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Random walks on homogeneous spaces and Diophantine approximation on fractals

  • David Simmons
  • Barak WeissEmail author
Article
  • 79 Downloads

Abstract

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in \({\mathbb {R}}^d\). As examples, we show that for any self-similar fractal \(\mathcal {K}\subseteq {\mathbb {R}}^d\) satisfying the open set condition (for instance any translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on \(\mathcal {K}\) is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. We prove analogous results for matrix approximation, and for the case of fractals defined by Möbius transformations.

Notes

Acknowledgements

The first-named author was supported in part by the EPSRC Programme Grant EP/J018260/1. The second-named author was supported by ERC starter Grant DLGAPS 279893. The authors are grateful to Yves Benoist and Jean-François Quint for useful discussions, and to Alex Eskin for useful comments and encouraging remarks.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkHeslington, YorkUK
  2. 2.Tel Aviv UniversityTel AvivIsrael

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