Skip to main content
×
×
Home

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

  • Jonathan Jordan (a1)
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.

Copyright
References
Hide All
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. 124 (Birkhäuser, Basel, 2003).
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.
3. Chung F. R. K., Spectral graph theory, CBMS Regional Conference Series, Volume 92 (American Mathematical Society, Providence, RI, 1997).
4. Fukushima M. and Shima T., On a spectral analysis for the Sierpiński gasket, Potent. Analysis 1 (1992), 135.
5. Jordan J. H., Spectrum of the Laplacian of an asymmetric fractal graph, Proc. Edinb. Math. Soc. 49 (2006), 101113.
6. Lindstrøm T., Brownian motion on nested fractals, Memoirs of the American Mathematical Society, Volume 420 (American Mathematical Society, Providence, RI, 1990).
7. Malozemov L. and Teplyaev A., Self-similarity, operators and dynamics, Math. Phys. Analysis Geom. 6 (2003), 201218.
8. Rammal R. and Toulouse G., Random walks on fractal structures and percolation clusters, J. Phys. Lett. 44 (1983), L13–L22.
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), 275311.
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).
11. Sabot C., Spectral analysis of a self-similar Sturm-Liouville operator, Indiana Univ. Math. J. 54 (2005), 645668.
12. Shima T., On eigenvalue problems for the random walks on the Sierpiński pre-gaskets, Jpn. J. Ind. Appl. Math. 8 (1991), 127141.
13. Shima T., On eigenvalue problems for Laplacians on PCF self-similar sets, Jpn. J. Ind. Appl. Math. 13 (1996), 123.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 83 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st January 2018. This data will be updated every 24 hours.