# The baker's map with a convex hole

## TL;DR

This paper investigates the properties of convex holes in the baker's map, identifying conditions under which the survivor set has zero or positive Hausdorff dimension, and extending previous one-dimensional results to higher dimensions.

## Contribution

It characterizes dimension and cycle traps for convex holes in the baker's map and extends prior work from one-dimensional cases to higher dimensions.

## Key findings

- Interior convex holes are not dimension traps.
- Positive Hausdorff dimension can occur for large measure holes.
- Critical convex holes mark the boundary of trap properties.

## Abstract

We consider the baker's map $B$ on the unit square $X$ and an open convex set $H\subset X$ which we regard as a hole. The survivor set $\mathcal J(H)$ is defined as the set of all points in $X$ whose $B$-trajectories are disjoint from $H$. The main purpose of this paper is to study holes $H$ for which $\dim_H \mathcal J(H)=0$ (dimension traps) as well as those for which any periodic trajectory of $B$ intersects $\overline H$ (cycle traps). We show that any $H$ which lies in the interior of $X$ is not a dimension trap. This means that, unlike the doubling map and other one-dimensional examples, we can have $\dim_H \mathcal J(H)>0$ for $H$ whose Lebesgue measure is arbitrarily close to one. Also, we describe holes which are dimension or cycle traps, critical in the sense that if we consider a strictly convex subset, then the corresponding property in question no longer holds. We also determine $\delta>0$ such that $\dim_H \mathcal J(H)>0$ for all convex $H$ whose Lebesgue measure is less than $\delta$. This paper may be seen as a first extension of our work begun in [3, 4, 6, 7, 13] to higher dimensions.