1 8 Ju l 2 00 3 CONJUGACIES FOR TILING DYNAMICAL SYSTEMS

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes. * Research supported in part by NSF Vigre Grant DMS-0091946 ** Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158 I. Notation and main results We begin with a definition of tiling dynamical systems, in sufficient generality for this work. Let A be a nonempty finite collection of compact connected sets in the Euclidean space E, sets with dense interior and boundary of zero volume. Let X(A) be the set of all tilings of E by congruent copies, which we call tiles, of the elements of the “alphabet” A. We assume X(A) is nonempty, which is automatic for the special class of substitution tiling systems on which we will concentrate below. We label the “types” of tiles by the elements of A. We endow X(A) with the metric m[x, y] ≡ sup n≥1 1 n mH [Bn ∩ ∂x,Bn ∩ ∂y], (1) where Bn denotes the open ball of radius n centered at the origin O of E , and ∂x the union of the boundaries of all tiles in x. (A ball centered at a is denoted Bn(a).) The Hausdorff metric mH is defined as follows. Given two compact subsets P and Q of E, mH [P,Q] = max{m̃(P,Q), m̃(Q,P )}, where m̃(P,Q) = sup p∈P inf q∈Q ‖p− q‖, (2) with ‖w‖ denoting the usual Euclidean norm of w. Under the metric m two tilings are close if they agree, up to a small Euclidean motion, on a large ball centered at the origin. The converse is also true for tiling systems with finite local complexity (as defined below): closeness implies agreement, up to small Euclidean motion, on a large ball centered at the origin [see RaS1]. Although the metric m depends on the location of the origin, the topology induced by m is Euclidean invariant. A sequence of tilings converges in the metric m if and only if its restriction to every compact subset of E converges in mH . It is not hard to show [RW] that X(A) is compact and that the natural action of the connected Euclidean group GE on X(A), (g, x) ∈ GE ×X(A) 7→ g[x] ∈ X(A), is continuous. To include certain examples it is useful to generalize the above setup, to use what is sometimes called “colored tiles”. To make the generalization we assign a “color” from some finite set to each element of A, represented on each tile by a “color marking”, a line segment in the interior of the tile, of different length for different colors. We then redefine ∂x as the union of the tile boundaries and color markings in the tiling x. Definition 1. A tiling dynamical system is the action of GE on a closed, GE-invariant subset of X(A). We emphasize the close connection between such dynamical systems and subshifts. A subshift with Z-action is the natural action of Z on a compact, Zinvariant subset X of BZd , for some nonempty finite set B. If we associate with each element of B a “colored” unit cube in E, the face-to-face tilings of E by those arrays of such cubes corresponding to the subshift X gives a tiling dynamical system which is basically the suspension of the subshift X (but with rotations of the entire tiling also permitted).