Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T21:05:27.922Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Classical measure theory

Published online by Cambridge University Press:  12 August 2010

Jonathan Jordan
Jonathan Jordan
Affiliation:
Department of Probability and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, UK ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

fractal graphsLaplacianspectral decimationpentagasket

MSC classification

Secondary: 28A80: Fractals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 731 - 746
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1. Adams, B., Smith, S. A., Strichartz, R. S. and Teplyaev, A., The spectrum of the Laplacian on the pentagasket, in Fractals in Graz 2001, Trends in Mathematics, pp. 1–24 (Birkhäuser, Basel, 2003).Google Scholar
2. Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B. and Teplyaev, A., Vibration nodes of 3n-gaskets and other fractals, J. Phys. A41 (2008), 015101.Google Scholar
3. Chung, F. R. K., Spectral graph theory, CBMS Regional Conference Series, Volume 92 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
4. Fukushima, M. and Shima, T., On a spectral analysis for the Sierpiński gasket, Potent. Analysis 1 (1992), 1–35.CrossRefGoogle Scholar
5. Jordan, J. H., Spectrum of the Laplacian of an asymmetric fractal graph, Proc. Edinb. Math. Soc. 49 (2006), 101–113.CrossRefGoogle Scholar
6. Lindstrøm, T., Brownian motion on nested fractals, Memoirs of the American Mathematical Society, Volume 420 (American Mathematical Society, Providence, RI, 1990).CrossRefGoogle Scholar
7. Malozemov, L. and Teplyaev, A., Self-similarity, operators and dynamics, Math. Phys. Analysis Geom. 6 (2003), 201–218.CrossRefGoogle Scholar
8. Rammal, R. and Toulouse, G., Random walks on fractal structures and percolation clusters, J. Phys. Lett. 44 (1983), L13–L22.CrossRefGoogle Scholar
9. Sabot, C., Integrated density of states and self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension, Annales Inst. H. Poincaré B37 (2001), 275–311.CrossRefGoogle Scholar
10. Sabot, C., Spectral properties of self-similar lattices and iteration of rational maps, Mémoires de la Société Mathématique de France, Volume 92 (Société Mathématique de France, Paris, 2003).CrossRefGoogle Scholar
11. Sabot, C., Spectral analysis of a self-similar Sturm-Liouville operator, Indiana Univ. Math. J. 54 (2005), 645–668.CrossRefGoogle Scholar
12. Shima, T., On eigenvalue problems for the random walks on the Sierpiński pre-gaskets, Jpn. J. Ind. Appl. Math. 8 (1991), 127–141.CrossRefGoogle Scholar
13. Shima, T., On eigenvalue problems for Laplacians on PCF self-similar sets, Jpn. J. Ind. Appl. Math. 13 (1996), 1–23.CrossRefGoogle Scholar