Rough analysis, as undertaken and popularised by Robert Strichartz and Jun Kigami,
deals with the construction of a Laplacian and the study of associated problems
on certain fractal sets, embedded in some Euclidean space, thus naturally
exploiting the Euclidean topology. In this talk, we generalise this study to
abstract, totally disconnected, metric measure unilateral shift spaces. In
particular, we discuss the construction of a Laplacian as a renormalised limit
of difference operators defined on finite sets that approximate the entire space.
We further propose a weak definition of this Laplacian, analogous to the one in
calculus, by choosing test functions as those which have finite energy and vanish
on various (appropriately defined) boundary sets. We then define the Neumann
derivative of functions on these boundary sets and establish a relationship between
the three important concepts in our analysis so far, namely, the Laplacian, the
bilinear energy form and the Neumann derivative of a function.
This is a joint work with my doctoral student, Sharvari Neetin Tikekar.