Thomas Jordan

The shrinking target problem for certain self-affine systems.

The shrinking target problem looks at a dynamical system $T:X\to X$, a sequence of shrinking (in some suitable sense) sets $\{B_n\}$ and the set of points $x$ for which $T^n(x)\in B_n$ for infinitely many $n\in\N$. A typical problem is to look at whether the sets have full measure for some natural invariant measure and in the case where the set has zero measure to look at the Hausdorff dimension. We look at the second problem for a self-affine dynamical system in $\R^2$. We take the set of $B_n$ to be geometric balls. It turns out that how the dimension of the shrinking target set behaves depends on the centre of the ball. The techniques we need to use to the study these sets come from the study of Bernoulli convolutions and include both the transversality technique and the more recent results of Hochman and Shmerkin. This is joint work with Henna Koivusalo.