Hodge-de rham theory of k-forms on carpet type fractals

Jason Bello, Yiran Li, Robert S. Strichartz

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages23-62
Number of pages40
Edition9783319132297
DOIs
StatePublished - 2015

Publication series

NameApplied and Numerical Harmonic Analysis
Number9783319132297
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Bibliographical note

Funding Information:
† Research supported by the National Science Foundation, grant DMS - 1162045

Funding Information:
∗ Research supported by the National Science Foundation through the Research Experiences for Undergraduates Program at Cornell,grant DMS-1156350

Keywords

  • Analysis on fractals
  • Harmonic 1-forms
  • Hodge-de rham theory
  • K-forms
  • Magic carpet
  • Sierpinski carpet

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