The Dimension Of The Boundary Of Integral Self-affine Tile

We first give a tool, pseudo-norm, which can absorb the non-uniform contractilityof A^-1's in different directions, and some properties of it. We define the Hausdorffdimension and box dimension by the pseudo-norm instead of Euclidean norm, and then we introduce the definition of self-affine tile and self-similar tile. By using the tool we attain an estimate of the Hausdorff dimension of the boundary of a integral self-affine tile, and if the maximum and minimum moduli of the eigenvalues of A are equal, we get the exact Hausdorff dimension of the boundary of the integral self-affine tile. Finally, we show two examples, one is self-similar tile, and the other is a tile whose expandent matrix A has two eigenvalues with equal moduli.