Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T02:38:21.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

Published online by Cambridge University Press:  20 November 2018

Sze-Man Ngai
Sze-Man Ngai*
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, U.S.A. email: smngai@georgiasouthern.edu
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 set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords

28A8035P2035J0543A0547A75spectral dimensionfractalLaplacianself-similar measureiterated function system with overlapssecond-order self-similar identities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 3 , 01 June 2011 , pp. 648 - 688
Copyright
Copyright © Canadian Mathematical Society 2011

References

[BNT] Bird, E. J., Ngai, S.-M., and Teplyaev, A., Fractal Laplacians on the unit interval. Ann. Sci. Math. Québec 27(2003), no. 2, 135–168.Google Scholar
[D] Davies, E. B., Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995.Google Scholar
[DL] Deng, Q.-R. and Lau, K.-S., Open set condition and post-critically finite self-similar sets. Nonlinearity 21(2008), no. 6, 1227–1232. doi:10.1088/0951-7715/21/6/004 Google Scholar
[E] Erdʺos, P., On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61(1939), 974–976. doi:10.2307/2371641 Google Scholar
[F1] Falconer, K., Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Ltd., Chichester, 1990.Google Scholar
[F2] Falconer, K., Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997.Google Scholar
[FLN] Fan, A.-H., Lau, K.-S., and Ngai, S.-M., Iterated function systems with overlaps. Asian J. Math. 4(2000), no. 3, 527–552.Google Scholar
[Fe] Feng, D.-J., The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195(2005), no. 1, 24–101.Google Scholar
[FL W] Feng, D.-J., Lau, K.-S., and Wang, X.-Y., Some exceptional phenomena in multifractal formalism. II. Asian J. Math. 9(2005), no. 4, 473–488.Google Scholar
[FO] Feng, D.-J. and Olivier, E., Multifractal analysis of weak Gibbs measures and phase transition—application to some Bernoulli convolutions. Ergodic Theory Dynam. Systems 23(2003), no. 6, 1751–1784.Google Scholar
[HLN] Hu, J., Lau, K.-S., and Ngai, S.-M., Laplace operators related to self-similar measures on Rd. J. Funct. Anal. 239(2006), no. 2, 542–565. doi:10.1016/j.jfa.2006.07.005 Google Scholar
[H] Hutchinson, J. E., Fractals and self-similarity. Indiana Univ. Math. J. 30(1981), no. 5, 713–747. doi:10.1512/iumj.1981.30.30055 Google Scholar
[JY] Jin, N. and Yau, S. S. T., General finite type IFS and M-matrix. Comm. Anal. Geom. 13(2005), no. 4, 821–843.Google Scholar
[K] Kigami, J., Analysis on fractals. Cambridge Tracts in Mathematics, 143, Cambridge University Press, Cambridge, 2001.Google Scholar
[KL] Kigami, J. and Lapidus, M. L., Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 158(1993), no. 1, 93–125. doi:10.1007/BF02097233 Google Scholar
[L] Lapidus, M. L., Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture. Trans. Amer. Math. Soc. 325(1991), no. 2, 465–529. doi:10.2307/2001638 Google Scholar
[LP] Lapidus, M. L. and Pomerance, C., The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. (3) 66(1993), no. 1, 41–69. doi:10.1112/plms/s3-66.1.41 Google Scholar
[La1] K.-S., Lau Fractal measures and mean p-variations. J. Funct. Anal. 108(1992), no. 2, 427–457. doi:10.1016/0022-1236(92)90031-D Google Scholar
[La2] K.-S., Lau, Dimension of a family of singular Bernoulli convolutions. J. Funct. Anal. 116(1993), no. 2, 335–358. doi:10.1006/jfan.1993.1116 Google Scholar
[LN1] K.-S., Lau and Ngai, S.-M., Lq-spectrum of the Bernoulli convolution associated with the golden ratio. Studia Math. 131(1998), no. 3, 225–251.Google Scholar
[LN2] K.-S., Lau and Ngai, S.-M., Multifractal measures and a weak separation condition. Adv. Math. 141(1999), no. 1, 45–96. doi:10.1006/aima.1998.1773 Google Scholar
[LN3] K.-S., Lau and Ngai, S.-M., Second-order self-similar identities and multifractal decompositions. Indiana Univ. Math. J. 49(2000), no. 3, 925–972.Google Scholar
[LN4] K.-S., Lau and Ngai, S.-M., A generalized finite type condition for iterated function systems. Adv. Math 208(2007), no. 2, 647–671. doi:10.1016/j.aim.2006.03.007 Google Scholar
[L WC] K.-S., Lau Wang, J., and C.-H., Chu Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures. Studia Math. 117(1995), no. 1, 1–28.Google Scholar
[L W] Lau, K.-S. and Wang, X.-Y., Some exceptional phenomena in multifractal formalism. I. Asian J. Math. 9(2005), no. 2, 275–294.Google Scholar
[Mi] Minc, H., Nonnegative matrices. Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.Google Scholar
[NS1] Naimark, K. and Solomyak, M., On the eigenvalue behaviour for a class of operators related to self-similar measures on Rd. C. R. Acad. Sci. Paris Sér. I Math. 319(1994), no. 8, 837–842.Google Scholar
[NS2] Naimark, K. and Solomyak, M., The eigenvalue behaviour for the boundary value problems related to self-similar measures on Rd. Math. Res. Lett. 2(1995), no. 3, 279–298.Google Scholar
[N W] Ngai, S.-M. and Wang, Y., Hausdorff dimension of self-similar sets with overlaps. J. London Math. Soc. (2) 63(2001), no. 3, 655–672. doi:10.1017/S0024610701001946 Google Scholar
[PS1] Peres, Y. and Solomyak, B., Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3(1996), no. 2, 231–239.Google Scholar
[PSS] Peres, Y., Schlag, W., and Solomyak, B., Sixty years of Bernoulli convolutions. In: Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., 46, Birkhäuser, Basel, 2000, pp. 39–65.Google Scholar
[So] Solomyak, B., On the random seriesP±，n (an Erdʺos problem). Ann. of Math. (2) 142(1995), no. 3, 611–625. doi:10.2307/2118556 Google Scholar
[SV] Solomyak, M. and Verbitsky, E., On a spectral problem related to self-similar measures. Bull. London Math. Soc. 27(1995), no. 3, 242–248. doi:10.1112/blms/27.3.242 Google Scholar
[S] Strichartz, R. S., Differential equations on fractals. A tutorial. Princeton University Press, Princeton, NJ, 2006.Google Scholar
[STZ] Strichartz, R. S., Taylor, A., and Zhang, T., Densities of self-similar measures on the line. Experiment. Math. 4(1995), no. 2, 101–128.Google Scholar
[W] Weyl, H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(1912), no. 4, 441–479. doi:10.1007/BF01456804 Google Scholar