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On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals. (English) Zbl 0798.58049

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.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A30 Ergodic theorems, spectral theory, Markov operators
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