Dynamical Systems Seminar

A system is said to have historic behaviour if there is a positive Lebesgue measure set of points having non convergent Birkhoff averages. The question of knowing whether systems with historic behaviour are abundant in some families of dynamics has recently regained attention, with the recent works of Kiriki and Soma, and Berger's definition of (local) emergence, which measures how big is the set of accumulation points of Birkhoff averages. In this talk, I will present two examples of systems with historic behaviour. The first one, obtained with Guarino and Santiago, is a modification of Bowen's eye example in which the set of points with historic behaviour is of positive Lebesgue measure but nowhere dense. The second one, in collaboration with Andersson, is the study of reparametrized linear flows of the two torus with two fixed points; we obtain some Diophantine conditions on the flow's parameters under which the system has/has not historic behaviour.