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“Laplacians” on finitely ramified, graph directed fractals

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Abstract.

A finitely ramified graph directed fractal is approximated by an adapted sequence of increasingly refined graphs. The scaling problem for a corresponding sequence of “discrete Laplacians” is rephrased via a renormalization map comparing two subsequent graphs. A limit set dichotomy for this map is proved: The forward orbit always accumulates at periodic points, even if the corresponding models are disconnected. Thus these periodic points can be numerically approximated by an iteration scheme based on the Schur complement. This allows to turn numerical information via the “short-cut test” into theorems on the existence of Laplacians on fractals.

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Authors and Affiliations

  1. Faculty of Mathematics, Bielefeld University, 33501, Bielefeld, Germany

    Volker Metz

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  1. Volker Metz
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Correspondence to Volker Metz.

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Mathematics Subject Classification (2000): 31C25, 60J45, 65N55, 47J10, 28A80

Research supported by the DFG research group “Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.

The author is greatful to the anonymous referees for pointing out flaws in two original definitions.

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Metz, V. “Laplacians” on finitely ramified, graph directed fractals. Math. Ann. 330, 809–828 (2004). https://doi.org/10.1007/s00208-004-0571-9

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  • DOI: https://doi.org/10.1007/s00208-004-0571-9

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