Fukushima, Masatoshi; Shima, Tadashi On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals. (English) Zbl 0798.58049 Commun. Math. Phys. 163, No. 3, 461-471 (1994). Summary: We consider a general finitely ramified fractal set called a nested fractal which is determined by \(N\) number of similitudes. Basic properties of the integrated density of states \({\mathcal N}(x)\) for the discrete Laplacian on the associated nested prefractal are investigated. In particular \(d{\mathcal N}\) is shown to be purely discontinuous if \(M < N\), where \(M\) is the number of branches of the inverse of the rational function involved in the spectral decimation method due to Rammal- Toulouse. Sierpinski gaskets and the modified Koch curve are special examples. Cited in 2 Documents MSC: 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37A30 Ergodic theorems, spectral theory, Markov operators Keywords:discontinuity; nested fractal; discrete Laplacian; Sierpinski gaskets; Koch curve PDFBibTeX XMLCite \textit{M. Fukushima} and \textit{T. Shima}, Commun. Math. Phys. 163, No. 3, 461--471 (1994; Zbl 0798.58049) Full Text: DOI References: [1] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat.6, 103–144 (1965) · Zbl 0127.03401 · doi:10.1007/BF02591353 [2] Fukushima, M.: Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In: Albeverio, S., Fenstad, J.E., Holden, H., Lindstrøm, T. (eds.), Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Vol.1. Cambridge: Cambridge Univ. Press, 1992, pp. 151–161 [3] Fukushima, M., Nakao, S., Kotani, S.: Random spectrum. Seminar on Probability, Vol.45, 1977 (in Japanese) [4] Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. Potential Analysis1, 1–35 (1992) · Zbl 1081.31501 · doi:10.1007/BF00249784 [5] Kigami, J., Lapidus, M.L.: Weyl’s problem for the spectral distributions of Laplacians on p.c.f self-similar fractals. Commun. Math. Phys.158, 93–125 (1993) · Zbl 0806.35130 · doi:10.1007/BF02097233 [6] Kusuoka, S.: Diffusion processes on nested fractals. Lecture Notes in Math. vol. 1567, Springer 1993 · Zbl 0807.90009 [7] Lindstrøm, T.: Brownian motion on nested fractals. Memoir AMS420, 1989 · Zbl 0688.60065 [8] Malozemov, L.: The difference Laplacian {\(\Delta\)} on the modified Koch curve. Russ. J. Math. Phys.3, no. 1 (1992) [9] Malozemov, L.: The integrated density of states for the difference Laplacian on the modified Koch graph. Commun. Math. Phys.156, 387–397 (1993) · Zbl 0786.58039 · doi:10.1007/BF02098488 [10] Rammal, R.: Spectrum of harmonic excitations on fractals. J. Physique45, 191–204 (1984) · doi:10.1051/jphys:01984004502019100 [11] Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Physique Lett.43, L13-L22 (1982) [12] Shima, T.: On eigenvalue problems for the random walks on the Sierpinski pre-gaskets. Japan J. Indus. Appl. Math.8, 127–141 (1991) · Zbl 0715.60088 · doi:10.1007/BF03167188 [13] Shima, T.: Lifschitz tails for random Schrödinger operators on nested fractals. Osaka J. Math.29, 749–770 (1992) · Zbl 0774.60060 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.