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 in (Catanzaro et al., 2022).
References
2022
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Persistence landscapes of affine fractals
Michael J. Catanzaro, Lee Przybylski , and Eric S. Weber
Demonstratio Mathematica, May 2022
We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.
