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Renormalisation of finitely ramified fractals

Published online by Cambridge University Press:  14 November 2011

Volker Metz
Volker Metz
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany, e-mail: [email protected]
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Transition probabilities are calculated which make the construction of diffusions on finitely ramified fractals straightforward. In contrast to former approaches using Brouwer's Fixed Point Theorem, we consider an approximation procedure based on the iteration of a nonlinear map L. Physically, this is done by ‘coarse-graining-renormalisation of finite electric resistor networks’. Mathematically, it is a convergence problem for quotients of Dirichlet forms on finite graphs. These graphs approximate finitely ramified fractals. The basic tool is a contraction theorem for the renormalisation map L which allows the use of known results about nested fractals for non-nested (p.c.f. self-similar) ones. In general, the above contraction is not strict because several linear independent fixed points occur.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 1085 - 1104
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Anderson, W. N. Jr and Trapp, G. E.. Shorted operators. II. SIAM J. Appl. Math. 28 (1975), 60–71.CrossRefGoogle Scholar
2Barlow, M. T.. Harmonic analysis on fractal spaces. Seminaire BOURBAKI, 44 eme annee, 1991–1992, n° 755, 345–68.Google Scholar
3Barlow, M. T.. Random walks, electrical resistance, and nested fractals. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, eds Elworthy, K. D. and Ikeda, N., Pitman Research Notes in Mathematics 283, 131–57 (Harlow: Longman Scientific, 1993).Google Scholar
4Barlow, M. T. and Perkins, E. A.. Brownian motion on the Sierinski gasket. Probab. Theory Related Fields 79 (1988), 543–628.CrossRefGoogle Scholar
5Bharucha-Reid, A. T.. Elements of the Theory of Markov Processes and Their Applications (New York: McGraw-Hill, 1960).Google Scholar
6Bollobás, B.. Graph Theory. An Introductory Course (Berlin: Springer, 1979).Google Scholar
7Bouleau, N. and Hirsch, F.. Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics 14 (New York: De Gruyter, 1991).CrossRefGoogle Scholar
8Doyle, P. G. and Snell, J. L.. Random Walks and Electric Networks, Carus Mathematical Monographs 22 (Washington: Mathematical Association of America, 1984).CrossRefGoogle Scholar
9Duffin, R. J.. Distributed and lumped networks. J. Math. Meek 8 (1959), 793–826.Google Scholar
10Fitzsimmons, P. J., Hambly, B. M. and Kumagai, T.. Transition density estimates for Brownian motion on affine nested fractals. Comm. Math. Phys. 165 (1994), 595–620.CrossRefGoogle Scholar
11Fukushima, M.. Dirichlet Forms and Markov Processes (Amsterdam: North-Holland, 1980).Google Scholar
12Fukushima, M. and Shima, T.. On a spectral analysis for the Sierpinski gasket. Potential Anal. 1 (1992), 1–35.CrossRefGoogle Scholar
13Gantmacher, F. R.. The Theory of Matrices. Vol. I (New York: Chelsea, 1960).Google Scholar
14Hattori, K., Hattori, T. and Watanabe, H.. Gaussian field theories on general networks and the spectral dimension. Progr. Theoret. Phys. Suppl. 92 (1987), 108–43.CrossRefGoogle Scholar
15Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–47.CrossRefGoogle Scholar
16Kigami, J.. Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335 (1993), 721–55.Google Scholar
17Krebs, W. B.. A diffusion defined on a fractal state space. Stochastic Process. Appl. 37 (1991), 199–212.CrossRefGoogle Scholar
18Kumagai, T.. Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, eds Elworthy, K. D. and Ikeda, N., Pitman Research Notes in Mathematics 283, 219–47 (Harlow: Longman, 1993).Google Scholar
19Kusuoka, S.. A diffusion process on a fractal. In Probabilistic Methods in Mathematical Physics, eds Itô, K. and Ikeda, N., Proceedings Taniguchi Symposium, Katata 1985, 251–74 (Amsterdam: Kino Kuniya-North Holland, 1987).Google Scholar
20Kusuoka, S.. Diffusion processes on nested fractals. In Statistical Mechanics and Fractals, eds Dobrushin, R. L. and Kusuoka, S., Lecture Notes in Mathematics 1567, 39–98 (Berlin: Springer, 1993).CrossRefGoogle Scholar
21Lindstram, T.. Brownian Motion on Nested Fractals, Memoirs of the American Mathematical Society 83, No. 420 (Providence, R.I.: American Mathematical Society, 1990).Google Scholar
22Markham, T. L.. Nonnegative matrices whose inverses are M-matrices. Proc. Amer. Math. Soc. 36 (1972), 326–30.Google Scholar
23Meadows, R. G.. Electric Network Analysis (Harmondsworth: Penguin Books, 1972).Google Scholar
24Metz, V.. Potentialtheorie auf dem Sierpinski gasket. Math. Ann. 289 (1991), 207–37.CrossRefGoogle Scholar
25Okada, M., Sekiguchi, T. and Shiota, Y.. Heat kernels on infinite graph networks and deformed Sierpinski gaskets. Japan J. Appl. Math. 7 (1990), 527–43.CrossRefGoogle Scholar
26Oshima, Y.. Lectures on Dirichlet forms (Lecture notes at Erlangen University, 1988).Google Scholar