# Markov Random Walks on Homogeneous Spaces and Diophantine Approximation on Fractals

## TL;DR

This paper establishes criteria for the almost sure equidistribution of random walks on homogeneous spaces, extending to Markovian cases, and applies these results to demonstrate generic Diophantine approximation properties on fractals.

## Contribution

It introduces new criteria for equidistribution of Markov-dependent random walks on homogeneous spaces and applies these to Diophantine approximation on fractals.

## Key findings

- Almost every point on certain fractals is well approximable.
- The results extend measure classification to Markovian random walks.
- Application to Diophantine approximation on self-similar sets.

## Abstract

In a 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.