Markov random walks on homogeneous spaces and Diophantine approximation on fractals

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In the first part, using the recent measure classification results of Eskin–Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space G / Γ G/\Gamma . Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons–Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so, in particular, well approximable.