Portal de Periódicos da CAPES

Hodge-de Rham theory on fractal graphs and fractals

2013; American Institute of Mathematical Sciences; Volume: 13; Issue: 2 Linguagem: Inglês

10.3934/cpaa.2014.13.903

1553-5258

Skye Aaron, Zach Conn, Robert Strichartz, Yu Han,

Topological and Geometric Data Analysis

We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and 3-dimensional Sierpinski gasket (SG$^3$), but the method is expected to be effective for many PCF fractals, and also infinitely ramified fractals such as the Sierpinski carpet (SC). Our approach is to construct k-forms and de Rham differential operators $d$ and $\delta$ for a sequence of graphs approximating the fractal, and then pass to the limit with suitable renormalization, in imitation of Kigami's approach on constructing Laplacians on functions. One of our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on functions. We give explicit construction of harmonic 1-forms for our examples. We also prove that the measures on line segments provided by 1-forms are not absolutely continuous with respect to Lebesgue measures.