Markov random walks on homogeneous spaces and Diophantine approximation on fractals
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- by Roland Prohaska and Cagri Sert PDF
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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.References
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Additional Information
- Roland Prohaska
- Affiliation: Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
- ORCID: 0000-0001-8563-9978
- Email: roland.prohaska@math.ethz.ch
- Cagri Sert
- Affiliation: Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
- Address at time of publication: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
- MR Author ID: 1216698
- ORCID: 0000-0003-0616-9969
- Email: cagri.sert@math.uzh.ch
- Received by editor(s): August 16, 2019
- Received by editor(s) in revised form: April 7, 2020
- Published electronically: August 28, 2020
- Additional Notes: The second-named author was supported by SNF grants 152819 and 178958.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8163-8196
- MSC (2010): Primary 37A50; Secondary 60G50, 37A45, 28A80
- DOI: https://doi.org/10.1090/tran/8181
- MathSciNet review: 4169685