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 ⩽ G ⩽ F ⇏ p( G) ⩾ p( H). Moreover, p( G) = p( H) ⇔ [ G: H] < ∞.
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