Dynamical Systems Days

December 3 to 14, 2007
Universidad Católica del Norte, Antofagasta, Chile

Description

 

The main goal of this event is to initiate a collaboration of Chilean researchers and post-graduate students with foreign mathematicians, who can propose new directions of research related to Dynamical Systems. The workshop consists of four mini courses and a limited number of talks in order to have plenty of time for informal discussions.

The meeting will be held at Universidad Católica del Norte in Antofagasta, Chile from December 3 to 14, 2007.

We invite everyone to participate. If you are interested in coming please register at the web page:
www.dynamicalsystems.cl/conferences/dynamics007/register
For more information please write an e-mail to anillo@ucn.cl.

Inscription is 40.000 CLP (about 80 USD). It is not obligatory, but we are asking everyone who can to pay it.

We hope to see you soon in Antofagasta.

Organizing Committee


Juan Rivera-Letelier (UCN)
Bernardo San Martín (UCN)
Bartłomiej Skorulski (UCN)
Carlos Vásquez (UCN)

Morning, First Week
Afternoon, First Week
Morning, Second Week
Afternoon, Second Week

Minicourses

VIVIANE BALADI
Dependence of SRB measures on chaotic dynamics

MARIA JOSE PACIFICO
Lorenz-like attractors

ALEJANDRO KOCSARD
Introduction to the Cohomological Theory of Dynamical Systems

ALAIN CHENCINER
Conservative Dynamics and the  Calculus of Variations

Talks

Flavio Abdenur
Ergodic Theory of C^1-Generic Non-Conservative Systems

Lorenzo Díaz
Non-generic measures for generic diffeomorphism

Neil Dobbs
Pesin-Ledrappier theory for cusp maps

Tomasz Downarowicz
Entropy properties of smooth maps (with introduction to the theory of symbolic extensions)

Vanderlei Horita
Destroying horseshoes via heterodimensional cycles

Roberto Markarian
Slow decay of correlations in billiard systems

Alejandra Rodrigues Hertz
A new criterion for ergodicity and non-uniform hyperbolicity

Federico Rodriguez Hertz
Non-uniform measure rigidity

Raul Ures
Recent advances in partially Hiperbolic dynamics

 

Mini-courses

VIVIANE BALADI
Dependence of SRB measures on chaotic dynamics

SRB measures are invariant probability measures associated to many interesting chaotic dynamics, and which often have strong statistical features (exponential decay of correlations, limit theorems…).
In this course, we shall consider smooth families of (mostly discrete time) chaotic dynamical systems f_t having a unique SRB measure mu_t, and study the dependency o  mu_t on t. We shall start with the toy model of smooth expanding maps, for which mu_t has a smooth density which depends smoothly on t. We shall see that explicit formulas can be given for the derivatives of mu_t in this case. Then we shall recall (without proofs) results of Katok-Knieper-Pollicott-Weiss and Ruelle for  smooth uniformly hyperbolic dynamics, in particular Ruelle´s formula for the derivative of mu_t (as a distribution). The heart of the course will be devoted to the study of piecewise expanding unimodal maps, where we recently discovered with Daniel Smania that the SRB measure is differentiable if and only if f_t is tangent to the topolological class of f_0. In this case, we also obtain a formula for the derivative of the SRB measure.

ALEJANDRO KOCSARD
Introduction to the Cohomological Theory of Dynamical Systems

Abstract in PDF Format

MARIA JOSE PACIFICO
Lorenz-like attractors.

In these lettures we shall see:
1. the equations of Lorenz and its main properties.
2. the geometric model for the equations of Lorenz
3. singular-hyperbolic attractors
4. a brief discussion about the proof that the flow generated by the above equations support a chaotic attractor with zero volume.
5. establish the equivalence between the geometrical model, the  flow generated by the equations of Lorenz and singular-hyperbolic  attractors.
6.  establish some topological properties displayed by  singular-hyperbolic attractors.

Notes in PDF format

ALAIN CHENCINER
Conservative dynamics and the calculus of variations.

Starting with elementary calculus of variations and Legendre transform, it is shown how the mathematical structures of conservative dynamics (Poincaré-Cartan integral  invariant, symplectic structure, hamiltonian form of the equations) arise from the simple computation of the variations of an action integral. The study of simple examples of integrable geodesic flows on the 2-torus then leads to the notion of Lagrangian submanifolds and to the Hamilton-Jacobi equation, whose relation to the Hamiltonian vector field is the first step of the duality between particles and waves.

Notes in PDF format

Talks


Flavio Abdenur
Ergodic Theory of C^1-Generic Non-Conservative Systems

We survey and discuss some recent progress (and also some ongoing work) on the ergodic theory of C^1-generic non-conservative systems. Topics include the existence of interesting invariant measures, the existence of physical attractors, and the relation between topological entropy and horseshoes.

Lorenzo Díaz

Non-generic measures for generic diffeomorphism

we discuss some recent resutls about homoclinic classes of $C^-1$-generic
diffeomorphis prove that non-hyperbolic homoclinic classes of these diffeomorphism
support an ergodic measure with uncountable support and zero Lyapunov
exponent.

Neil Dobbs

Pesin-Ledrappier theory for cusp maps

Pesin theory in the non-invertible setting was developed by Ledrappier for $C^{1+varepsilon}$ multimodal maps of the interval with non-flat critical points. This work can be extended to maps with flat critical points and with singularities where the derivative tends to infinity. As a corollary, the presence of very flat critical points precludes the existence of finite absolutely continuous probability measures.

Tomasz Downarowicz
Entropy properties of smooth maps
 (with introduction to the theory of symbolic extensions)

Notes in PDF format

Vanderlei Horita
Destroying horseshoes via heterodimensional cycles

We propose a model for the destruction of three-dimensional horseshoes emph{via} heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call emph{generating}.

Joint work with L. J. D´i az, I. Rios, and M. Sambarino

Roberto Markarian

Slow decay of correlations in billiard systems

We will present
a) some recent results on polynomial decay of
correlations in billiard systems and
b) a general strategy for estimating correlation functions
for smooth systems with singulariaties and weak hyperbolicity.



Alejandra Rodrigues Hertz
A new criterion for ergodicity and non-uniform hyperbolicity

We find a new criterion to establish ergodicity through the description of the ergodic components of an invariant measure. This criterion applies, for instance, to establish the C^1 Pugh-Shub conjecture about stable ergodicity for 2-dimensional center bundle. It is also used to prove uniqueness of SRB measures for transitive surface diffeomorphisms. The talk is aimed at a broad audience, and should be accessible to anyone with elementary notions in Pesin theory.

Federico Rodriguez Hertz
Non-uniform measure rigidity

Abstract: In this talk we shall present some recent results in
non-uniform measure rigidity for higher rank abelian actions. Measure
rigidity deals with the scarsity of invariant measures. For example,
Furstenberg raised the problem of studiyng the invariant measures for
the commuting endomorphisms of the circle, $ imes 2$ and $ imes 3$,
the unique invariant ergodic measures should be Lebesgue and the ones
supported over periodic orbits. This conjecture was solved in the case
the measure is assumed to be of possitive entropy with respect to an
element of the action in 1988 by R. Lyons and in 1990 by D. Rudolph.

This conjecture generalizes to a broad context. We will discuss this
generalizations and also some of the ingredients of its proof in the
simplest cases.

This is part of joint works with Boris Kalinin and Anatole Katok.

Raul Ures
Recent advances in partially Hiperbolic dynamics


Abstract in PDF

Participants

  • VIVIANE BALADI
  • MARIA JOSE PACIFICO
  • ALEJANDRO KOCSARD
  • ALAIN CHENCINER

 

  • FLAVIO ABDENUR
  • RAMÓN CORREA
  • LORENZO DÍAZ
  • NEIL DOBBS
  • VANDERLEI HORITA
  • JAN KIWI
  • ROBERTO MARKARIAN
  • ANDRÉS NAVAS
  • ALEJANDRA RODRIGUEZ
  • FEDERICO RODRIGUEZ
  • MARTÍN SAMBARINO
  • RAÚL URES
  • RICHARD URZÚA
  • JOSÉ VIETEZ
JORNADAS DE SISTEMAS DINAMICOS
ANTOFAGASTA
3 al 14 de DICIEMBRE 2007
UNIVERSIDAD CATOLICA DEL NORTE

 

El Comité Organizador de las Jornadas de Sistemas Dinámicos Antofagasta, tiene el agrado de invitar a Profesores y Estudiantes interesados en el área, a participar de este evento que se desarrollará en el auditorium K 120 A del campus central de nuestra Casa de Estudios, ubicado en Avenida Angamos 0610, Antofagasta.
Las Jornadas de Sistemas Dinamicos forman parte de las actividades consideradas dentro del proyecto Anillo de Sistemas Dinamicos de Baja Dimensión. El espiritu de este evento es motivar el trabajo tanto de los investigadores locales como de nuestros estudiantes de postgrado en nuevos topicos de investigación dentro del area de Sistemas Dinámicos y fomentar la interacción con matematicos extranjeros. Para esto hemos contemplado la realización de cuatro cursillos y algunas conferencias que permitan generar grupos de discusión en los topicos tratados.

CURSILLISTAS

VIVIANE BALADI :
MARIA JOSE PACIFICO : Do atrator caotico de Lorenz aos atratores hiperbolico-singulares
ALEJANDRO KOCSARD : Introducción a la teoría homológica de los sistemas dinámicos.
ALAIN CHENCINER :

PARTICIPANTES CONFIRMADOS

FLAVIO ABDENUR
RAMÓN CORREA
LORENZO DÍAZ
NEIL DOBBS
VANDERLEI HORITA
JAN KIWI
ROBERTO MARKARIAN
ANDRÉS NAVAS
ALEJANDRA RODRIGUEZ
FEDERICO RODRIGUEZ
MARTÍN SAMBARINO
RAÚL URES
RICHARD URZÚA
JOSÉ VIETEZ

La organización de este evento contempla algunas becas que cubrirán gastos de estadía a estudiantes avanzados en el área de sistemas dinámicos. Los estudiantes interesados en becas deben escribir a anillo@ucn.cl antes del 16 de Noviembre señalando su intención de participar e indicando el nombre y direccion electrónica de un Profesor auspiciador.
Los investigadores interesados en participar de las Jornadas, sirvanse confirmar y coordinar su participación en anillo@ucn.cl.
Mas información en la pagina web del evento: www.dynamicalsystems.cl

COMITÉ ORGANIZADOR

JUAN RIVERA
BERNARDO SAN MARTÍN
BARTEK SKORULSKI
CARLOS H. VÁSQUEZ

AUSPICIAN

Proyecto Anillo en Sistemas Dinámicos de Baja Dimensión.
Universidad Católica del Norte
PROSUL

Av. Ejercito 0115 Antofagasta
(56-55) 350400
Habitacion Single $35.000 mas IVA
Habitacion Doble $35.000 mas IVA
incluye desayuno y estacionamiento
Todas las reservaciones deben ser hechas con 10 dias de anticipacion

Av. Grecia 1000 Antofagasta
56-55) 419111
Habitacion Single $31.800 IVA incluido
Habitacion Doble $39.800 IVA incluido
desayuno incluido
Todas las reservaciones deben ser hechas con 7 dias de anticipacion

Baquedano 508 Antofagasta
(56-55) 224814 – 222939
Habitacion Single $29.070 IVA incluido
Habitacion Doble $22.360 IVA incluido
desayuno y estacionamiento incluido
Todas las reservaciones deben ser hechas con 4 dias de anticipacion

Av. Grecia 1490
(56-55) 228888
De Lunes a Jueves en la noche
$42.500 mas IVA Habitaciones Single y Doble
De Viernes a Domingo en la noche
$35.000 mas IVA Habitaciones Single y Doble
Incluye desayuno y estacionamiento
Todas las reservaciones deben ser hechas con 7 dias de anticipacion

Prat 867 Antofagasta
(56-55) 268063
Habitacion Single $22.305 IVA incluido
Habitacion Doble $24.411 IVA incluido
Habitacion Triple $31.324 IVA incluido
Suite $35.000 IVA incluido
incluye desayuno y estacionamiento
Todas las reservaciones deben ser hechas con 7 dias de anticipacion

  • Hotel Florencia

Jose Toribio Medina 030 Antofagasta
(56-55) 775685
Habitacion Single $35.000 mas IVA
Habitacion Doble $40.000 mas IVA
incluye desayuno y estacionamiento
Todas las reservaciones deben ser hechas al menos con 2 dias de anticipacion

How to get to the workshop

From the airport you can take Taxi. The cost of it to the Hotel El Tatio or to the university is 9000 CLP (pesos chilenos). You can also use a transfer (about 3000 CLP).

Please notice that there are two major universities in Antofagasta, Univerisdad Catolica del Norte (UCN) and Universidad de Antofahasta. The workshop is held in the first one.

Here is the map where you can find the Hotel El Tato and the univeristy (click to enlarge):

 click aquí para agrandar / Click here to enlarge

 

From the hotel you can walk to UCN (aprox. 15 min.) or at Avenida Angamos you can take colectivo which has a number which finishes with 3. It costs 400-450 CLP. However, if you do not speak Spanish, we strongly recommend you to walk (at least for the first days).