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Symbolic Dynamics: Entropy = Dimension = Complexity

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Abstract

Let d be a positive integer. Let G be the additive monoid ℕd or the additive group ℤd. Let A be a finite set of symbols. The shift action of G on A G is given by S g(x)(h) = x(g+h) for all g, h ∈ G and all x ∈ A G. A G-subshift is defined to be a nonempty closed set X ⊆ A G such that S g(x)∈X for all g ∈ G and all x ∈ X. Given a G-subshift X, the topological entropy ent(X) is defined as usual (Ruelle Trans. Am. Math. Soc. 187, 237–251, 1973). The standard metric on A G is defined by ρ(x, y) = \(2^{-|F_{n}|}\) where n is as large as possible such that x↾F n = y↾F n . Here F n = {0, 1, … , n}d if G = ℕd, and F n = {−n, … , −1, 0, 1, … , n}d if G = ℤd. For any X ⊆ A G the Hausdorff dimension dim(X) and the effective Hausdorff dimension effdim(X) are defined as usual (Hausdorff Math. Ann. 79, 157–179 1919; Reimann 2004; Reimann and Stephan 2005) with respect to the standard metric. It is well known that effdim(X) = sup x∈X lim inf n K(x↾F n )/|F n | where K denotes Kolmogorov complexity (Downey and Hirschfeldt 2010). If X is a G-subshift, we prove that ent(X) = dim(X) = effdim(X), and ent(X) ≥ limsup n K(x↾F n )/|F n | for all x ∈ X, and ent(X) = limn K(x↾F n )/|F n | for some x ∈ X.

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Notes

  1. In particular, the sequence F n with n = 0, 1, 2, … is a Følner sequence for G.

  2. Instead of log2 we could use logb for any fixed b > 1, for instance b = e or b = 10. The base b = 2 is convenient for information theory, where entropy is measured in bits.

  3. Our Φ for A G is obtained as follows. Let \(\#:A^{*}\to \mathbb {N}\) be a standard Gödel numbering of A ∗. In other words, for each σ ∈ A ∗ let #(σ) be a numerical code for σ from which σ can be effectively recovered. Let a be a fixed symbol in A. Define Φ: ℕ → A G by letting Φ(#(σ))=x σ ∈A G where x σ ∈⟦σ⟧ and x σ (g) = a for all g ∈ G∖dom(σ).

  4. Here are the details. Define L 1 = {υ j ∣j = 1, 2, … } where υ j ∈K 1 is chosen inductively so that υ i ∩υ j = ∅ for all i < j and |υ j | is as large as possible. Then for all τ ∈ K 1 there exists υ ∈ L 1 such that τ∩υ≠∅ and |τ| ≤ |υ|. From this it follows that \(|\bigcup L_{1}|\ge |\bigcup K_{1}|/3^{d}\).

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Acknowledgments

We thank the anonymous referees for comments which led to improvements in this paper. In particular, the proof of Lemma 5.1 was suggested by the first referee and is much simpler than our original proof.

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    Stephen G. Simpson

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Simpson, S.G. Symbolic Dynamics: Entropy = Dimension = Complexity. Theory Comput Syst 56, 527–543 (2015). https://doi.org/10.1007/s00224-014-9546-8

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