The baker’s map with a convex hole

We consider the baker's map B on the unit square X and an open convex set which we regard as a hole. The survivor set 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 (dimension traps) as well as those for which any periodic trajectory of B intersects (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 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 such that 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 Clark (2016 Discrete Continuous Dyn. Syst. A 6 1249–69; Clark 2016 PhD Dissertation The University of Manchester; Glendinning and Sidorov 2015 Ergod. Theor. Dynam. Syst. 35 1208–28; Hare and Sidorov 2014 Mon.hefte Math. 175 347–65; Sidorov 2014 Acta Math. Hung. 143 298–312) to higher dimensions.