The baker’s map with a convex hole

Abstract

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.

Open maps: Small and large holes with unusual properties

“…Remark 4.13. It is shown in [8] that if we restrict our class of holes to convex ones for the baker's map, then there exists δ > 0 such that for any H of Lebesgue measure less than δ, we have dim H J (H) > 0. It would be interesting to establish an analogous result for hyperbolic toral automorphisms.…”