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Zbl 1068.31004
Metz, Volker
``Laplacians'' on finitely ramified, graph directed fractals. (English)
[J] Math. Ann. 330, No. 4, 809-828 (2004). ISSN 0025-5831; ISSN 1432-1807/e

In the paper reviewed [J. Funct. Anal. 220, 118--156 (2005; Zbl 1068.31005)], the author developed the ``short-cut test", an algorithmic approach to the existence and uniqueness of ``Laplacians" on finitely ramified graph directed fractals. In the present paper the author establishes a result which can be used to find periodic points among the accumulation points of the short-cut test and to justify the numerical aspects of the test. A finitely ramified graph directed fractal can be approximated by an adapted sequence of increasingly refined graphs. The scaling problem for the sequence of discrete Laplacians is reformulated via a renormalization map comparing two subsequent graphs. The main result of this paper is a limit set dichotomy for this map: The forward orbit always accumualtes at periodic points, even if the corresponding models are disconnected.
[Renming Song (Urbana)]

MSC 2000:: *31C25 Dirichlet spaces
60J45 Probabilistic potential theory
65N55 Multigrid methods; domain decomposition (BVP of PDE)
47J10 Nonlinear eigenvalue problems
28A80 Fractals

Keywords: ergodicity; Dirichlet forms; existence and uniqueness of ``Laplacians" on finitely ramified graph directed fractals

Citations: Zbl 1068.31005

Cited in: Zbl 1068.31005

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