Elementary fractal geometry. 3. Complex Pisot factors imply finite type

C Bandt - Discrete & Computational Geometry, 2024 - Springer
Self-similar sets require a separation condition to admit a nice mathematical structure. The
classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive …

Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

D Kong, B Sun - Indagationes Mathematicae, 2024 - Elsevier
Abstract Given ρ∈(0, 1/4], the four corner Cantor set E⊂ R 2 is a self-similar set generated
by the iterated function system {(ρ x, ρ y),(ρ x, ρ y+ 1− ρ),(ρ x+ 1− ρ, ρ y),(ρ x+ 1− ρ, ρ y+ 1 …

Elementary fractal geometry. 5. Weak separation is strong separation

C Bandt, MF Barnsley - arXiv preprint arXiv:2404.04892, 2024 - arxiv.org
For self-similar sets, there are two important separation properties: the open set condition
and the weak separation condition introduced by Zerner, which may be replaced by the …

Conditional bi-Lipschitz equivalence of self-similar sets

Q Jia, C Chen, Y Ma, L Lei, K Jiang - Chaos, Solitons & Fractals, 2021 - Elsevier
Let A, B be two non-empty sets in R. If there exists a bi-Lipschitz map ϕ between A and B,
then we say A and B are bi-Lipschitz equivalent. If the function ϕ satisfies some initial extra …

Resonance between self-similar sets and their univoque sets

C Chen, Y Ma, L Lei, M Gareeb, K Jiang - Fractals, 2021 - World Scientific
Let K be a self-similar set in ℝ. Generally, if the iterated function system (IFS) of K has some
overlaps, then some points in K may have multiple codings. If an x∈ K has a unique coding …