Greens Functions On PCF Self-Similar Sets
- Author(s): Kloster, Frank Alexander
- Advisor(s): Lapidus, Michel L
- et al.
In the area of fractal analysis, many details about the analytic structure of certain post-critically-finite (p.c.f.) self-similar structures such as the Sierpinski gasket. These include details about its Laplacian, Green's function, and solutions to differential equations. While general techniques have been proposed, many examples have yet been worked out, such as the Hata tree-like structure. Here, we work out and discuss said analytic structure for these examples. While the technical details are significantly more advanced, several fascinating patterns tend to originate, some of which are of a completely different nature than the analytic structure of the Sierpinski gasket. Many of these examples can be further followed up, such as working through solutions to specific differential equations.