For C^k quasi-periodic Schrodinger operators in the local perturbative regime, we prove that the length of the corresponding spectral gap has a polynomial decay upper bound w.r.t. its label. This is based on a refined quantitative reducibility theorem for C^k quasi-periodic SL(2,R) cocycles. As an application, we are able to show the homogeneity of the spectrum.

We deal with germs of nvolutions: homeomorphisms $R$ that are their own inverse. We are interested in germs of homeomorphisms $F$ that are $R$-reversible: $F \circ R = R \circ F^{-1}$. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries of a given germ are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries.

Previous results on network ODE-equivalence and minimality can be used to partition the set of coupled cell networks into classes, according to the associated (network) dynamics. We apply this methodology to homogeneous coupled cell networks with asymmetric inputs. Ongoing work with Manuela Aguiar (Porto) and Pedro Soares (Gdańsk U. of Technology, Poland).

We study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For random processes, this problem corresponds to the longest common substring problem and we will explain how the growth rate of the longest common substring is linked with the Renyi entropy. We will also extend these studies to the realm of random dynamical systems.

Consideramos certas classes de transformações expansoras por
partes em espaços de dimensão finita que possuem probabilidades
invariantes absolutamente contínuas com respeito à medida de Lebesgue.
Deduzimos uma fórmula para a entropia de tais probabilidades. Usando
essa fórmula, apresentamos condições suficientes para a continuidade
dessa entropia em certas famílias parametrizadas. Aplicamos os
resultados a uma família de tendas bidimensionais. Trabalho em
colaboração com Antonio Pumariño.

For a vector field, heteroclinic cycle is a cyclic sequence of equilibria and connecting trajectories. A heteroclinic network is a connected union of finitely many heteroclinic cycles. It is possible to assign a graph to a heteroclinic network by identifying equilibria and connections in the network with nodes and edges in the graph.

Time crystals are crystalline structures in the time domain. In contrast to the usual crystals which manifest in spatial dimensions, the formation of a time crystal or even a proposal of such, can be tricky if not controversial.

The reason is closely related to the time translation symmetry. Time translation symmetry not only underlies the invariance of laws of physics, but also in the standard dynamical framework, is directly related to the con- servation of energy. It was thought until very recently that time translation symmetry could not be spontaneously broken.

We present a method of combining coupled cell systems to get dynamics supporting robust simple heteroclinic networks given by the product of robust simple heteroclinic networks (cycles). We consider coupled cell networks, with no assumption on symmetry, and combine them via the join operation. Assuming that the dynamics of the component networks supports robust simple heteroclinic cycles or networks, we show that the join dynamics realizes a more complex heteroclinic network given by the product of those cycles or networks.

We propose an approach to construct Bernoulli trials {Xi}i∈N combining dependence and independence periods, and we call it Bernoulli sequence with random dependence (BSRD). The structure of dependence in the past 

Patients infected with the human immunodeficiency virus (HIV) are more vulnerable to develop various types of cancer, in particular, Hodgkin’s lymphoma, Kaposi’s sarcoma and vulvar cancer. Furthermore, cancers progression tends to be more aggressive in HIV-positive individuals than in HIV-negative ones. In this work, we develop mathematical models to describe the dynamics of cancer growth and HIV infection, when chemotherapy and treatment for HIV, namely, highly active antiretroviral therapy (HAART) are included.