An Algorithm to Estimate the Hausdorff Dimension of Self-Affine Sets

AbstractWe present an algorithm, based on Falconer's results in [4,6], to effectively estimate the Hausdorff dimension of self-affine sets in Rn: For a given finite set of contracting non-singular linear maps T1,…,Tm, we obtain a decreasing sequence of computable real numbers converging to Falconer's dimension d. For almost all (a1,…,am) ϵ Rmn, the number d is the Hausdorff dimension of the unique nonempty compact subset F satisfying F = Umi=1(Ti(F)+ ai). Similarly, we obtain an increasing sequence of computable real numbers converging to Falconer's lower bound d- which is indeed a lower bound for the Hausdorff dimension of F if the sets Ti(F) are disjoint for 1 ≤ i ≤ m