Shrinking targets and eventually always hitting points for interval maps

Abstract

We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of 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 $n$. We derive necessary and sufficient conditions on the shrinking rate of the targets for $\mathcal{E}_{\text{ah}}$ to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauss map and correspondingly for the maximal digits in continued fractions expansions. In the case of the doubling map we also compute the packing dimension of $\mathcal{E}_{\text{ah}}$ complementing already known results on the Hausdorff dimension of $\mathcal{E}_{\text{ah}}$.