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Combinatorial and analytical problems for fractals and their graph approximations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs.

The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.

In paper I, we study the number of spanning trees on the sequence of graphs approximating a self-similar set admitting spectral decimation.

In paper II, we study harmonic functions on p.c.f. self-similar sets. Unlike the standard Dirichlet problem and harmonic functions in Euclidean space, harmonic functions on these sets may be locally constant without being constant in their entire domain. In that case we say that the fractal has a degenerate harmonic structure. We prove that for a family of variants of the Sierpinski gasket the harmonic structure is non-degenerate.

In paper III, we investigate properties of the Kusuoka measure and the corresponding energy Laplacian on the Sierpinski gaskets of level k.

In papers IV and V, we establish a connection between the discrete combinatorial graph Laplacian determinant and the regularized determinant of the fractal itself. We establish that for a certain class of p.c.f. fractals the logarithm of the regularized determinant appears as a constant in the logarithm of the discrete combinatorial Laplacian.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2019. , p. 37
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 112
Keywords [en]
Fractal graphs, energy Laplacian, Kusuoka measure
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-369918ISBN: 978-91-506-2739-8 (print)OAI: oai:DiVA.org:uu-369918DiVA, id: diva2:1271631
Public defence
2019-02-15, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2019-01-23 Created: 2018-12-17 Last updated: 2019-01-23
List of papers
1. Connections between discrete and regularized determinants on fractals
Open this publication in new window or tab >>Connections between discrete and regularized determinants on fractals
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-369455 (URN)
Available from: 2018-12-17 Created: 2018-12-17 Last updated: 2018-12-17
2. Non-degeneracy of the harmonic structure on Sierpiński gaskets
Open this publication in new window or tab >>Non-degeneracy of the harmonic structure on Sierpiński gaskets
2019 (English)In: Journal of Fractal Geometry, ISSN 2308-1309, Vol. 6, no 2, p. 143-156Article in journal (Refereed) Published
Abstract [en]

We prove that the harmonic extension matrices for the two dimensional level-k Sierpiński gasket are invertible for every k ≥ 2. This has been previously conjectured to be true by Hino in [10] and [11] and tested numerically for k ≤ 50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.

Keywords
Harmonic structure, harmonic extension matrices, energy Laplacian, Sierpinski gasket, prefractal graphs
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-369738 (URN)10.4171/JFG/73 (DOI)000467078300003 ()
Available from: 2018-12-17 Created: 2018-12-17 Last updated: 2019-05-28Bibliographically approved
3. Regularized Laplacian determinants of self-similar fractals
Open this publication in new window or tab >>Regularized Laplacian determinants of self-similar fractals
2018 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 108, no 6, p. 1563-1579Article in journal (Refereed) Published
Abstract [en]

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.

Place, publisher, year, edition, pages
SPRINGER, 2018
Keywords
Regularized determinant, Fractal Laplacian, Spectral zeta functions, Sierpinski gasket
National Category
Other Physics Topics
Identifiers
urn:nbn:se:uu:diva-356383 (URN)10.1007/s11005-017-1027-y (DOI)000431317300009 ()
Note

Correction in: Lett Math Phys (2018) 108:1581. https://doi.org/10.1007/s11005-018-1081-0

Available from: 2018-08-15 Created: 2018-08-15 Last updated: 2018-12-17Bibliographically approved
4. Counting spanning trees on fractal graphs and their asymptotic complexity
Open this publication in new window or tab >>Counting spanning trees on fractal graphs and their asymptotic complexity
2016 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 49, no 35, article id 355101Article in journal (Refereed) Published
Abstract [en]

Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem 3.4. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski gasket, a non-post critically finite analog of the Sierpinski gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.

Keywords
fractal graphs, spanning trees, spectral decimation, asymptotic complexity
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-304155 (URN)10.1088/1751-8113/49/35/355101 (DOI)000381302500005 ()
Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2018-12-17Bibliographically approved
5. The Kusuoka measure and the energy Laplacian on level-k Sierpinski gaskets
Open this publication in new window or tab >>The Kusuoka measure and the energy Laplacian on level-k Sierpinski gaskets
(English)In: Article in journal (Refereed) Accepted
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-369925 (URN)
Available from: 2018-12-17 Created: 2018-12-17 Last updated: 2018-12-17

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