Dynamical Systems

Sistemas Dinámicos de Santiago Dynamical Day

Event Date: Aug 20, 2018 in Dynamical Systems, Seminars

Primera Sesión:  14:30 hrs. Speaker:          Mike Todd (University of St Andrews, United Kingdom). Title:   Phase transitions and limit laws. Abstract:         The `statistics’ of a dynamical system is the collection of statistical limit laws it satisfies.  This starts with Birkhoff’s Ergodic Theorem, which is about averages of some observable along orbits: this is a pointwise result, for typical points for a given invariant measure.  Then we can look for forms of Central Limit Theorem, Large Deviations and so on: these are about how averages fluctuate, globally, with respect to the...

Read More

Desvíos rotacionales para mapas del toro y aplicaciones.

Event Date: Aug 13, 2018 in Dynamical Systems, Seminars

ABSTRACT:   El número de rotación de Poincaré es sin duda alguna el invariante más importante en el estudio dinámico de homeomorfismos del círculo (que preservan orientación). En general, estos sistemas exhiben lo que llamamos “desvíos rotacionales uniformemente acotados”, es decir, cualquier órbita de un homeomorfismo de este tipo siempre se mantiene a distancia uniformemente acotada de la órbita de la rotación rígida correspondiente. Esta importante propiedad tiene implicaciones profundas en dinámica unidimensional. En dimensiones superiores, en analogía con la teoría de...

Read More

Sensitive dependence of geometric Gibbs measures at positive temperature

Event Date: Jul 23, 2018 in Dynamical Systems, Seminars

ABSTRACT:   In this talk we give the main ideas of the construction of  the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states diverge at positive temperature. These are the first examples of divergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel....

Read More

Morse theory for the action functional and a Poincare-Birkhoff theorem for flows

Event Date: Jul 09, 2018 in Dynamical Systems, Seminars

ABSTRACT: The goal of this talk is twofold. Firstly I would like to explain how pseudo-holomorphic curves can be used to study Morse theory of the action functional from classical mechanics. Then I will move to applications, focusing on a generalization of the Poincare-Birkhoff theorem for Reeb flows on the three-sphere.  

Read More

Analogies between the geodesic flow on a negatively curved manifold and countable Markov shifts

Event Date: Jul 09, 2018 in Dynamical Systems, Seminars

ABSTRACT: By the work of Bowen and Ratner it is known that the geodesic flow on a compact negatively curved manifold can be modeled as a suspension flow over a subshift of finite type. Unfortunately, a symbolic representation is not available if the manifold is non-compact. In this talk I will briefly explain some recent developments on the study of the thermodynamic formalism of the geodesic flow on non-compact negatively curved manifolds. Surprisingly some of the methods used to understand the geodesic flow have consequences to the theory of countable Markov shifts. I will explain such...

Read More

Norm and pointwise averages of multiple ergodic averages and applications

Event Date: Jul 05, 2018 in Dynamical Systems, Seminars

ABSTRACT: Via the study of multiple ergodic averages for a single transformation, Furstenberg, in 1977, was able to provide an ergodic theoretical proof of Szemerédi’s theorem, i.e., every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions. We will present some recent developments in the area for more general averages, e.g., for multiple commuting transformations with iterates along specific classes of integer valued sequences. We will also get numerous applications of the aforementioned study to number theory, as we will present the...

Read More