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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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