The Dimension Of Boundary Of Self-affine Tile

In P.Duvall and J.Keesling's essay, they gave out an method to compete the Hausdorff dimension of the boundary of a self-similar tile:dim_H ((?)T) = logλ/logc1/c is the contraction factor and A is the largest eigenvalue of the contact matrix.But this method can only be used in the case that the attractor T = T(A,D) is a self similar tile.It's out of use when the matrix A is a self affine. We introduce the "pseudo-metric" from [2], this pseudo-metric has two good proportions:1) For any x∈R^d ,ω|-(Ax) = q^1/dω|-(x),|detA |= q >1∈R.2) There exists constantη>0,for any x, y∈R^d ,thenω|-(x + y)≤ηmax{ω|-(x),ω|-(y)}.We use two proportions above to give out a similar result to extend the method of calculating the Hausdorff dimension of the boundary from the self similar tile to the self affine tile.Theorem 0.1 A is a self affine matrix, | det A |= q > 1∈R, if T = T(A,D) satisfying VEC , there exist a constant a > 0, such thattheω(x) is the Hausdorff metricω(x) ,it's determined by pseudo-metric, the definition of VEC and T_n will be introduced in the second part.Theorem 0.2 T = T(A,D) is a self affine tile, |detA|=q >1∈R , the contact matrix C has the largest eigenvalueλ.If T = T(A,D) satisfies the VEC, thenAnd several examples are presented to illustrate the general theory in the last part of this paper.