Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-07-04240-7
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?)

  • 1. M. T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995), Springer Verlag, Berlin, 1998, pp. 1-121. MR 1668115 (2000a:60148)
  • 2. M. T. Barlow and J. Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc. (2) 56 (1997), 320-332. MR 1489140 (99b:35162)
  • 3. M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Fields 79 (1988), 543-623. MR 966175 (89g:60241)
  • 4. A. F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, no. 132, Springer Verlag, 1991. MR 1128089 (92j:30026)
  • 5. J. D. Biggins and N. H. Bingham, Near-constancy phenomena in branching processes, Math. Proc. Camb. Philos. Soc. 110 (1991), 545-558. MR 1120488 (93d:60136)
  • 6. R. P. Boas, Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627 (16:914f)
  • 7. G. Doetsch, Handbuch der Laplace-Transformation. Band I: Theorie der Laplace-Transformation, Birkhäuser Verlag, Basel, 1971, Verbesserter Nachdruck der ersten Auflage 1950, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 14. MR 0344807 (49:9546)
  • 8. S. Dubuc, Etude théorique et numérique de la fonction de Karlin-McGregor, J. Analyse Math. 42 (1982), 15-37. MR 729400 (85g:30040)
  • 9. M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal. 1 (1992), 1-35. MR 1245223 (95b:31009)
  • 10. P. J. Grabner, Functional iterations and stopping times for Brownian motion on the Sierpinski gasket, Mathematika 44 (1997), 374-400. MR 1600494 (99b:60128)
  • 11. G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. MR 0185094 (32:2564)
  • 12. J. Jorgenson and S. Lang, Basic analysis of regularized series and products, Lecture Notes in Mathematics, vol. 1564, Springer-Verlag, Berlin, 1993. MR 1284924 (95e:11094)
  • 13. S. Karlin and J. McGregor, Embeddability of discrete time simple branching processes into continuous time branching processes, Trans. Amer. Math. Soc. 132 (1968), 115-136. MR 0222966 (36:6015)
  • 14. J. Kigami, A harmonic calculus for p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), 721-755. MR 1076617 (93d:39008)
  • 15. J. Kigami, Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets, J. Funct. Anal. 156 (1998), 170-198. MR 1632976 (99g:35096)
  • 16. -, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • 17. J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), 93-125. MR 1243717 (94m:58225)
  • 18. B. Krön, Green functions on self-similar graphs and bounds for the spectrum of the Laplacian, Ann. Inst. Fourier (Grenoble) 52 (2002), 1875-1900. MR 1954327 (2003k:60180)
  • 19. B. Krön and E. Teufl, Asymptotics of the transition probabilities of the simple random walk on self-similar graphs, Trans. Amer. Math. Soc. 356 (2004), 393-414. MR 2020038 (2004k:60130)
  • 20. M. Kuczma, On the Schröder equation, Rozprawy Mat. 34 (1963), 50. MR 0173875 (30:4082)
  • 21. M. L. Lapidus, Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions, Topol. Methods Nonlinear Anal. 4 (1994), 137-195. MR 1321811 (96g:58196)
  • 22. M. L. Lapidus and M. van Frankenhuysen, Fractal geometry and number theory, Birkhäuser Boston Inc., Boston, MA, 2000, Complex dimensions of fractal strings and zeros of zeta functions. MR 1726744 (2001b:11079)
  • 23. T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., vol. 420, Amer. Math. Soc., 1990. MR 988082 (90k:60157)
  • 24. L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs, J. Funct. Anal. 129 (1995), 390-405. MR 1327184 (96e:60114)
  • 25. -, Self-similarity, operators and dynamics, Math. Phys. Anal. Geom. 6 (2003), 201-218. MR 1997913 (2004d:47012)
  • 26. H. Mellin, Die Dirichlet'schen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht, Acta Math. 28 (1903), 37-64.
  • 27. S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242-256. MR 0031145 (11:108b)
  • 28. F. Oberhettinger, Tables of Mellin transforms, Springer-Verlag, New York, 1974. MR 0352890 (50:5376)
  • 29. R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals, Encyclopedia of Mathematics and its Applications, vol. 85, Cambridge University Press, Cambridge, 2001. MR 1854469 (2002h:33001)
  • 30. R. Rammal and G. Toulouse, Random walks on fractal structures and percolation clusters, J. Physique Lettres 44 (1983), L13-L22.
  • 31. S. Rosenberg, The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, vol. 31, Cambridge University Press, Cambridge, 1997. MR 1462892 (98k:58206)
  • 32. C. Sabot, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal. 173 (2000), 497-524. MR 1760624 (2001j:35216)
  • 33. T. Shima, On eigenvalue problems for the random walk on the Sierpinski pre-gaskets, Japan J. Appl. Ind. Math. 8 (1991), 127-141. MR 1093832 (92g:60094)
  • 34. T. Shima, The eigenvalue problem for the Laplacian on the Sierpinski gasket, Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser., vol. 283, Longman Sci. Tech., Harlow, 1993, pp. 279-288. MR 1354159 (96m:31015)
  • 35. -, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math. 13 (1996), 1-23. MR 1377456 (97f:28028)
  • 36. R. S. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal. 164 (1999), 181-208. MR 1695571 (2000f:35032)
  • 37. -, Fractafolds based on the Sierpinski gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 4019-4043. MR 1990573 (2004b:28013)
  • 38. -, Laplacians on fractals with spectral gaps have nicer Fourier series, Math. Res. Lett. 12 (2005), 269-274. MR 2150883 (2006e:28013)
  • 39. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, no. 46, Cambridge University Press, 1995. MR 1342300 (97e:11005b)
  • 40. A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Funct. Anal. 159 (1998), 537-567. MR 1658094 (99j:35153)
  • 41. -, Spectral zeta function of symmetric fractals, Fractal geometry and stochastics III (C. Bandt, U. Mosco, and M. Zähle, eds.), Progr. Probab., vol. 57, Birkhäuser, Basel, 2004, pp. 245-262. MR 2087144 (2005h:28028)
  • 42. -, Spectral zeta functions of fractals and the complex dynamics of polynomials, available at http://arxiv.org/pdf/math.SP/0505546, 2005.
  • 43. E. Teufl, On the asymptotic behaviour of analytic solutions of linear iterative functional equations, Aequationes Math. (2006), to appear.
  • 44. G. Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954. MR 0061658 (15:861a)

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: https://doi.org/10.1090/S0002-9947-07-04240-7
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

American Mathematical Society