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Partially motivated by the study of I. Binder, N. Makarov, and S. Smirnov BMS03 on dimension spectra of polynomial Cantor sets, we initiate the investigation on some general harmonic measures, inspired by Sullivan's dictionary, for distance-expanding dynamical systems. Let f X X be an open distance-expanding map on a compact metric space (X, ). A Gromov hyperbolic tile graph associated to the dynamical system (X, f) is constructed following the ideas from M. Bonk, D. Meyer BM17 and P. Ha\"issinsky, K. M. Pilgrim HP09. We consider a class of one-sided random walks associated with (X, f) on. They induce a Martin boundary of the tile graph, which may be different from the hyperbolic boundary. We show that the Martin boundary of such a random walk admits a surjection to X. We provide a class of examples to show that the surjection may not be a homeomorphism. Such random walks also induce measures on X called harmonic measures. When is a visual metric, we establish an equality between the fractal dimension of the harmonic measure and the asymptotic quantities of the random walk.
Li et al. (Sun,) studied this question.