An Embedding Theorem for a Class of Z² MSFTs
by
Samuel J. Lightwood
University of Maryland

DEFINITION (uniformly filling): A Z^2 SFT, X, is uniformly filling if there exists M, k>0 such that for all large R and all words w locally allowed on an M-neighborhood of the boundary of an RxR square, the following holds: there is a locally allowed word defined on the entire square which agrees with w on an (M-k)-neighborhood of the boundary of the square. (This definition has several equivalent formulations.) DEFINITIONS: For H a rank two subgroup of Z^2, let Fix_X(H) denote the fixed point set of H. For nonzero p in Z^2, let X_p denote the set of points in X such that x=px. Let h_1(X_p) denote the one-dimensional directional entropy of the subsystem X_p in the direction perpendicular to p. EMBEDDING THEOREM: Suppose X is a Z^2 subshift and Y is a uniformly filling Z^2 MSFT such that (1) h(X)<h(Y); (2) |Fix_X(H)| |Fix_Y(H)| for all H; (3) for all p, Y_p is mixing and h_1(X_p) < h_1(Y_p). Then there exists an embedding of X into Y. REMARK: Conditions 2 and 3 can be modified to get a sharper (but harder to check) result, in which necessary coherent embeddings of rank-one periodic subsystems are simply assumed to exist. (They can exist and possibly violate the statement of (3) for finitely many p.) THE PROOF. I. APERIODIC REGIONS. The proof is a marker construction attempting to parallel Krieger's approach with the Z case. A clopen set is used to construct primary marker sets for each point x in X. Roughly, for a given point x, a lattice site is within distance m of a marker unless there is periodicity of order less than m throughout an MxM square around the site. A code for the aperidic regions is constructed using three ingredients: 1) Entropy, 2) A tiling construction using Voronoi tiles on a region called the land, and 3) A finite tiling by finite tree prototiles of the boundaries of the Voronoi tiles as they sit in the tiling of the land. The finite tree prototiles are used to code Y allowed words on the thickened boundaries of the Voronoi tiles of the land. The entropy is then used to injectively code X allowed words on each Voronoi tile into Y allowed words on each Voronoi tile inside the thickened boundary. II. PERIODIC REGIONS. Codes for periodic regions (sea) are built by a variation of Krieger's argument applied to Z x Z_g systems (or taken from assumptions in the ßharper" formulation). III. INTERMEDIATE REGIONS. The intermediate regions (coast) are arranged to occur in uniformly flat ribbons from which a second clopen marking set is used to produce secondary marker sets for each x in X. A lattice site in a strip will be within distance n of a marker unless there is periodicity on order less than n through an NxN square around the site. Coding in the aperiodic intermediate regions is done again with an analogous set of three ingredients. Coding the periodic intermediate regions requires a map between periodic words of different sizes and periods. IV. INJECTIVITY. To guarantee injectivity, three different marking symbols (primary, secondary and tertiary) are used in image points. The mixing and uniform filling properties are used repeatedly. V. CHOICE OF PARAMETERS. There are various interdependencies and complications in the construction. The choice of n affects the choice of the flat neighborhoods, which in turn affects the choice of the clopen set for the primary markers. In the end one chooses the parameters in the order m, then n, then M, then N. This gives rise to very different scales: m << n << M < N. (but sometimes M << N.) A GOOD CLASS: The uniformly filling mixing SFT's appear to be a very natural class of Z^2 SFT's to which one can extend parts of the Z-theory.

Date received: March 18, 1998