Shrinking target problem for Bedford-McMullen carpets

Given a dynamical system $T:X\to X$ with an ergodic invariant measure $\mu$, for any set $B, \mu(B)>0$ $\mu$-almost every trajectory will visit $B$ infinitely many times. However, it is an interesting question to ask a more general situation: instead of one set $B$ we take a sequence of sets $B_n$ (usually - a decreasing sequence) and ask about the set of points $\Gamma = \{x\in X; T^n(x)\in B_n\ {\rm i.o.}\}$. This kind of question (first posed by Hill and Velani) is called {\it shrinking target problem}.

I'm going to present the following result (joint with Balazs Barany). Let $X$ be the Bedford-McMullen carpet (a simplest kind of self-affine IFSs). Let $T$ be the natural expanding map preserving $X$. Let $B_n$ be either a sequence of $\alpha n$-level cylinders or of geometric balls of diameter $e^{-\alpha n}$ (for some positive $\alpha$). In the latter case we need in addition some assumptions about the centers of the balls. We are then able to calculate the Hausdorff dimension of the shrinking target set $\Gamma$.