Aline Melo (UFC)
Title: Continuity of the Lyapunov exponent for Markov cocycles.
Abstract: An important problem in ergodic theory concerns the regularity of the Lyapunov exponents as a function of the data. In this talk, we establish the joint Hölder continuity of the maximal Lyapunov exponent as a function of the Markov cocycle and the transition kernel. Our approach provides a more computable Hölder exponent. This is a joint work with Ao Cai, Marcelo Durães and Silvius Klein.
Andressa Paola Cordeiro (UFRGS)
Title: Entropy of Markov tree-shifts given by different matrices
Abstract: Tree-shifts are discrete dynamical systems with characteristics of both one-dimensional and two-dimensional shifts. Two definitions of entropy for these systems were introduced in the past few years as a generalization of the entropy of one-dimensional dynamical systems. In this talk, we analyse some relations between both definitions and present the entropy of some examples, whose transitions in each direction are given by different matrices. These results are a joint work in development with Alexandre Tavares Baraviera.
Hellen de Paula (UFMG)
Title: Generalized entropy on the border of chaos
Abstract: A classical problem in dynamical systems is to measure the complexity of a map in terms of its orbits, and one of the main concepts used to achieve this goal is entropy. Nonetheless, many interesting families of dynamical systems have every element with zero-entropy. One of these are Morse-Smale diffeomorphisms. I want to show how we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces. We also apply our technique to compute the dispersion of the orbits of maps on the border of chaos with mild dissipation.
Luis Pedro Piñeyrúa (UFC)
Title: Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms.
Abstract: We introduce the notion of fibered lifted partially hyperbolic diffeomorphisms and we prove that any partially hyperbolic diffeomorphism isotopic to a fibered lifted one where the isotopy take place inside partially hyperbolic systems is dynamically coherent. Moreover we prove some global stability result: every two partially hyperbolic diffeomorphisms in the same connected component of a fibered lifted partially hyperbolic diffeomorphisms, are leaf conjugate. Joint work with Martín Sambarino.
Ulisses Lakatos (UFF)
Título: Onset of chaos in actions on the sphere
Abstract: The action of transitive groups of homeomorphisms encode the symmetries of a manifold with no privileged reference frames. In the case of the unit sphere S², the rotations group SO(3) forms the only such group which is also compact. However, the hierarchy of intermediate closed subgroups between SO(3) and the full group of homeomorphisms is still not completely understood. In particular, it is not known which conditions are sufficient to ensure that such a group contains a map of positive topological entropy. We review a recent result, joint with F. Tal, according to which any proper C¹ extension of PSL(2,C) contains a chaotic map. Time allowing, we also hint at results of a similar nature which may be pursued in the near future.
Zheng Bian (ICMC-USP)
Title: Mean-field reduction for random systems heterogeneously coupled on complex networks
Abstract: Consider a complex network on N nodes; on each node sits a random dynamical system, and node systems interact pairwise if there is an edge connecting the pair of nodes. This gives rise to an N-dimensional system with emergent behaviour due to interactions. The hubs, i.e., very well-connected nodes, experience a non-negligible and predictable net effect from the interactions; this predictable emergent behaviour is referred to as its mean-field reduction. We present abstract results for such reduction on a general class of networks and for a general class of node dynamics, as well as some concrete examples. This is joint work with Tiago Pereira (ICMC-USP) and Jeroen Lamb (Imperial College London, UK).