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VIBRATION SPECTRA OF THE m-TREE FRACTAL

DANIEL J. FORD

Department of Mathematics, University of Connecticut, 196 Auditorium Rd U-3009, Storrs, CT 06269, USA

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and

BENJAMIN STEINHURST

Department of Mathematics, University of Connecticut, 196 Auditorium Rd U-3009, Storrs, CT 06269, USA

Corresponding author.

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https://doi.org/10.1142/S0218348X1000483XCited by:4 (Source: Crossref)

Abstract

We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.

Keywords:

References

N. Bajorinet al., J. Phys. A 41, 015101 (2008), DOI: 10.1088/1751-8113/41/1/015101. Crossref, Google Scholar
N. Bajorinet al., Fractals 16(3), 243 (2008), DOI: 10.1142/S0218348X08004010. Link, Web of Science, Google Scholar
T. Shima, Jpn. J. Ind. Appl. Math. 13, 1 (1996), DOI: 10.1007/BF03167295. Crossref, Google Scholar
D. Zhou, J. Fourier Anal. Appl 16, 76 (2010), DOI: 10.1007/s00041-009-9087-8. Crossref, Web of Science, Google Scholar
D. Zhou, Pac. J. Math 241(2), 369 (2009), DOI: 10.2140/pjm.2009.241.369. Crossref, Web of Science, Google Scholar
J. Kigami, J. Funct. Anal. 128, 48 (1995), DOI: 10.1006/jfan.1995.1023. Crossref, Web of Science, Google Scholar
J. Kigami , Analysis on Fractals , Cambridge Tracts in Mathematics ( Cambridge University Press , 2001 ) . Crossref, Google Scholar
R. Strichartz , Differential Equations on Fractals: A Tutorial ( Princeton University Press , Princeton , 2006 ) . Crossref, Google Scholar
J. Kigami, Trans. Am. Math. Soc. 335(2), 721 (1993), DOI: 10.2307/2154402. Web of Science, Google Scholar
V. Nerashevych , Self-Similar Groups , Mathematical Surveys and Monographs 117 ( American Mathematical Society , Providence, RI , 2005 ) . Crossref, Google Scholar
L. Barthold and R. Grigorchuk, C. R. Acad. Sci. Paris Sér. I Math. 331(6), 429 (2000). Crossref, Google Scholar
L. Bartholdi, R. Grigorchuk and V. Nekrashevych, Fractals in Braz 2001, Trends in Mathematics, eds. P. Brabner and W. Woess (Birkhäuser, Basel, Switzerland, 2002) pp. 25–118. Google Scholar
J. Kigami, J. Funct. Anal. 204, 399 (2003), DOI: 10.1016/S0022-1236(02)00149-0. Crossref, Web of Science, Google Scholar
M. Fukushima and T. Shima, Potential Anal. 1, 1 (1992), DOI: 10.1007/BF00249784. Crossref, Google Scholar
J. Kigami and M. Lapidus, Commun. Math. Phys. 158(1), 93 (1993), DOI: 10.1007/BF02097233. Crossref, Web of Science, Google Scholar
K. Falconer , Fractal Geometry: Mathematical Foundations and Applications ( John Wiley & Sons Ltd. , Chichester , 1990 ) . Google Scholar
S. Roman , Advanced Linear Algebra ( Springer Science+Business Media , New York , 2008 ) . Crossref, Google Scholar