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2019 The Kusuoka measure and the energy Laplacian on level-$k$ Sierpiński gaskets
Anders Öberg, Konstantinos Tsougkas
Rocky Mountain J. Math. 49(3): 945-961 (2019). DOI: 10.1216/RMJ-2019-49-3-945

Abstract

We extend and survey results in the theory of analysis on fractal sets from the standard Laplacian on the Sierpinski gasket to the energy Laplacian, which is defined weakly by using the Kusuoka energy measure. We also extend results from the Sierpinski gasket to level-$k$ Sierpinski gaskets, for all $k\geq 2$. We observe that the pointwise formula for the energy Laplacian is valid for all level-$k$ Sierpinski gaskets, $SG_k$, and we provide a proof of a known formula for the renormalization constants of the Dirichlet form for post-critically finite self-similar sets along with a probabilistic interpretation of the Laplacian pointwise formula. We also provide a vector self-similar formula and a variable weight self-similar formula for the Kusuoka measure on $SG_k$, as well as a formula for the scaling of the energy Laplacian.

Citation

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Anders Öberg. Konstantinos Tsougkas. "The Kusuoka measure and the energy Laplacian on level-$k$ Sierpiński gaskets." Rocky Mountain J. Math. 49 (3) 945 - 961, 2019. https://doi.org/10.1216/RMJ-2019-49-3-945

Information

Published: 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07088345
MathSciNet: MR3983309
Digital Object Identifier: 10.1216/RMJ-2019-49-3-945

Subjects:
Primary: 28A80

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 3 • 2019
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