Symbolic Dynamics: Entropy = Dimension = Complexity

Let d be a positive integer. Let G be the additive monoid a"center dot (d) or the additive group a"currency sign (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 a G and all x a A (G) . A G-subshift is defined to be a nonempty closed set X aS dagger A (G) such that S (g) (x)aX for all g a G and all x a 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 rho(x, y) = where n is as large as possible such that xa dagger 3/4F (n) = ya dagger 3/4F (n) . Here F (n) = {0, 1, aEuro broken vertical bar , n} (d) if G = a"center dot (d) , and F (n) = {-n, aEuro broken vertical bar , -1, 0, 1, aEuro broken vertical bar , n} (d) if G = a"currency sign (d) . For any X aS dagger 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 (xaX) lim inf (n) K(xa dagger 3/4F (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) a parts per thousand yen limsup (n) K(xa dagger 3/4F (n) )/|F (n) | for all x a X, and ent(X) = lim (n) K(xa dagger 3/4F (n) )/|F (n) | for some x is an element of X.