The complex dimensions of every Sierpiński Carpet modification of dust type

Abstract

We investigate modified Sierpiński Carpet fractals, constructed by dividing a square into a square $n\times n$ grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we produce “dust type” carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the fractal zeta functions, first introduced by Michel Lapidus, Goran Radunović, and Darko Žubrinić in their book (2017), from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpiński Carpet modification of dust type.