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Approximation of energy forms on finitely ramified fractals by discrete graphs and related metric measure spaces

  • Our goal is to approximate energy forms on suitable fractals by discrete graph energies and certain metric measure spaces, using the notion of quasi-unitary equivalence. Quasi-unitary equivalence generalises the two concepts of unitary equivalence and norm resolvent convergence to the case of operators and energy forms defined in varying Hilbert spaces. More precisely, we prove that the canonical sequence of discrete graph energies (associated with the fractal energy form) converges to the energy form (induced by a resistance form) on a finitely ramified fractal in the sense of quasi-unitary equivalence. Moreover, we allow a perturbation by magnetic potentials and we specify the corresponding errors. This aforementioned approach is an approximation of the fractal from within (by an increasing sequence of finitely many points). The natural step that follows this realisation is the question whether one can also approximate fractals from outside, i.e., by a suitable sequence of shrinking supersets. We partly answer this question by restricting ourselves to a very specific structure of the approximating sets, namely so-called graph-like manifolds that respect the structure of the fractals resp. the underlying discrete graphs. Again, we show that the canonical (properly rescaled) energy forms on such a sequence of graph-like manifolds converge to the fractal energy form (in the sense of quasi-unitary equivalence). From the quasi-unitary equivalence of energy forms, we conclude the convergence of the associated linear operators, convergence of the spectra and convergence of functions of the operators – thus essentially the same as in the case of the usual norm resolvent convergence.

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Author:Jan Simmer
Referee:Olaf Post
Advisor:Olaf Post
Document Type:Doctoral Thesis
Date of completion:2021/02/19
Publishing institution:Universität Trier
Granting institution:Universität Trier, Fachbereich 4
Date of final exam:2020/08/12
Release Date:2021/03/17
Tag:Analysis on fractals
GND Keyword:Approximation; Fraktal
Number of pages:101
Institutes:Fachbereich 4
Licence (German):License LogoCC BY-NC-ND: Creative-Commons-Lizenz 4.0 International

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