Schmidt's Game on Certain Fractals

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.