BME Fractal Geometry Seminar

Suppose \((X,d)\) is a metric space equipped with a Borel probability measure and suppose \((B_i)_{i \in \mathbb{N}}\) is a sequence of measurable sets in \(X\). Suppose \(T: X \to X\) is a measure-preserving transformation and consider the set: \[ \mathcal{B} = \left\{x \in X : T^n x \in B_n \text{ for infinitely many } n \in \mathbb{N}\right\}. \] This is a shrinking target set. The terminology "shrinking targets" was first introduced by Hill and Velani in 1995. Since then, shrinking target problems have received a great deal of interest, especially with regards to studying the measure-theoretic and dimension-theoretic properties of shrinking target sets. In this talk, I will discuss some recent work with Thomas Jordan (Bristol, UK) and Ben Ward (York, UK) where we establish the Hausdorff dimension of a shrinking target set where our "targets" (the \(B_n\)) are rectangles and \(X\) is a self-similar carpet.