Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure

Teplyaev, Alexander

Canadian journal of mathematics, Tome 60 (2008) no. 2, pp. 457-480

Résumé

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.

Détail

DOI : 10.4153/CJM-2008-022-3

Mots-clés : 28A80, 31C25, 53B99, 58J65, 60J60, 60G18, fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric

Bibliographie

[1] [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] [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] [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] [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] [5] Barlow, M. T. and Bass, R. F., Stability of parabolic Harnack inequalities. Trans. Amer.Math. Soc. 356(2004), 1501–1533. Google Scholar

[6] [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] [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] [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] [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] [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] [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] [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] [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] [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] [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] [16] Hveberg, K., Injective mapping systems and self-homeomorphic fractals. Ph.D. Thesis, University of Oslo, 2005. Google Scholar

[17] [17] Kameyama, A., Distances on topological self-similar sets and the kneading determinants. J. Math. Kyoto Univ. 40(2000), 601–672. Google Scholar

[18] [18] Kigami, J., A harmonic calculus on the Sierpiński spaces. Japan J. Appl. Math. 6(1989), no. 2, 259–290. Google Scholar

[19] [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] [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] [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] [22] Kigami, J., Analysis on Fractals. Cambridge Tracts in Mathematics 143, Cambridge University Press, 2001. Google Scholar

[23] [23] Kigami, J., Harmonic analysis for resistance forms. J. Funct. Anal. 204(2003), no. 2, 399–444. Google Scholar

[24] [24] Kigami, J., Local Nash inequality and inhomogeneity of heat kernels. Proc. LondonMath. Soc. 89(2004), no. 2, 525–544. Google Scholar

[25] [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] [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] [27] Kuchment, P., Quantum graphs. I. Some basic structures. Waves Random Media 14(2004), no. 1, S107–S128. Google Scholar

[28] [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] [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] [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] [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] [32] Malozemov, L. and Teplyaev, A., Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6(2003), no. 3, 201–218. Google Scholar

[33] [33] Metz, V., The cone of diffusions on finitely ramified fractals. Nonlinear Anal. 55 (2003), no. 6, 723–738. Google Scholar

[34] [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] [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] [36] Strichartz, R. S., Analysis on fractals. Notices Amer. Math. Soc 46(1999), no. 10, 1199–1208. Google Scholar

[37] [37] Strichartz, R. S., Taylor approximations on Sierpiński type fractals. J. Funct. Anal. 174(2000), no. 1, 76–127. Google Scholar

[38] [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] [39] Strichartz, R. S., Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton, NJ, 2006. Google Scholar

[40] [40] Teplyaev, A., Spectral Analysis on Infinite Sierpiński Gaskets, J. Funct. Anal. 159(1998), no. 2, 537–567. Google Scholar

[41] [41] Teplyaev, A., Gradients on fractals. J. Funct. Anal. 174(2000), no. 1, 128–154. Google Scholar

[42] [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