This is the Dynamical Systems Group closing activity of 2019. It consists of three
40min talks, described below:

Ana Rodrigues (Exeter): Recent advances on fair measures and fair entropy

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. Roth) about   transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps.

Armando Castro (UFBA): Espaços Anisotrópicos e Cones convexos com aplicações a linear response

Nessa palestra, falaremos de artigo recentemente submetido em que constru\'\i mos um esquema abstrato para provar uma propriedade de gap espectral para operadores positivos. Entre as possíveis aplicações, destacaremos a diferenciabilidade de
medidas de equilíbrio com respeito à din\^amica.

Paulo Varandas (FCT-CMUP/UFBA): Minimality and points with historic behavior

The celebrated Birkhoff ergodic theorem is among the most useful tools in ergodic theory. Dual to the law of large number, it ensures that C\`esaro averages of continuous observables are almost everywhere convergent and that the limit is explicitly determined by the invariant measure. It turns out that the set of points where the convergence fails (also known as points with historic behavior) can have large topological complexity. In this talk I will describe a  simple and sufficient condition for the existence of a Baire residual set of points exhibiting historic behavior. This criterium is satisfied not only by dynamics satisfying some gluing orbit property as it also applies to a minimal and non-uniquely ergodic maps and open classes of robustly transitive partially hyperbolic diffeomorphism. This is a joint work with M. Carvalho.