Hodge-de Rham Theory of K-Forms on Carpet Type Fractals

Abstract

We outline a Hodge-de Rham theory of k-forms (for k = 0,1,2) on two fractals: the Sierpinski Carpet (SC) and a new fractal that we call the Magic Carpet (MC), obtained by a construction similar to that of SC modified by sewing up the edges whenever a square is removed. Our method is to approximate the fractals by a sequence of graphs, use a standard Hodge-de Rham theory on each graph, and then pass to the limit. While we are not able to prove the existence of the limits, we give overwhelming experimental evidence of their existence, and we compute approximations to basic objects of the theory, such as eigenvalues and eigenforms of the Laplacian in each dimension, and harmonic 1-forms dual to generators of 1-dimensional homology cycles. On MC we observe a Poincare type duality between the Laplacian on 0-forms and 2-forms. On the other hand, on SC the Laplacian on 2-forms appears to be an operator with continuous (as opposed to discrete) spectrum. 2010 Mathematics Subject Classification. Primary: 28A80