The topological Hausdorff dimension and transport properties of Sierpiński carpets

Abstract In this Letter, the analytical expression of topological Hausdorff dimension D t H is derived for some kinds of infinitely ramified Sierpinski carpets. Furthermore, we deduce that the Hausdorff dimension of the union of all self-avoiding paths admitted on the infinitely ramified Sierpinski carpet has the Hausdorff dimension D H s a = D t H . We also put forward a phenomenological relation for the fractal dimension of the random walk on the infinitely ramified Sierpinski carpet. The effects of fractal attributes on the transport properties are highlighted. Possible applications of the developed tools are briefly outlined.

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