Self-Similarity,Operators and Dynamics |
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Authors: | Malozemov Leonid Teplyaev Alexander |
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Affiliation: | (1) Countrywide Securities Corporation, 4500 Park Granada, Calabasas, CA, 91302, U.S.A;(2) Department of Mathematics, University of California, Riverside, CA, 92521, U.S.A |
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Abstract: | We construct a large class of infinite self-similar (fractal, hierarchical or substitution) graphs and show, under a certain strong symmetry assumption, that the spectrum of the Laplacian can be described in terms of iterations of an associated rational function (so-called 'spectral decimation'). We prove that the spectrum consists of the Julia set of the rational function and a (possibly empty) set of isolated eigenvalues which accumulate to the Julia set. In order to obtain our results, we start with investigation of abstract spectral self-similarity of operators. |
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Keywords: | infinite graphs self-similar graphs fractal graphs hierarchical graphs substitution graphs Laplacian spectral decimation self-similar spectrum Julia set complex dynamics |
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