Adriana da Luz
Star vector fields and centralizers
In this talk we will focus on the centralizers of star vector fields. We aim to add to the understanding of the following paradigm : The existence of even some weak form of hyperbolicity can prevent the existence of symmetries and therefore force a trivial centralizer. We will analyze this problem from two new points of view. We will allow for more flexible notions of symmetries, and we will be studying star vector fields. An important point here is that these vector fields have two extra difficulties: First, the star property does not a prior give any hyperbolicity on the wandering orbits, that do play a role in finding possible symmetries. Second, we are considering vector fields that might have singularities accumulated by non wandering orbits. The talk is based on a work in progress with Bruno Santiago.
Alfonso Artigue
On continuum-wise minimality
We say that a homeomorphism of a compact metric space is ’cw-minimal’ if all the proper closed invariant subsets are zero-dimensional. H. Kato proved that if such system is in addition cw-expansive, then it is transitive, sensitive and minimal subsets are dense. We will show that independently of cw-expansivity, cw-minimality implies transitivity and either sensitivity or minimality (assuming the local connectedness of the space). We will also give an example of a homeomorphism of the two-dimensional torus which is cw-minimal and has only one minimal subset (a fixed point).
Amie Wilkinson
Exponential mixing for geodesic flows on some incomplete negatively curved surfaces
Cristina Lizana
Robust transitivity and domination for endomorphisms displaying critical points
We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism.
This allows us to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation. This is a joint work with R. Potrie, E. Pujals and W. Ranter.
Dominik Kwietniak
Constructing minimal systems using semicocycles
At least since the time of Denjoy, it has been known that interesting
examples of dynamical systems can be obtained by appropriate
enrichment of simple dynamics. In his classic example, Denjoy started
with an irrational rotation of a circle. Over the years, a similar
idea has been used many times under different names, reflecting
different approaches to the construction: almost automorphic systems,
almost 1-1 extensions, or semicocycle extensions.
This talk will concentrate on the latter: we will show how to use
semicocycles to address a particular case of a notorious question:
what kinds of dynamics can be realised by minimal homeomorphisms on
manifolds?
Enrique Pujals
A geometrical approach to the dynamics of some infinite dimensional ODE and their nonlinear perturbations (with some applications to PDEs).
One of the successes of the “geometrical approach” to dynamics comes from understanding dynamical properties of certain linear models and later showing that those dynamical properties are robust once nonlinear perturbations are considered. In the talk, we will discuss that perspective in the context of certain infinite dimensional linear ODE and some linear PDEs. This is a work in progress with Patricia Cirilo, Bryce Gollobit and Felipe Hikari
Javier Correa
Orders of growth and generalized entropy
During my pos-doc at UFRJ under Zezé supervision, I came up with the notion of generalized entropy. Although it may be far from this conference’s topics, I think it is a nice example of the fruitful space that Zezé created to develop dynamical systems at UFRJ. The idea of this talk is to explain what is generalized entropy and to tell the results obtained so far with Enrique Pujals and Hellen de Paula.
José Vieitez
Zezé’s contributions on expansiveness
Sérgio Romaña Ibarra
Anosov Geodesic Flows Imply Geometries Without Conjugate Points
In this lecture we will discuss the proof of a problem left by R. Mañé. More specifically, we will show that every complete manifold with curvature bounded below and with Anosov geodesic flow has no conjugate points. This result had already been proven by R. Mañé in the case of finite volume and by Klingenberg in the compact case. Our proof covers these cases as well.A joint work with Italo Melo (UFPI).
Stefano Galatolo
Self consistent transfer operators in a weak and not so weak coupling regime. Invariant measures, convergence to equilibrium, linear response.
We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling.
We consider a large class of self consistent transfer operators and prove general statements about existence and uniqueness of invariant measures, speed of convergence to equilibrium, statistical stability and linear response, mostly in a “weak coupling” or weak nonlinearity regime. We apply the general statements to examples of different nature: coupled expanding maps, coupled systems with additive noise, systems made of different maps coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps.
Stefano Marmi
The Yoccoz–Birkeland livestock population model coupled with random price dynamics
Vitor Araujo
On the statistical stability of families of attracting sets
We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions can be verified for all singular-hyperbolic and sectional-hyperbolic attracting sets of C² vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions aim to avoid as much as possible the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincaré sections.
Welington Cordeiro
Mean dimension expansive dynamical systems
As it is well known, by a Mañé’s result, there are no expansive dynamical systems on infinite dimension compact metric spaces. Kato proved that several generalizations of expansivity cannot occur in systems defined on infinite-dimensional compact metric spaces. Recently, expansive like properties in the context of systems defined on infinite dimension spaces are having the attention of researchers of the Dynamical Systems Theory. We are going to involve Mean dimension theory, which was introduced by Gromov, and developed systematically by Lindenstrauss and Weiss, and define the Mean Dimension Expansive Systems notion. Our definition is inspired by the definition of Bowen of the Entropy Expansive dynamical systems. During the talk, we will also present some basic results for these systems and show some examples of systems defined on infinite-dimensional compact metric spaces satisfying our definition