# Dynamical Day

3 December 2018

Facultad de Matemáticas, UC, Santiago

The aim of this meeting is to gather a group of mathematicians working on dynamical systems to discuss new developments in the area.

Jairo Bochi

Italo Cipriano

Godofredo Iommi

Jan Kiwi

Mario Ponce

14:30 -15:30 Felipe Riquelme

15:45 -16:45 Jairo Bochi

16:50- 17:50 Natalia Jurga

Speaker: Felipe RiquelmeSpeaker: Felipe RiquelmeTitle: Ground states at zero temperature in negative curvature

Abstract

Let $X$ be the unit tangent bundle of a complete negatively curved Riemannian manifold and let $(g_t):X o X$ be its associated geodesic flow. After the work of R. Bowen and D. Ruelle, it is well know that, if $X$ is compact, then any H»older-continuous potential $F:X o mathbb{R}$ admits an unique equilibrium measure. Moreover, there is a fair enough description of some properties of the pressure map $tmapsto P(tF)$ such as its regularity and its asymptotic behavior. For non-compact situations, the existence of equilibrium measures has been successfully studied over the last years. Moreover, regularity properties of the pressure map have been established in recent works by G. Iommi, F. Riquelme and A. Velozo.

In this talk we will be interested on the study of ground states at zero temperature for positive H»older-continuous potentials. More precisely, for $F:X omathbb{R}$ a positive potential going to 0 through infinity, we will study the asymptotic behavior of the equilibrium state $m_{tF}$ for the potential $tF$ as $t o+infty$. Indeed, we will show precise constructions of potentials having convergence/divergence to ergodic/non-ergodic ground states. This is a joint work with Anibal Velozo.

Speaker: Jairo BochiTitle: Emergence

Abstract:

I will talk about ongoing work with Pierre Berger.

Topological entropy is a way of quantifying the complexity of a dynamical system. It involves counting how many segments of orbit of some length $t$ can be distinguished up to some fine resolution $epsilon$. If we are allowed to disregard a set of orbits of small measure, then we are led to the concept of metric entropy. Now suppose we don´t care emph{when} a piece of orbit visits a certain region of the space, but only emph{how often}. Pursuing this idea, we are led to fundamentally new ways of quantifying dynamical complexity. This program was initiated by Berger a couple of years ago.

The first new concept that I´ll explain is emph{topological emergence} of a dynamical system: the bigger it is, the more different statistical behaviors are allowed by the system. We will explain how topological emergence is bounded from above in terms of the dimension of the ambient space. I´ll also present examples of dynamical systems where this bound is essentially attained.

Then we´ll come to another key concept: emph{metric emergence} of a dynamical system with respect to a reference measure. Roughly speaking, it quantifies how far from ergodic our system is. (To draw a comparison, topological emergence quantifies how far from uniquely ergodic the system is.) KAM theory reveals that non-ergodicity is somewhat typical among conservative dynamical systems, and metric emergence provides a way of measuring the complexity of the KAM picture. I´ll present examples and questions.

Speaker: Natalia JurgaTitle: Rigorous estimates on the top Lyapunov exponent for random matrix products

We study the Lyapunov exponent of random matrix products of positive $2 imes 2$ matrices and describe an efficient algorithm for its computation, which is based on the Fredholm theory of determinants of trace-class linear operators. Moreover, we obtain rigorous bounds on the error term in terms of two constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the average amount of projective contraction of the positive cone under the action of the matrices. This is joint work with Ian Morris from the University of Surrey.

This events is supported by CONICYT through the proyecto Anillo ACT172001 «New trends in ergodic theory» and by the Facultad de Matemáticas, Pontificia Universidad Católica de Chile