Abstract
By introducing twists into the iterated function system that defines the Sierpinski gasket, we are able to construct a unique regular energy form that satisfies a self–similar identity with any prescribed projective weights. Our construction is explicit (involving finding a root of a 4th order polynomial), and we are able to find explicitly a polynomial identity for the algebraic variety containing the smooth manifold of admissible weights. Without the twists, there are obstructions to existence, and a complete description due to Sabot is quite complicated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blasiak, A., Strichartz, R., Ugurcan, B.: Spectra of self–similar Laplacians on the Sierpinski gasket with twists. (in preparation)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Hambly, B.M., Metz, V., Teplyaev, A.: Self–similar energies on p.c.f. self–similar fracctals. Proc. London Math. Soc. (in press)
Kigami, J.: A harmonic calculus on the Sierpinski spaces. Jpn. J. Appl. Math. 8, 259–290 (1989)
Kigami, J.: Harmonic calculus on p.c.f. self–similar sets. Trans. Amer. Math. Soc. 335, 721–755 (1993)
Kigami, J.: Analysius on Fractals. Cambridge University Press, New York (2001)
Kigami, J.: Harmonic analysis for resistance forms. J. Funct. Anal. 204, 399–444 (2003)
Lindstrφm, T.: Brownian motion on nested fractals. Mem. Amer. Math. Soc. 20 (1990)
Metz, V.: How many diffusions exist on the Vicsik snowflake. Acta Appl. Math. 32, 227–241 (1993)
Metz, V.: Hilbert projective metric on cones of Dirichlet forms. J. Funct. Anal. 127, 438–455 (1995)
Metz, V.: Renormalization contracts on nested fractals. J. Reine Angew. Math. 459, 161–175 (1996)
Metz, V.: Shorted operators: an application in potential theory. Linear Algebra Appl. 264, 439–455 (1997)
Metz, V.: The cone of diffusions on finitely ramified fractals. Nonlinear Anal. 55, 723–738 (2003)
Metz, V.: The Laplacian on the Hany fractal. Arab. J. Sci. Eng. Sect. C Theme Issues 28 (IC), 199–211 (2003)
Metz, V.: The short–cut test. J. Funct. Anal. 222, 118–156 (2005)
Peirone, R.: Convergence and uniqueness problems for Dirichlet forms on fractals. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)3–B, 431–460 (2000)
Peirone, R.: Existence of eigenforms on fractals with three vertices. (in press)
Sabot, C.: Existence and uniqueness of diffusions on finitely ramified self–similar fractals. Ann. Sci. École Norm. Super. 30, 605–673 (1997)
Strichartz, R.: Analysis on fractals. Notices Amer. Math. Soc. 46, 1199–1208 (1999)
Strichartz, R.: Function spaces on fractals. J. Funct. Anal. 198, 43–83 (2003)
Strichartz, R.: Differential Equations on Fractals: A Tutorial. Princeton University Press (2006)
Strichartz, R.: Periodic and almost periodic function on infinite Sierpinski gaskets. (preprint)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of R. S. Strichartz supported by the National Science Foundation, grant DMS–0140194.
Rights and permissions
About this article
Cite this article
Cucuringu, M., Strichartz, R.S. Self–similar Energy Forms on the Sierpinski Gasket with Twists. Potential Anal 27, 45–60 (2007). https://doi.org/10.1007/s11118-007-9047-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-007-9047-3