Café
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Café
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Café
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10:50- 11:50
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Solange Aranzubia
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Dante Carrasco
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Marco Uribe
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Alvaro Rovella
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Bernardo San Martín
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12:00- 13:00
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Marcelo Viana
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Jacob Palis
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Carlos Gustavo Moreira (Brasil)
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Lorenzo Díaz
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Andres Navas
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13:00-15:00
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Almuerzo
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Almuerzo
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Almuerzo
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15:00-16:00
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Roberto Markarian
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Milton Jara
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Jan Kiwi
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*Mario Ponce (Divulgación escolares)
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16:00-16:15
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Café
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Café
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Café
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16:15-17:15
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Jairo Bochi
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Alejandro Maass
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Servet Martinez
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*Andrés Navas (Divulgación escolares)
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Nicolás Alvarado (Santiago)
Solange Aranzubía (Santiago)
Alma Armijo (Santiago)
Jairo Bochi (Santiago)
Sebastián Burgos (Santiago)
Dante Carrasco (Concepción)
Alvaro Castañeda (Santiago)
Jonathan Conejeros (Santiago)
Erik Contreras (Santiago)
Lorenzo Diaz (Rio de Janeiro)
Enzo Fuentes
Gerardo Honorato (Valparaíso)
Tadashii Horta (Valparaíso)
Ignacio Huerta (Santiago)
Godofredo Iommi (Santiago)
Milton Jara (Rio de Janeiro)
Jan Kiwi (Santiago)
José López (Valparaíso)
Roberto Markarian (Montevideo)
Vanessa Matus de la Parra (Valparaíso)
Carlos Gustavo Moreira (Rio de Janeiro)
María Isabel Huanca (La Paz)
Alejandro Maass (Santiago)
Servet Martínez (Santiago)
Ruo Moore (Santigao)
Andrés Navas (Santiago)
Bárbara Núñez (Valparaíso)
Gabriel Núñez (Montevideo)
Eduardo Oregón (Santiago)
Maria José Pacifico (Rio de Janeiro)
Jacob Palis (Rio de Janeiro)
Sebastián Pérez
Mario Ponce (Santiago)
Enrique Pujals (Rio de Janerio)
Sebastián Ramírez (Valparaíso)
Ricardo Reyes (Antofagasta)
Felipe Riquelme (Valparaíso)
Cristóbal Rivas (Santiago)
Alvaro Rovella (Montevideo)
Radu Saghin (Valparaíso)
Bernardo San Martín (Antofagasta)
Michael Schraudner (Santiago)
Samuel Tomás (La Paz)
Siming Tu (Santiago)
Marco Uribe (Concepción)
Francisco Valenzuela (Valparaíso)
Carlos Vásquez (Valparaíso)
Samuel Vega (Valparaíso)
Renato Velozo (Santiago)
Marcelo Viana (Rio de Janeiro)
Kendry Vivas (Antofagasta)
Bruno Yemini (Valparaíso)
Felipe Riquelme – Pontificia Universidad Católica de Valparaíso.
Conferencista |
Titulo |
Abstracts |
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Servet Martínez |
Measure evolution of cellular automata and of finitely anticipative transformations |
The evolution of cellular automata and of finitely anticipative transformations is studied by using right sets. These are the sets of symbols that are compatible with a past of a position and the respective coordinate of the transformation. We show under some suitable conditions, that if the entropy converges to zero then the right sets increase towards the whole alphabet. This is a work in common with Pierre Collet. |
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Milton Jara |
The cut-off phenomenon for stochastically perturbed dynamical systems. |
We consider stochastic perturbations of a differential equation with an hyperbolic, attracting fixed point. We show thermalization as the strength of the noise tends to 0, that is, that the process converges to a local equilibrium measure in a time window of fixed size around a time that diverges with the strength of the noise. In the case on which the fixed point is unique and the differential equation is strongly coercive, this implies the so-called cut-off phenomenon. In that case, we also give necessary and sufficient conditions for the existence of profile cut-off, that is, that the distance to equilibrium of the stochastic process converges, when properly rescaled, to a universal function.Joint work with Gerardo Barrera (Edmonton) |
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Nelda Jaque and Bernardo San Martín |
Entorpy for impulsive semi-flow. |
In this talk we study a notion of entropy for not necessarily continuous semiflows on compact metric spaces, by using a family of semimetrics and the usual notions of spanning and separated sets. These semimetrics measure the proximity of two orbits allowing a small time lag. We prove that this alternative notions in the case of continuous semiflows agree with the classical one introduced by Bowen (1971). Finally, we prove that these notions of entropy are well defined for regular impulsive semiflows. We also prove that the definition by using separated sets, entropy is smaller than or equal to the $ au$-entropy introduced by Alves, Carvalho and V´asquez (2015). |
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Marcelo Vianna |
Continuity of Lyapunov exponents |
I will report on some recent work with Bocker, Malheiro, Avila, Eskin, Yang and Tal concerning the way Lyapunov exponents of linear cocycles depend on the underlying data, especially the cocycle and the invariant probability distribution. |
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Jan Kiwi |
Irreducibility of complex cubic polynomials with a periodic critical point |
The space of monic centered complex cubic polynomials with marked critical points is isomorphic to $C^2$. For each $n ge 1$, the locus $S_n$ formed by all polynomials with a specified critical point periodic of exact period n forms an affine algebraic set. We prove that $S_n$ is irreducible, thus giving an affirmative answer to a question posed by Milnor in the early 90´s. This is a join work with Matthieu Arfeux (PUCV, Valparaiso). |
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Andrés Navas |
Conjugaciones en dimensión 1 |
En esta charla abordaremos temas como la clausura de la clase de conjugación de un grupos de difeomorfismos del círculo, con énfasis en aspecto de regularidad. Tangencialmente revisaremos las distribuciones invariantes por difeomorfismos. |
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Marco Uribe |
Principal Poincaré Pontryagin function associated to some families of Morse real Polynomial |
It is known that the principal Poincaré Pontryagin function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is also an Abelian integral in non-generic cases. In non-generic cases it is an iterated integral. We give in a special case a precise description of the principal Poincaré Pontryagin function, an iterated integral of length at most 2, involving logarithmic functions with only 1 ramification at a point at infinity. We extend this result to some non-isomonodromic families of real Morse polynomials. This is joint work with Michèlle Pelletier (Université de Bourgogne). |
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Felipe Riquelme (curso 2) |
Dinámica del flujo geodésico en curvatura negativa |
El objetivo de este minicurso es estudiar propiedades dinámicas del flujo geodésico en variedades no compactas a curvatura negativa. Entre estas propiedades destacamos la ergodicidad de medidas invariantes, mixing topológico y mixing de medidas, y existencia de medidas de máxima entropía. Específicamente, durante la primera sesión se introducirán conceptos geométricos “base”, tomando como ejemplo fundamental la geometría de superficies hiperbólicas. Se discutirá el rol de la curvatura negativa para describir al conjunto no-errante del flujo geodésico y a las variedades fuertemente estables e inestables. Esto nos permitirá construir explícitamente medidas invariantes por el flujo. Durante la segunda sesión estudiaremos la ergodicidad y el mixing del flujo geodésico considerando las medidas construidas en la primera sesión. Luego recordaremos la definición de entropía métrica y topológica. Precisaremos el valor exacto de la entropía topológica y estableceremos criterios de existencia de medidas de máxima entropía. Finalmente, si el tiempo lo permite, se introducirán conceptos que permitirán hablar de la falla de la semi-continuidad superior de la entropía respecto a la topología débil*. |
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Maria José Pacífico |
Bifurcating mechanisms deriving from a singular horseshoe. |
We show some consequences of the discovery of singular horseshoe in the scenario of bifurcation theory. |
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Dante Carrasco |
Sobre Dinámica Topológica: Expansividad, Transitividad Robusta y Entropía |
Actualmente, muchas propiedades topológicas que presentan ciertos sistemas dinámicos son de gran interés para su estudio, tal es el caso de la expansividad. Dicha propiedad está enfocada tanto para homeomorsmos como para flujos [4, 11]. Existen muchas variantes y generalizaciones en relación a esta propiedad dinámica. En esta dirección, mostraremos que el atractor de Rovella es K∗-expansivo [5], tal como sucede con el atractor geométrico de Lorenz, [2, 9]. Además, se darán otros ejemplos geométricos que satisfacen la propiedades de la K∗-expansividad y cuyas construcciones son motivadas por lo de la herradura singular [10] y del atractor geométrico de Lorenz [1, 8], así como otras extensiones de expansividad para flujos en el contexto medible y expansividad sobre otros espacios en la dirección de los 2-métricos [6] y en espacios métricos tipo fuzzy [7]. La expansividad para difeomorfismos en el sentido medible también será analizada [3] |
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Jairo Bochi |
Flexibility of Lyapunov Exponents |
It was proved by Dolgopyat and Pesin that any compact smooth Riemannian manifold admits ergodic volume-preserving smooth diffeomorphisms. I will discuss the following question: What are the possible Lyapunov spectra (with respect to the volume measure) of ergodic diffeomorphisms? The strategy to answer the question is to begin with a diffeomorphism whose exponents are far apart and them mix them carefully using a deformation of Baraviera-Bonatti type. I will explain how to implement this strategy in the case of Anosov diffeomorphisms. Another question is: Which are the possible Lyapunov spectra in a fixed homotopy class of volume-preserving Anosov diffeos? I will discuss the possibility of an «exotic» Anosov example. This talk is based on joint work with A. Katok and F. Rodriguez Hertz. |
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Carlos Gustavo Moreira |
Symbolic dynamica and fractal geometry: the geometry of the Lorenz-like parameter spaces and of the Markov and Lagrange spectra |
We will present results which describe (fractal) geometrical properties of parameter spaces of Lorenz-like maps on intervals – the so-called lexicographical world and Milnor-Thurston world, and also of the classical Markov and Lagrange spectra (and generalizations), related to Diophantine approximations. Despite being objects appearing in quite diffetent contexts, they share similar intrincated fractal geometrical features. The proofs of these results are deeply related with symbolic dynamics and bifurcations of them. |
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Solange Aranzubia |
A Formula for the Boundary of Chaos in the Lexicographical Scenario and Applications to the Bifurcation Diagram of the Standard Two Parameter Family of Quadratic Increasing-Increasing Lorenz maps |
The Geometric Lorenz Attractor has been a source of inspiration for many mathematical studies. Most of these studies deal with the two or one dimensional representation of its first return map. A one dimensional scenario (the increasing-increasing one’s) can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics of any member of the standard family can be modeled by a subshift in the Lexicographical model of two symbols. These subshifts can be considered as the maximal invariant set for the shift map in some interval, in the Lexicographical model. For all of these subshifts, the lower extreme of the interval is a minimal sequence and the upper extreme is a maximal sequence. The Lexicographical world (LW) is precisely the set of sequences (lower extreme, upper extreme) of all of these subshifts. In this scenario the topological entropy is a map from LW onto the interval [0, log 2]. The boundary of chaos (that is the boundary of the set of (a, b) ∈ LW such that htop(a, b) > 0) is given by a map b = χ(a), which is defined by a recurrence formula. In this talk we will show an explicit formula for the value χ(a) for a in a dense set contained in the set of minimal sequences. Moreover, we apply this computation to determine regions of positive topological entropy for the standard quadratic family of contracting increasing-increasing Lorenz maps. |
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Lorenzo Diaz |
Productos-torcidos no-hiperbólicos: aproximación ergódica, exponentes de Lyapunov y aplicaciones |
En el contexto de sistemas (biyecciones) robustamente transitivos y no-hiperbólicos definidos como productos torcidos com fibra $mathbb{S}^1$estudiaremos la topología del espaio de las medidas ergódicas. Presentaremos el (un) análisis multifractal de los conjuntos de exponentes de Lyapunov (referentes a la fibra). Introduciremos un estudio axiomático e veremos aplicaciones. Trabajos en conjunto con K. Gelfert y M. Rams. |
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Roberto Markarian |
Ergodicidad de billares hiperbólicos |
Se presentarán resultados sobre la prueba de la propiedad de Bernoulli en una amplia clase de billares no poligonales con exponentes de Liapunov no nulos. Trabajo conjunto con Gianluigi Del Magno (Salvador, Brasil) |
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Alvaro Rovella |
Puntos periódicos para mapas del anillo |
Se dan condiciones para que un mapa de grado d>1 del anillo abierto A tenga puntos periódicos, tantos como un mapa del mismo grado del círculo. Se muestran varios ejemplos de mapas sin puntos periódicos. Las técnicas usadas se aplican a mapas de grado d de la esfera de dimensión 2, para obtener ejemplos y algunas condiciones suficientes para el crecimiento exponencial de la cantidad de puntos de periodo n. |
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Alejandro Mass |
Eigenvalues of finite rank Cantor minimal systems and applications |
In this talk I will review the last necessary and sufficient conditions to be an eigenvalue of a minimal Cantor system and how these conditions allow to solve some classical questions in the ergodic theory of this kind of systems. |
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Los interesados en asistir al coloquio pueden enviar un correo electrónico antes del 15 de Septiembre a Cecilia Marín:
para inscribirse.
If you are interested in attending the Colloquim please send and email before the 15th of September to Ceciclia Marín:
cecilia.marin.g@gmail.com
to register.
CENA
El día Lunes 25 de septiembre a las 20 horas en lugar a confirmar, se llevará a cabo una cena para celebrar el cumpleaños de Rafael.
Aquellos que deseen acompañar a Rafael en esta celebración podrán inscribirse con Cecilia Marin al correo cecilia.marin.g@gmail.com.
El costo de la cena es de $35.000. (excluye quienes paguen inscripción)