# Shrinking targets and eventually always hitting points for interval maps

## TL;DR

This paper investigates the measure and dimension of points that consistently hit shrinking targets in various interval maps, providing conditions for full or zero measure and calculating packing dimensions.

## Contribution

It establishes necessary and sufficient conditions for the measure of the set of eventually always hitting points in several interval maps, including the doubling and Manneville-Pomeau maps.

## Key findings

- Full measure conditions for the set of hitting points.
- Zero measure conditions for the set of hitting points.
- Dimension calculations for the set in the doubling map.

## 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 $\mathcal{E}_{\text{ah}}$.