Schmidt's Game on Certain Fractals
Lior Fishman
Abstract
We construct (α,β) and α-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 RN satisfying the open set condition then for any countable collection of non-singular affine transformations Λi:RN RN, dimK=dimK ( ∞i=1(Λi(BA))) where BA is the set of badly approximable vectors in RN.
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