We develop the concepts of ontogenesis and phylogenesis in the context of discrete
dynamical systems, in particular, iterated interval maps. The topological dynamics of a
continuous interval is characterised by the kneading invariant. Using symbolic dynamics,
we show how to deﬁne and study systematic changes in this type of systems. Moreover,
the study of a particular type of interactions between a large number of discrete dynamical
systems with common characteristics - a population - gives us the possibility of studying
We consider random perturbations of general non-uniformly expanding maps, possibly having a non-degenerate critical set, and discuss their
mixing rates along random orbits.
In particular, we prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the
critical set at a certain time decays in a (stretched) exponential fashion for almost all random orbits, then the decay of correlations
along random orbits is stretched exponential, up to some waiting time.
We construct networks from simple robust heteroclinic cycles in R4.
Under appropriate assumptions only very few ways exist by which cycles can be joined together in a network.
Those of types B and C have been known and investigated in the last decades, while type A networks are mostly absent from the literature.
Solutions of functional equations may have exotic properties: they may be singular, fractal or even everywhere discontinuous. In particular, those properties may be present in solutions of conjugacy problems in dynamical systems or in the Barnsley's fractal interpolation for which we provide an explicit formula showing how, in certain cases, a solution can be constructively determined.
Fernandez and Demers studied the statistical properties of the Manneville-Pomeau
map with the physical measure when a hole is put in the system, overcoming some of
the problems caused by subexponential mixing. I’ll discuss the same setup, but with a
class of natural equilibrium states. We ﬁnd conditionally invariant measures and give
precise information on the transitions between the fast exponentially mixing, the slow
exponentially mixing and the subexponentially mixing phases. This is joint work with
Alessandro Margheri(*) (University of Lisbon and CMAF)
Friday, 27 February, 2015 - 14:30
Apresentamos alguns resultados sobre a existência de órbitas periódicas no
problema restringido dos três corpos circular e plano com dissipação e sobre a dinâmica do
problema de Kepler com dissipação linear.
Apresentaremos o conceito de estabilidade estatística e ofereceremos uma visita
guiada por alguns resultados ilustrando o conceito, em contextos como as transformações
quadráticas, as transformações de Viana, os difeomorﬁsmos de Hénon ou os ﬂuxos de
Lorenz. Consideraremos mais detalhadamente um resultado recente com A. Pumariño
e E. Vigil, onde daremos condições suﬁcientes para a estabilidade estatística de certas
classes de transformações multidimensionais expansoras por partes.
In this joint work with Isabel Rios we prove the existence and uniqueness of equilibrium states associated to Holder continuous potentials with small variation for a family of partially hyperbolic systems. This family related to the three-dimensional horseshoe introduced by Díaz, Horita, Sambarino and Rios in 2009. The ergodic properties of this model were studied by Leplaideur, Oliveira and Rios in 2011. In this last work, the authors proved the existence of a regular potential (with big variation) which has two mutually singular equilibrium states.
Armando Castro Jr. (Universidade Federal da Bahia - UFBA, Brasil)
Friday, 30 January, 2015 - 14:30
In this talk, after a brief survey on linear response formula results, we will focus
in some developments that we have carried out at UFBA. Such developments have implications
in the regularity of Lyapunov exponents, Hausdorff Dimension and Decay and
Large Deviations rates of non-uniformly expanding systems.
This is a joint work with T. Bomfim and P. Varandas.