The Hausdorff Dimension of General Sierpinski Carpets

We refer to R as a general Sierpίήski carpet, after Mandelbrot [4], since Sierpiήski's universal curve is a special case of this construction [6]. It is clear that R = {J[fi(R)9 where r = \R\ and the ft are affine maps contracting R by a factor of n horizontally and m vertically. When n = m these maps are actually similarity transformations, and a well known argument shows the dimension of R is log r/ log n (following e.g. Beardon [1]). If n > m, however, a different approach is required, essen tially because squares are stretched into narrow rectangles under itera tion of the maps /*. Our method relies on elementary probability theory to address the general case, and we obtain the following result.