Schmidt's Game on Certain Fractals

TL;DR

This paper explores Schmidt's game on fractals, constructing winning sets on measures supported on fractals and deriving Hausdorff dimensions, especially for attractors of certain iterated function systems.

Contribution

It introduces methods to construct Schmidt's game winning sets on fractals supported by specific measures and relates these to Hausdorff dimensions for attractors of contracting similarities.

Findings

Constructed (\alpha,eta) and \alpha-winning sets on fractals.

Derived Hausdorff dimension formulas for intersections with affine transformations.

Proved dimension invariance under certain affine transformations on fractal attractors.

Abstract

We construct (\alpha ,\beta) and \alpha -winning sets in the sense of Schmidt's game, played on the support of certain measures (very friendly and awfully friendly measures) and show how to derive the Hausdorff dimension for some. In particular we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of R^N satisfying the open set condition then for any countable collection of non-singular affine transformations \Lambda_i:R^N \to R^N, dimK=dimK\cap (\cap ^{\infty}_{i=1}(\Lambda_i(BA))) where BA is the set of badly approximable vectors in R^N.