Abstract

Let G be (N d,+) or (Z d,+) where d ≥ 1. Let A be a finite set of symbols. We endow A with the discrete topology and A G with the product topology. The shift action of G on A G is given by (S g x)(h) = x(g + h) for g, h ∈ G and x ∈ A G. A subshift is a nonempty set X ⊆ A G which is topologically closed and shift-invariant, i.e., x ∈ X implies S g x ∈ X for all g ∈ G. Symbolic dynamics is the study of subshifts. If X ⊆ A G and Y ⊆ B G are subshifts, a shift morphism from X to Y is a continuous mapping Φ: X → Y such that Φ(S g x) = S g Φ(x) for all x ∈ X and g ∈ G. By compactness, any shift morphism Φ is given by a block code, i.e., a finite mapping φ: A F → B where F is a finite subset of G and Φ(x)(g) = φ(S g x↾F) for all x ∈ X and g ∈ G. 2 Some new (!?!) results on subshifts: Let d be a positive integer, let A be a finite set of symbols, and let X be a nonempty subset of A G where G is N d or Z d. The Hausdorff dimension, dim(X), and the effective Hausdorff dimension, effdim(X), are defined as usual with respect to the standard metric ρ(x, y) = 2 −|Fn| where n is as large as possible such that x↾Fn = y↾Fn. Here Fn is {1,..., n} d if G = N d, or {−n,..., n} d if G = Z d. We first state some old results. 1. effdim(X) = sup x∈X effdim(x). 2. effdim(x) = liminf n→∞

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