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
dimHJ(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 dimHJ(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 dimHJ(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 [3, 4, 6, 7,
13] to higher dimensions.Comment: 31 pages, 10 figure