LXIII Colloquium: Celebrating Rafael Labarca´s 60th Birthday

Conferencista

Titulo

Abstracts

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.

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)

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). 

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.   

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).

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.

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).

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*.

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.

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 homeomor􏰀smos 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]

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. 

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.

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.

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.

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)

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.

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.