Symbolic Dynamics: Entropy = Dimension = Complexity

Abstract

Let d be a positive integer. Let G be the additive monoid ? d or the additive group ? d . 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 ?(x, y) = 2 ? | F n | $2^{-|F_{n}|}$ where n is as large as possible such that x?F n = y?F n . Here F n = {0, 1, ? , n} d if G = ? d , and F n = {?n, ? , ?1, 0, 1, ? , n} d if G = ? d . 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) = lim n K(x?F n )/|F n | for some x ? X.