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  • Proceedings of the Edinburgh Mathematical Society, Volume 53, Issue 3
  • October 2010, pp. 731-746

The spectra of the Laplacians of fractal graphs not satisfying spectral decimation

  • Jonathan Jordan (a1)
  • DOI: http://dx.doi.org/10.1017/S0013091508000898
  • Published online: 12 August 2010
Abstract
Abstract

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (ℂ2)3 to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory.

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1.B. Adams , S. A. Smith , R. S. Strichartz and A. Teplyaev , The spectrum of the Laplacian on the pentagasket, in Fractals in Graz 2001, Trends in Mathematics, pp. 124 (Birkhäuser, Basel, 2003).

4.M. Fukushima and T. Shima , On a spectral analysis for the Sierpiński gasket, Potent. Analysis 1 (1992), 135.


7.L. Malozemov and A. Teplyaev , Self-similarity, operators and dynamics, Math. Phys. Analysis Geom. 6 (2003), 201218.

8.R. Rammal and G. Toulouse , Random walks on fractal structures and percolation clusters, J. Phys. Lett. 44 (1983), L13–L22.

9.C. Sabot , Integrated density of states and self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension, Annales Inst. H. Poincaré B37 (2001), 275311.

11.C. Sabot , Spectral analysis of a self-similar Sturm-Liouville operator, Indiana Univ. Math. J. 54 (2005), 645668.

12.T. Shima , On eigenvalue problems for the random walks on the Sierpiński pre-gaskets, Jpn. J. Ind. Appl. Math. 8 (1991), 127141.

13.T. Shima , On eigenvalue problems for Laplacians on PCF self-similar sets, Jpn. J. Ind. Appl. Math. 13 (1996), 123.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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