Level sets of harmonic functions on the Sierpiński gasket
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpiński gasket. We introduce a parameter, calledeccentricity, which classifies these functions up to affine transformationsh→ah+b. We describe three (presumably) distinct measures that describe how the eccentricities are distributed in the limit as we subdivide the gasket into smaller copies (cells) and restrict the harmonic function to the small cells. One measure simply counts the number of small cells with eccentricity in a specified range. One counts the contribution to the total energy coming from those cells. And one counts just those cells that intersect a fixed generic level set. The last measure yields a formula for the box dimension of a generic level set. All three measures are defined by invariance equations with respect to the same iterated function system, but with different weights. We also give a construction for a rectifiable curve containing a given level set. We exhibit examples where the curve has infinite winding number with respect to some points.
- [Ba]Barlow, M. T.,Diffusion on Fractals, Lecture Notes in Math.,1690, Springer-Verlag, Berlin-Heidelberg, 1998.
- [BDEG]Barnsley, M. F., Demoko, S. G., Elton, J. H. andGeronimo, J. S., Invariant measures for Markov processes arising from place-dependent probabilities,Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 367–394.
- [BST]Ben-Bassat, O., Strichartz, R. andTeplyaev, A., What is not in the domain of the Laplacian on a Sierpiński gasket type fractal,J. Funct. Anal. 166 (1999), 197–217. CrossRef
- [Bo]Bougerol P., Limit theorems for products of random matrices, inProducts of Random Matrices with Applications to Schrödinger Operators (Bougerol, P. and Lacroix, J., authors), pp. 1–180, Birkhäuser, Basel-Boston, 1985.
- [DSV]Dalrymple, K., Strichartz, R. S. andVinson, J. P., Fractal differential equations on the Sierpiński gasket,J. Fourier Anal. Appl. 5 (1999), 203–284.
- [H]Hennion, H., Sur un théorème spectral et son application aux noyaux lipschitziens,Proc. Amer. Math. Soc. 118 (1993), 627–634.
- [KL]Keller, G. andLiverani, C., Stability of the spectrum for transfer operators,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 141–152.
- [K1]Kigami, J., A harmonic calculus on the Sierpiński spaces,Japan J. Appl. Math. 8 (1989), 259–290.
- [K2]Kigami, J., Harmonic metric and Dirichlet form on the Sierpiński gasket, inAsymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Sanda/Kyoto, 1990) (Elworthy, K. D. and Ikeda, N., eds.), pp. 201–218, Pitman Research Notes in Math.283, Longman, Harlow, 1993.
- [K3]Kigami, J., Effective resistances for harmonic structures on p.c.f. self-similar sets,Math. Proc. Cambridge Philos. Soc. 115 (1994), 291–303.
- [K4]Kigami, J.,Analysis on Fractals, Cambridge Univ. Press, Cambridge, 2001.
- [Ku]Kusuoka, S., Dirichlet forms on fractals and products of random matrices,Publ. Res. Inst Math. Sci. 25 (1989), 659–680.
- [Ö]Öberg, A., Approximation of invariant measures for iterated function systems,Ph. D. thesis, Umeå University, 1998.
- [R]Rachev, S. T.,Probability Metrics and the Stability of Stochastic Models, Wiley, Chichester, 1991.
- [S1]Strichartz, R., Analysis on fractals,Notices Amer. Math. Soc. 46 (1999), 1199–1208.
- [S2]Strichartz, R., Taylor approximations on Sierpiński gasket type fractals,J. Funct. Anal. 174 (2000), 76–127. CrossRef
- [T]Teplyaev, A., Gradients on fractals,J. Funct. Anal. 174 (2000), 128–154. CrossRef
- Level sets of harmonic functions on the Sierpiński gasket
Arkiv för Matematik
Volume 40, Issue 2 , pp 335-362
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Author Affiliations
- 1. Department of Mathematics, Uppsala University, P. O. Box 480, SE-751 06, Uppsala, Sweden
- 2. Mathematics Department Malott Hall, Cornell University, 14853, Ithaca, NY, USA
- 3. Mathematics Department, University of North Texas, 76203, Denton, TX, USA