| Résume | The term \emph{shrinking target problems} in dynamical systems describes a class of questions which seek to understand the recurrence behavior of typical orbits of a dynamical system. The standard ingredients of such questions are a probability measure preserving dynamical system $(X,\mu, T)$ and a sequence of targets $\{B_m\}_{m\in \mathbb{N}}$ with $B_m\subset X$ and $\mu(B_m)\to 0$. Classical questions in this area focus on the set of points whose $n$-th iterate under $T$ lies in the $n$-th target for infinitely many $n$. We study a uniform analogue of these questions, the so-called \emph{eventually always hitting points}. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. I will talk about necessary and sufficient conditions on the shrinking rate of the targets for the set of eventually always hitting points to be of full or zero measure. In particular, I will present some recent dichotomy results for some interval maps obtained in collaboration with Mark Holland, Maxim Kirsebom, and Tomas Persson. |
