Random walks on homogeneous spaces and Diophantine approximation on fractals

David Simmons, Barak Weiss

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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 Rd. As examples, we show that for any self-similar fractal K⊆ Rd 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 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.

Original languageEnglish
Pages (from-to)337-394
Number of pages58
JournalInventiones Mathematicae
Volume216
Issue number2
DOIs
StatePublished - 1 May 2019
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics

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