Group Members
Samantha Forshay, Leng Mawi, Matthew Peeks.
Supervisors
Chris Hayes, Luke Rogers, Sasha Teplyaev.
Overview
A celebrated result in analysis and probability on fractals is the construction of a diffusion on the standard Sierpinski Carpet by Barlow and Bass. One key part of their argument is a pair of upper and lower estimates for the resistances of precarpets: if \(K_n\) denotes the level \(n\) approximation of the carpet and \(E_n\) is the minimal Dirichlet energy of a function that is identically 1 on one side of the carpet and identically 0 on the other side, then there are constants \(0<c\leq C< \infty\) so that \(c\rho^n\leq E_n\leq C\rho^n\). Estimates for \(\rho\) are known but the exact value is not.
The Sierpinski-type carpets to which the preceding estimates of resistance have been extended are all self-similar. By contrast, in the setting of post-critically finite fractals, resistance scaling has been successfully studied also in the self-affine case, initially by Fitzsimmons, Hambly and Kumagai.
The goal of this project was to investigate what aspects of the Barlow-Bass approach to resistance estimation on carpets could be extended to the self-affine case, and to make numerical computations of the behavior of resistance in this setting and its dependence on the affine scalings.