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3 Demonstratio Mathematica

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Persistence landscapes of affine fractals

Catanzaro Michael J., Przybylski Lee, Weber Eric S.

Demonstratio Mathematica

2022

Vol. 55, nr 1

163--192

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.

Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]

Byars Allison, Camrud Evan, Harding Steven N., McCarty Sarah, Sullivan Keith, Weber Eric S.

Demonstratio Mathematica

2021

Vol. 54, nr 1

85--109

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

On diffeomorphic actions on cantor sets

Waliszewska A.

Demonstratio Mathematica

1998

Vol. 31, nr 2

417-428