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“Laplacians” on finitely ramified, graph directed fractals. (English) Zbl 1068.31004
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.

MSC:
31C25Dirichlet spaces
60J45Probabilistic potential theory
65N55Multigrid methods; domain decomposition (BVP of PDE)
47J10Nonlinear spectral theory, nonlinear eigenvalue problems
28A80Fractals
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