Schmidt’s game on fractals

Abstract

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute 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, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ^N → ℝ^N,

$$ \dim K = \dim K \cap \left( {\bigcap\limits_{i = 1}^\infty {(\Lambda _i (BA))} } \right) $$

where BA is the set of badly approximable vectors in ℝ^N.