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

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 ; \Delta \Delta \Delta ; Tm , we obtain a decreasing sequence of computable real numbers converging to Falconer's dimension d. For almost all (a 1 ; \Delta \Delta \Delta ; am ) 2 R mn , the number d is the Hausdorff dimension of the unique non-empty compact subset F satisfying F = S 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 \Gamma which is indeed a lower bound for the Hausdorff dimension of F if the sets T i (F ) are disjoint for 1 i m. 1 Introduction An iterated function system (IFS) is given by a finite collection of contracting maps S 1 ; \Delta \Delta \Delta ; Sm on R n , i.e. with jS i (x) \Gamma S i (y)j c i jx \Gamma yj for some 0 ! c i ! 1, all 1 i m and all x; y 2 R ..