Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T14:25:22.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective resistances for harmonic structures on p.c.f. self-similar sets

Published online by Cambridge University Press:  24 October 2008

Jun Kigami
Jun Kigami
Affiliation:
Department of Mathematics, College of General Education, Osaka University, Toyonaka 560, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 291 - 303
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barlow, 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 Math. 283 (Longman, 1993), pp. 131–157.Google Scholar
[2]Barlow, M. T. and Bass, R. F.. On the resistance of the Sierpinski carpet. Proc. Roy. Soc. London A 431 (1990), 354–360.Google Scholar
[3]Barlow, M. T. and Bass, R. F.. Transition densities for Brownian motion on the Sierpinski carpet. Prob. Th. Rel. Fields 91 (1992), 307–330.CrossRefGoogle Scholar
[4]Barlow, M. T. and Perkins, E. A.. Brownian motion on the Sierpinski gasket. Prob. Th. Rel. Fields 79 (1988), 542–624.CrossRefGoogle Scholar
[5]Fukushima, M.. Dirichlet Forms and Markov Processes (North-Holland/Kodansya, 1980).Google Scholar
[6]Fukushima, M.. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In Ideas and Methods in Mathematical Analysis, Stochastics and Application, vol. 1 (eds. Albeverio, , Fenstad, , Holden, and Lindstrøm, ) (Cambridge University Press, 1992).Google Scholar
[7]Fukushima, M. and Shima, T.. On a spectral analysis for the Sierpinski gasket. Potential Analysis 1 (1992), 1–35.CrossRefGoogle Scholar
[8]Goldstein, S.. Random walks and diffusions on fractals. In Percolation theory and ergodic theory of infinite particle systems (ed. Kester, H.) (Springer, 1987).Google Scholar
[9]Hutchinson, J. E.. Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[10]Kigami, J.. A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math. 6 (1989), 259–290.CrossRefGoogle Scholar
[11]Kigami, J.. Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 355 (1993), 721–755.Google Scholar
[12]Kigami, J.. Hausdorff dimensions of self-similar sets and shortest path metrics. (To appear in J. Math. Soc. Japan.)Google Scholar
[13]Kigami, J.. Harmonic calculus on limits of networks and its application to dendrites. (Preprint 1992.)Google Scholar
[14]Kigami, J. and Lapidus, M. L.. Weyl's problem for the spectral distributions of Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys. 158 (1993), 93–125.CrossRefGoogle Scholar
[15]Kumagai, T.. Estimates of the transition densities for Brownian motion on nested fractals. (To appear in Prob. Th. Rel. Fields.)Google Scholar
[16]Kumagai, T.. Regularity, closedness and spectral dimension of the Dirichlet forms on p.c.f. self-similar sets. (To appear in J. Math. Kyoto Univ.)Google Scholar
[17]Kusuoka, S.. A diffusion process on a fractal. In Probabilistic Methods on Mathematical Physics, Proc. of Taniguchi International Symp. (eds. Ito, K. & Ikeda, N.) Kinokuniya 1987, pp. 251–274.Google Scholar
[18]Kusuoka, S.. Dirichlet forms on fractals and products of random matrices. Publ. RIMS. 25 (1989), 659–680.CrossRefGoogle Scholar
[19]Kusuoka, S. and Zhou, X. Y.. Dirichlet forms on fractals: Poincaré constant and resistance. Prob. Th. Rel. Fields 93 (1992), 169–196.CrossRefGoogle Scholar
[20]Lindstrøm, T.. Brownian motion on nested fractals. Mem. Amer. Math. Soc. 420 (1990).Google Scholar
[21]Moran, P. A. P.. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Phil. Soc. 42 (1946), 15–23.CrossRefGoogle Scholar
[22]Shima, T.. On eigenvalue problems for the random walks on the Sierpinski pre-gaskets. Japan J. Indust. Appl. Math. 8 (1991), 124–141.CrossRefGoogle Scholar