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

Abstract We present an algorithm, based on Falconer's results in [4,6], to effectively estimate the Hausdorff dimension of self-affine sets in R n : For a given finite set of contracting non-singular linear maps T 1 ,…,T m , we obtain a decreasing sequence of computable real numbers converging to Falconer's dimension d. For almost all ( a 1 ,…, a m ) ϵ R mn , the number d is the Hausdorff dimension of the unique nonempty compact subset F satisfying F = U m i =1 ( T i ( F )+ a i ). 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 T i (F) are disjoint for 1 ≤ i ≤ m.

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