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Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure

Published online by Cambridge University Press:  20 November 2018

Alexander Teplyaev
Alexander Teplyaev*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs CT 06269-3009, U.S.A. e-mail: teplyaev@math.uconn.edu
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Abstract

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We define sets with finitely ramified cell structure, which are generalizations of post-critically finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords

28A8031C2553B9958J6560J6060G18fractalsself-similarityenergyresistanceDirichlet formsdiffusionsquantum graphsgeneralized Riemannian metric
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 457 - 480
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B. and Teplyaev, A., Vibration modes of 3n-gaskets and other fractals. To appear in J. Phys. A: Math. Theor. http://www.math.uconn.edu/˜teplyaev/research/ Google Scholar
[2] Barlow, M. T., Diffusions on fractals. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1690, Springer, Berlin, 1998, pp. 1–121.Google Scholar
[3] Barlow, M. T. and Bass, R. F., Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51(1999), no. 4, 673–744.Google Scholar
[4] Barlow, M. T. and Bass, R. F., Random walks on graphical Sierpinski carpets. In: RandomWalks and Discrete Potential Theory. Sympos. Math.39, Cambridge University Press, Cambridge, 1999, pp. 26–55.Google Scholar
[5] Barlow, M. T. and Bass, R. F., Stability of parabolic Harnack inequalities. Trans. Amer.Math. Soc. 356(2004), 1501–1533.Google Scholar
[6] Barlow, M. T., Bass, R. F. and Kumagai, T., Stability of parabolic Harnack inequalities on metric measure spaces J. Math. Soc. Japan, 58(2006), no. 2, 485–619.Google Scholar
[7] Barlow, M. T. and Hambly, B. M., Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets. Ann. Inst. H. Poincaré Probab. Statist. 33(1997), no. 5, 531–557.Google Scholar
[8] Ben-Bassat, O., Strichartz, R. S., and Teplyaev, A., What is not in the domain of the Laplacian on Sierpinski gasket type fractals. J. Funct. Anal. 166(1999), no. 2, 197–217.Google Scholar
[9] Bouleau, N. and Hirsch, F., Dirichlet forms and analysis on Wiener space. de Gruyter Studies in Mathematics 14, de Gruyter, Berlin, 1991.Google Scholar
[10] Fukushima, M., Oshima, Y., and Takada, M., Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics 19, de Gruyter, Berlin, 1994.Google Scholar
[11] Hambly, B.M., Heat kernels and spectral asymptotics for some random Sierpinski gaskets. In: Fractal Geometry and Stochastics, II. Progr. Probab. 46, Birkhäuser, Basel, 2000, pp. 239–267.Google Scholar
[12] Hambly, B.M., On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets. Probab. Theory Related Fields 117(2000), no. 2, 221–247.Google Scholar
[13] Hambly, B.M., Self-similar energies on post-critically finite self-similar fractals. J. LondonMath. Soc. 74(2006), no. 1, 93–112.Google Scholar
[14] Hino, M., On singularity of energy measures on self-similar sets. Probab. Theory Related Fields 132(2005), no. 2, 265–290.Google Scholar
[15] Hino, M. and Nakahara, K., On singularity of energy measures on self-similar sets. II. Bull. London Math. Soc. 38(2006), no. 6, 1019–1032.Google Scholar
[16] Hveberg, K., Injective mapping systems and self-homeomorphic fractals. Ph.D. Thesis, University of Oslo, 2005.Google Scholar
[17] Kameyama, A., Distances on topological self-similar sets and the kneading determinants. J. Math. Kyoto Univ. 40(2000), 601–672.Google Scholar
[18] Kigami, J., A harmonic calculus on the Sierpiński spaces. Japan J. Appl. Math. 6(1989), no. 2, 259–290.Google Scholar
[19] Kigami, J., Harmonic calculus on p.c.f. self-similar sets. Trans. Amer.Math. Soc. 335(1993), no. 2, 721–755.Google Scholar
[20] Kigami, J., Harmonic metric and Dirichlet form on the Sierpiński gasket. In: Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals 201–218, Pitman Res. Notes Math. Ser. 283, Longman Sci. Tech., Harlow, 1993, pp. 201–218.Google Scholar
[21] Kigami, J., Effective resistances for harmonic structures on p.c.f. self-similar sets. Math. Proc. Cambridge Philos. Soc. 115(1994), no. 2, 291–303.Google Scholar
[22] Kigami, J., Analysis on Fractals. Cambridge Tracts in Mathematics 143, Cambridge University Press, 2001.Google Scholar
[23] Kigami, J., Harmonic analysis for resistance forms. J. Funct. Anal. 204(2003), no. 2, 399–444.Google Scholar
[24] Kigami, J., Local Nash inequality and inhomogeneity of heat kernels. Proc. LondonMath. Soc. 89(2004), no. 2, 525–544.Google Scholar
[25] Kigami, J., Volume doubling measures and heat kernel estimates on self-similar sets. To appear in Memoirs of the American Mathematical Society. http://www-an.acs.i.kyoto-u.ac.jp/˜kigami/preprints.html Google Scholar
[26] Kigami, J., Measurable Riemannian goemetry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate. To appear in Matt. Ann. http://www-an.acs.i.kyoto-u.ac.jp/˜kigami/preprints.html Google Scholar
[27] Kuchment, P., Quantum graphs. I. Some basic structures. Waves Random Media 14(2004), no. 1, S107–S128.Google Scholar
[28] Kuchment, P., Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A. 38(2005), no. 22, 4887–4900.Google Scholar
[29] Kusuoka, S., Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci. 25(1989), no. 4, 659–680.Google Scholar
[30] Kusuoka, S., Lecture on diffusion processes on nested fractals. In: Statistical Mechanics and Fractals. Lecture Notes in Mathematics 1567, Springer-Verlag, Berlin, 1993, pp. 39–98.Google Scholar
[31] Kusuoka, S. and X Yin, Z., Dirichlet forms on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93 (1992), no. 2, 169–196.Google Scholar
[32] Malozemov, L. and Teplyaev, A., Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6(2003), no. 3, 201–218.Google Scholar
[33] Metz, V., The cone of diffusions on finitely ramified fractals. Nonlinear Anal. 55 (2003), no. 6, 723–738.Google Scholar
[34] Metz, V. and Sturm, K.-T., Gaussian and non-Gaussian estimates for heat kernels on the Sierpiński gasket. In: Dirichlet Forms and Stochastic Processes. de Gruyter, Berlin, 1995, pp. 283–289.Google Scholar
[35] Meyers, R., Strichartz, R., and Teplyaev, A., Dirichlet forms on the Sierpinski gasket. Pacific J. Math. 217(2004), no. 1, 149–174 Google Scholar
[36] Strichartz, R. S., Analysis on fractals. Notices Amer. Math. Soc 46(1999), no. 10, 1199–1208.Google Scholar
[37] Strichartz, R. S., Taylor approximations on Sierpiński type fractals. J. Funct. Anal. 174(2000), no. 1, 76–127.Google Scholar
[38] Strichartz, R. S., Fractafolds based on the Sierpiński gasket and their spectra. Trans. Amer.Math. Soc 355(2003), no. 10, 4019–4043.Google Scholar
[39] Strichartz, R. S., Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton, NJ, 2006.Google Scholar
[40] Teplyaev, A., Spectral Analysis on Infinite Sierpiński Gaskets, J. Funct. Anal. 159(1998), no. 2, 537–567.Google Scholar
[41] Teplyaev, A., Gradients on fractals. J. Funct. Anal. 174(2000), no. 1, 128–154.Google Scholar
[42] Teplyaev, A., Energy and Laplacian on the Sierpiński gasket. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1. Proc. Sympos. Pure Math. 72, American Mathematical Society, Providence, RI, 2004, pp. 131–154.Google Scholar