COMPUTING THE HAUSDORFF DIMENSION OF SUBSHIFTS USING MATRICES

On a subshift of finite type (SFT) we introduce a pseudometric d given by a nonnegative matrix B satisfying the cycle condition. We show tha t the Hausdorff dimension of this SFT with respect to d is given by th e Mauldin-Williams formula. If the ratio of the logarithms of any two nonzero entries of B is rational, we show that this Hausdorff dimensio n can be expressed essentially in terms of the logarithm of the specia l radius of a certain digraph. We apply our results to the Hausdorff d imension of the limit set of finitely generated free groups of isometr ies of infinite trees. To each finitely generated subgroup G of a give n finitely generated free group F, we attach an invariant rho(G), whic h gives the rate of growth of all words G of length l at most with res pect to a fixed set of minimal generators of F. We show that rho(G) is the spectral radius of a digraph Delta(G) induced by G. Then H less t han or equal to G less than or equal to F double right arrow rho(G) gr eater than or equal to p(H). Moreover, rho(G) = rho(H) double left rig ht arrow [G: H] < infinity. (C) 1998 Elsevier Science Inc.