Abstract :
On a subshift of finite type (SFT) we introduce a pseudometric d given by a nonnegative matrix B satisfying the cycle condition. We show that the Hausdorff dimension of this SFT with respect to d is given by the Mauldin-Williams formula. If the ratio of the logarithms of any two nonzero entries of B is rational, we show that this Hausdorff dimension can be expressed essentially in terms of the logarithm of the spectral radius of a certain digraph. We apply our results to the Hausdorff dimension of the limit set of finitely generated free groups of isometrics of infinite trees. To each finitely generated subgroup G of a given finitely generated free group F, we attach an invariant p(G), which gives the rate of growth of all words G of length l at most with respect to a fixed set of minimal generators of F. We show that p(G) is the spectral radius of a digraph Δ(G) induced by G. Then H less-than-or-equals, slant G less-than-or-equals, slant F not implies p(G) greater-or-equal, slanted p(H). Moreover, p(G) = p(H) left right double arrow [G: H] < ∞.
