On shrinking targets for piecewise expanding interval maps

I will speak about our results with Tomas Persson on the shrinking target problem for some classes of interval maps. Given an interval map $T:I\to I$ with invariant measure $\mu$ and a nonincreasing sequence $r_n \to 0$ we study, for $\mu$-almost every $x\in I$, the set of points $y$ such that the inequality $|T^nx - y| < r_n$ is satisfied for infinitely many $n$. The maps we consider are in general piecewise monotone and expanding with respect to some finite partition, plus have a summable decay of correlations for functions of bounded variation (the assumptions are rather abstract, but I'll present some examples). If the measure $\mu$ is not absolutely continuous with respect to the Lebesgue measure, we need in addition some assumptions about $\mu$.

Related problems were studied by Fan, Schmeling and Troubetzkoy and by Liao and Seuret. The novelty of our result is that we do not need to assume the Markov property.