Simple Infinitely Ramified Self-Similar Sets
Self-similar sets form a mathematically tractable class of fractals. Any family of contractive mappings f 1,..., f m generates a unique corresponding fractal set A. It is more difficult to find conditions for the f k which ensure that A has a nice structure. We describe a technique which allows us to determine self-similar sets with a particularly simple structure. Some of the resulting examples are known, like the Sierpiński gasket and carpet, some others seem to be new. A simple structure is necessary when we want to do classical analysis on A; for instance, define harmonic functions and a Laplace operator. So far, much analysis has been realized on a very small class of fractal spaces–essentially the relatives of the Sierpiński gasket. In this chapter, we discuss two classes of infinitely ramified fractals which seem to be more realistic from the point of view of physical modeling, and we give examples for which fractal analysis seems to be possible. A property of the boundaries for these fractal classes is verified.
KeywordsBrownian Motion Harmonic Function Heat Kernel Dirichlet Form Neighbor Type
Unable to display preview. Download preview PDF.
- 1.Bandt, C.: Self-similar measures. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin (2001)Google Scholar
- 6.Barlow, M.T.: Diffusion on fractals. Lecture Notes Math., Vol. 1690, Springer, Berlin (1998)Google Scholar
- 8.Barlow, M.T., Bass, R.F., Kumagai, T., Teplyaev, A.: Uniqueness of Brownian motion on Sierpiński carpets, Preprint, arXiv:0812.1802v1 (2008)Google Scholar
- 12.Grigor’yan, A., Kumagai, T.: On the dichotomy in the heat kernel two-sided estimates. In: P. Exner et al. (eds.) Analysis on Graphs and Applications. Proc. Symposia Pure Math. 77, Amer. Math. Soc., Providence, RI (2008)Google Scholar
- 15.Mesing, M.: Fraktale endlichen Typs. Ph.D. thesis, University of Greifswald, Germany (2007)Google Scholar