Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The zeta function of the Laplacian on certain fractals


Authors: Gregory Derfel, Peter J. Grabner and Fritz Vogl
Journal: Trans. Amer. Math. Soc. 360 (2008), 881-897
MSC (2000): Primary 30B50; Secondary 11M41, 37F10
Published electronically: September 24, 2007
MathSciNet review: 2346475
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the zeta function $ \zeta_\Delta$ of the Laplacian $ \Delta$ on self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30B50, 11M41, 37F10

Retrieve articles in all journals with MSC (2000): 30B50, 11M41, 37F10


Additional Information

Gregory Derfel
Affiliation: Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
Email: derfel@math.bgu.ac.il

Peter J. Grabner
Affiliation: Institut für Analysis und Computational Number Theory, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
Email: peter.grabner@tugraz.at

Fritz Vogl
Affiliation: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
Email: fvogl@osiris.tuwien.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04240-7
PII: S 0002-9947(07)04240-7
Keywords: Dirichlet series, Laplace operator, fractals, spectral decimation, complex dynamics
Received by editor(s): June 23, 2005
Received by editor(s) in revised form: November 4, 2005
Published electronically: September 24, 2007
Additional Notes: The second author was supported by the Austrian Science Fund project S9605
Article copyright: © Copyright 2007 American Mathematical Society