A Note on the Uniform Filling Property and Strong Irreducibility

Various separation properties of shift spaces have been defined in the literature. In [BPS], several were put into a chain of strict implications, leaving one open question: Is the uniform filling property (UFP) a strictly weaker condition than strong irreducibility (SI)? We given an example to show that it is, and we show that UFP and SI are equivalent for two-dimensional square filling shifts of finite type. 1. Background Definitions Let A = {1, . . . , n} be a finite alphabet. For D ⊆ Z2, y ∈ AD, v⃗ ∈ D and S ⊆ D, we will denote the symbol appearing in y at v⃗ as yv⃗ and the configuration of symbols appearing in y in the locations in S as yS. The two-dimensional, full n-shift is the set AZ2 together with the Z2-action defined by σv⃗(x)w⃗ = xv⃗+w⃗ for any x ∈ AZ2 and any v⃗, w⃗ ∈ Z2. A closed shift-invariant subset X of AZ2 together with the restriction of the Z2-action to X is called a Z2-shift space and is denoted by (X,Z2). Special notation for several particular subsets of Z2 will be helpful. Let Bm1,m2 = {(a1, a2) ∈ Z ∶ 0 ≤ ai ≤mi} be a rectangular block in the first quadrant. Similarly, denote the square block of side length 2m + 1 centered at the origin as Bm = {(a1, a2) ∈ Z ∶ ∣a1∣, ∣a2∣ ≤m}. Let Fr,k = Br−1,r−1/(Br−2k−1,r−2k−1 + (k, k)). Intuitively Fr,k is a square annulus with side length r and thickness k and with lower left corner at the origin. Recall that the maximum metric δ∞ on Z2 is given by δ∞(v⃗, w⃗) = max{∣vi −wi∣ ∶ 1 ≤ i ≤ 2} for v⃗, w⃗ ∈ Z2, and δ∞(S,T ) = min{δ∞(v⃗, w⃗) ∶ v⃗ ∈ S, w⃗ ∈ T} for non-empty subsets S,T ⊆ Z2.