DSNT Seminar: Shrinking Targets, Dynamical Coverings, Eventually Hitting Sets, and Related Problems

Abstract: The classical shrinking target problem involves a sequence of 'target' subsets A_n \subset X and a dynamic T: X \to X. The focus is on the set of all points R \subset X whose n-th iterate hits the n-th target, T^n(x) \in A_n, infinitely often. Numerous variants of this problem exist, including a 'dynamical covering problem', which can be viewed as a dynamical analogue of the Dvoretzky covering problem. Another version of these 'limsup set' problems involves replacing them with 'liminf sets'. There are few results on these latter two sets. I will provide an overview of such problems and, time permitting, present an upcoming result that examines the size of coverings for 'self-similar' dynamical systems [joint with Balázs Bárány (BME Budapest) and Henna Koivusalo (Bristol)]; as well as addressing a problem concerning self-affine liminf sets, [joint with Yu-Liang Wu (Oulu)].

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).