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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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.

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

First available in Project Euclid: 23 July 2019

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Zentralblatt MATH identifier

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

Kusuoka measure energy Laplacian Sierpiński gasket Laplacian pointwise formula


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.

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  • J. Azzam, M.A. Hall and R.S. Strichartz, Conformal energy, conformal Laplacian, and energy measures on the Sierpiński gasket, Trans. Amer. Math. Soc. 360 (2008), 2089–2131.
  • R. Bell, C.-W. Ho and R.S. Strichartz, Energy measures of harmonic functions on the Sierpiński gasket, Indiana University Mathematics Journal, 63 (2014), 831-868.
  • O. Ben-Bassat, R.S. Strichartz, Alexander Teplyaev, What Is Not in the Domain of the Laplacian on Sierpiński gasket Type Fractals, J. Funct. Anal. 166 (1999), 197–217.
  • B. Boyle, K. Cekala, D. Ferrone, N. Rifkin and A. Teplyaev, Electrical resistance of n-gasket fractal networks, Pacific J. Math. 233 (1) (2007), 15–40.
  • U. Freiberg and C. Thäle, Exact computation and approximation of stochastic and analytic parameters of generalized Sierpinski gaskets., Methodology and Computing in Applied Probability, (2013), 1-25.
  • B. M. Hambly and T. Kumagai, Asymptotics for the spectral and walk dimension as fractals approach Euclidean space, Fractals 10.04 (2002): 403-412.
  • M. Hino, Some properties of energy measures on Sierpinski gasket type fractals, J. Fractal Geom., vol. 3 (2016), 245-263.
  • M. Ionescu, E.P.J. Pearse, L.G. Rogers, H-Y. Ruan, R.S. Strichartz, The Resolvent Kernel for PCF Self-Similar Fractals, Trans. Amer. Math. Soc. 362(8) (2010), 4451–4479.
  • A. Johansson, A. Öberg and M. Pollicott, Ergodic Theory of Kusuoka Measures, J. Fract. Geom. 4 (2017), 185–214.
  • N. Kajino, Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, Potential Anal. 36, no. 1, (2012) 67-115.
  • N. Kajino Analysis and geometry of the measurable Riemannian structure on the Sierpi ski gasket, Fractal geometry and dynamical systems in pure and applied mathematics. I. Fractals in pure mathematics, 91-133, Contemp. Math., 600, Amer. Math. Soc., Providence, RI, 2013.
  • J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics (2001).
  • J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.
  • J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 209 (2003) 399-444.
  • J. Kigami, Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate, Math. Ann. 340 (2008), 781-804.
  • S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci, 25 (1989), 659-680.
  • L. Lovasz, Random walks on graphs: A survey, Combinatorics, Paul Erdos is Eighty, 2 (1993).
  • E.D. Mbakop, Analysis on Fractals, Worcester Polytechnique Institute (2009).
  • J. Needleman, R.S. Strichartz, A. Teplyaev, P.-L. Yung, Calculus on the Sierpiński gasket. I. Polynomials, exponentials and power series, J. Funct. Anal. 215 (2004), no. 2, 290–340.
  • L.G. Rogers, R.S. Strichartz and A. Teplyaev Smooth bumps, a Borel theorem and partitions of smooth functions on P.C.F. fractals, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1765-1790.
  • R.S. Strichartz, Differential Equations on Fractals, Princeton University Press (2006).
  • R.S. Strichartz and S.T. Tse, Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket, Analysis 30 (2010), 285–299.
  • R.S. Strichartz and Michael Usher, Splines on Fractals, Math. Proc. of the Cambridge Philos. Soc. 129 (2000), 331–360.
  • R.S. Strichartz, Solvability for differential equations on Fractals, Journal d'Analyse Mathmatique, 96 (2005), 247–267.
  • R.S. Strichartz, Some Properties of Laplacians on Fractals, J. Func. Anal. 164(2) (1999), 191–208.
  • A. Teplyaev, Energy and Laplacian on the Sierpiński gasket, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1. Proceedings of Symposia in Pure Mathematics 72, Amer. Math. Soc., (2004), 131–154.
  • A. Teplyaev Harmonic coordinates on fractals with finitely ramified cell structure Canad. J. Math. 60 (2008), no. 2, 457-480.
  • K. Tsougkas, Non-degeneracy of the harmonic structure on Sierpinski gaskets, to appear in the Journal of Fractal Geometry (2017).