This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_{\beta}(x)=\beta x \pmod 1$. Let $\mathcal{J}_{\beta} (a,b) := \{ x \in (0,1) : T_{\beta}^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\beta}$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \[ D_0(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \neq \emptyset \}, \] \[ D_1(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \text{ is uncountable} \}, \] \[ D_2(\beta) = \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \]Comment: 20 pages, 8 figures. Updated version has added examples and a small mistake in Lemma 3.4 has been fixe

We consider the baker's map B on the unit square X and an open convex set H ⊂ X which we regard as a hole. The survivor set 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 J (H) = 0 (dimension traps) as well as those for which any periodic trajectory of B intersects 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 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 δ > 0 such that dim H J (H) > 0 for all convex H whose Lebesgue measure is less than δ.This paper may be seen as a first extension of our work begun in [5,6,10,11,20] to higher dimensions.