{
  "id": "1204.5207",
  "version": "v3",
  "published": "2012-04-23T20:51:53.000Z",
  "updated": "2019-01-07T02:15:19.000Z",
  "title": "Spectral Analysis and Dirichlet Forms on Barlow-Evans Fractals",
  "authors": [
    "Benjamin Steinhurst",
    "Alexander Teplyaev"
  ],
  "comment": "v3: major revision",
  "categories": [
    "math.CA",
    "math.SP"
  ],
  "abstract": "We develop the foundation of the spectral analysis on Barlow-Evans projective limit fractals, or vermiculated spaces, which corresponds to symmetric Markov processes on these spaces. For some new examples, such as the generalized Laakso spaces and a Spierpinski P\\^ate \\`a Choux, one can develop a complete spectral theory, including the eigenfunction expansions that are analogous to Fourier series. Also, one can construct connected fractal spaces isospectral to the fractal strings of Lapidus and van Frankenhuijsen. Our work is motivated by recent progress in mathematical physics on fractals.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-06T14:30:33.000Z",
      "abstract": "We show that if a Barlow-Evans Markov process on a vermiculated space is symmetric, then one can study the spectral properties of the corresponding Laplacian using projective limits. For some examples, such as the Laakso spaces and a Spierpinski P\\^ate \\`a Choux, one can develop a complete spectral theory, including the eigenfunction expansions that are analogous to Fourier series. Also, one can construct connected fractal spaces isospectral to the fractal strings of Lapidus and van Frankenhuijsen.",
      "comment": "v2: minor fixes to language and layout",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2019-01-07T02:15:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L15",
      "28A80",
      "81Q35",
      "31C25",
      "34L10",
      "47A10",
      "60J35",
      "81Q12"
    ],
    "keywords": [
      "dirichlet forms",
      "barlow-evans fractals",
      "spectral analysis",
      "construct connected fractal spaces isospectral",
      "complete spectral theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.5207S"
    }
  }
}