Rocky Mountain Journal of Mathematics

The Kusuoka measure and the energy Laplacian on level-$k$ Sierpiński gaskets

Anders Öberg and Konstantinos Tsougkas

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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.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 3 (2019), 945-961.

Dates
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1563847242

Digital Object Identifier
doi:10.1216/RMJ-2019-49-3-945

Mathematical Reviews number (MathSciNet)
MR3983309

Zentralblatt MATH identifier
07088345

Subjects
Primary: 28A80: Fractals [See also 37Fxx]

Keywords
Kusuoka measure energy Laplacian Sierpiński gasket Laplacian pointwise formula

Citation

Öberg, Anders; Tsougkas, Konstantinos. The Kusuoka measure and the energy Laplacian on level-$k$ Sierpiński gaskets. Rocky Mountain J. Math. 49 (2019), no. 3, 945--961. doi:10.1216/RMJ-2019-49-3-945. https://projecteuclid.org/euclid.rmjm/1563847242


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