Markov random walks on homogeneous spaces and Diophantine approximation on fractals

Prohaska, Roland; Sert, Cagri

doi:10.1090/tran/8181

Markov random walks on homogeneous spaces and Diophantine approximation on fractals
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by Roland Prohaska and Cagri Sert PDF

Trans. Amer. Math. Soc. 373 (2020), 8163-8196 Request permission

Abstract:

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 $G/\Gamma$. 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.

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