Dynamical Systems

Royal measures are loosely Kronecker


Dominik Kwietniak

A nonhyperbolic ergodic measure is an ergodic invariant measure with one Lyapunov exponent equal zero. Gorodetski, Ilyashenko, Kleptsyn, and Nalsky constructed a nonhyperbolic ergodic measure for a skew product diffeomorphism of the three-dimensional torus. Inspired by this construction, Bonatti, Diaz and Gorodetski gave sufficient condi- tions for weak convergence of a sequence of measures supported on periodic orbits to an ergodic measure. A royal measure is a measure obtained through this scheme.

Mean Topological Dimension for topological dynamical systems


Fagner Bernardini Rodrigues

In this talk we intend to present the denition of mean topological dimension for a topological dynamical system. It is a topological invariant which was introduced by M. Gromov and exploited by some authors. This invariant enables one to assign a meaningful quantity to a dynamical system of infinite topological entropy. Also is possible to obtain a kind of variational principle as we intend to show.

Non-typical points for iterated function systems


Paulo Varandas

The celebrated Birkhoff's ergodic theorem asserts that from a probabilistic viewpoint the times averages of "almost all" points converge to a space average. Motivated by the application of iterated function systems (IFS) to model central dynamics of partially hyperbolic diffeomorphisms, we will describe mild conditions that ensure that Birkhoff non-typical points form a Baire generic subset. If time permits we will provide some applicationsof this result in a partial hyperbolicity context. This is a ongoing joint work with my postdoctoral student G. Ferreira (UFMA).

Cohomology and equidistribution for higher-rank Abelian actions on Heisenberg manifolds with applications to Theta sums


Salvatore Cosentino

I will present quantitative equidistribution results for the action of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group on a compact homogeneous nilmanifold. The results are based on the study of the cohomology of the action of such groups on the algebra of smooth functions on the nilmanifold, on tame estimates of the associated cohomological equations with respect to a suitable Sobolev grading, and on renormalization in an appropriate moduli space (a method applied by Forni to surface flows and by Forni and Flaminio to other parabolic flows).

Rotation number of contracted rotations


Arnaldo Nogueira

Let 0 < a < 1, 0 ≤ b < 1 and I = [0, 1). We call contracted rotation the interval map φ_{a,b}: x ∈ I → ax+b mod 1. Once the parameter a is fixed, we are interested in the family φ_{a,b}, where b runs on the interval I. We use the fact that, as in the case of circle homeomorphisms, any contracted rotation φa,b has a rotation number which depends only on the parameters a et b. We will discuss the dynamical and diophantine aspects of the subject.

Global Saddles for Planar Maps


Isabel Labouriau

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of D2-symmetric maps, for which we obtain a similar result for C1 homeomorphisms. Some applications to differential equations are also given.

This is joint work with B. Alarc ́on (UFF — Brasil) and S.B.S.D. Castro (CMUP) and is part of a project of using symmetries to obtain global stability results. 

Complete Poisson convergence and records


Jorge Freitas

We consider point processes of rare events that keep record of the number of extreme occurrences on a certain time frame and  also of the magnitude of the exceedances observed in every such occasion. We study the convergence of such point processes both in the presence and absence of clustering. As a result we prove the convergence of record times and record values point processes. 

Bifurcations in coupled cell systems associated to the valency


Pedro Soares

Coupled cell systems are dynamical systems which respect the structure of a network. In an homogeneous network the number of inputs directed to each cell is constant and it is called the network valency. In this talk, we focus on bifurcations in coupled cell systems with a condition associated to the network valency. Moreover, we address the lifting bifurcation problem that concerns whether all bifurcation branches are lifted from a small network to a bigger one. We present some results and examples about this problem.


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