Landscapes of fractals

Classical fractals, like Cantor’s middle-thirds set or the Mandelbrot set, can be expressed as the invariant set of an iterated function system acting on a metric space. We obtain approximations to the persistence landscape of the affine fractal by computing the persistence landscapes of the finite iterations of the system. We prove that there exists an affine transformation acting on the space of persistence landscapes which intertwines the action of the iterated function system. In some sense, this shows that the persistence landscape of a fractal is again a “fractal” due to its self-similarity. This project is joint work with Lee Przybylski and Eric Weber.