In this talk I will discuss how to compute the entropy following backwards trajectories in a way that at each step every preimage can be chosen with equal probability introducing fair measure and fair entropy (joint with M. Misiurewicz). I will discuss some advances on the study of fair entropy for non-invertible interval maps under the framework of thermodynamic formalism, showing that the fair measure is usually an equilibrium state (joint with Y. Zhang). I will then talk about some recent results (joint with S. and Z.

We present a deterministic discrete dynamical system which is used to classify and simulate complex irregular motion, in two or three dimension. The dynamical system is defined by a two-parameter family of bimodal interval maps which give directly the displacements through iteration. A trajectory is composed of patches of linear motions, through the plane, intertwined by changes of direction. The characterization of the movements is obtained from the topological classification of the interval map family.

Let f and g be piecewise smooth interval maps, with critical-singular sets, and A a cycle of intervals for f . We prove that A is a topological chaotic attractor if, and only if, A is a metric chaotic attractor. Let h|A be a topological conjugacy between f and g. We prove that, if h is differentiable at a single point p of the visiting set V , with non zero derivative, then h is smooth in A. Furthermore, the visiting set V is a residual set of A and, if the sets Cf and Cg are critical then V has µ full measure, for every expanding measure µ, with supp µ = A

In this talk, we discuss contributions to the thermodynamic formalism of partially hyperbolic attractors for non-singular endomorphisms. We consider a class of local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We present how to construct SRB measures for this class of maps and prove estimates for the measure of dynamical balls with respect to the volume measure and the SRB measure.

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.