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Infinite dimensional i.f.s. and smooth functions on the Sierpinski gasket
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 56, no 3, 1377-1404 p.Article in journal (Refereed) Published
Abstract [en]

We describe the infinitesimal geometric behavior of a large class of intrinsically smooth functions on the Sierpinski gasket in terms of the limit distribution of their local eccentricity, which is essentially the direction of the gradient. The distribution of eccentricicies is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. Continuity properties of the gradient are used to define a class of nearly harmonic functions which are well approximated by harmonic functions. The gradient is also used to identify the part of the Sierpinski gasket where a smooth function is nearly harmonic locally. We prove that for nearly harmonic functions the limit distribution is the same as that for harmonic functions found by Oberg, Strichartz and Yingst. In particular, we prove convergence in the Wasserstein metric. We consider uniform as well as energy weights.

Place, publisher, year, edition, pages
2007. Vol. 56, no 3, 1377-1404 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-95493ISI: 000247849000013OAI: oai:DiVA.org:uu-95493DiVA: diva2:169731
Available from: 2007-03-09 Created: 2007-03-09 Last updated: 2011-01-29Bibliographically approved
In thesis
1. A Study of Smooth Functions and Differential Equations on Fractals
Open this publication in new window or tab >>A Study of Smooth Functions and Differential Equations on Fractals
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers.

Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f.

In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions.

In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2007. 39 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 47
Keyword
Mathematical analysis, Analysis on fractals, p.c.f. fractals, Sierpinski gasket, Laplacian, differential equations on fractals, infinite dimensional i.f.s., invariant measure, harmonic functions, smooth functions, derivatives, products of random matrices, Matematisk analys
Identifiers
urn:nbn:se:uu:diva-7590 (URN)978-91-506-1920-1 (ISBN)
Public defence
2007-03-30, Häggsalen, Ångström Laboratory, Uppsala, 13:15
Opponent
Supervisors
Available from: 2007-03-09 Created: 2007-03-09Bibliographically approved

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