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

Markov Random Walks on Homogeneous Spaces and Diophantine Approximation on Fractals