On the strong separation condition for self-similar iterated function systems with random translations Article Swipe

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Simon Baker , Derong Kong , Zhiqiang Wang ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2401.14175

Given a self-similar iterated function system $Φ=\{ ϕ_i(x)=ρ_i O_i x+t_i \}_{i=1}^m$ acting on $\mathbb{R}^d$, we can generate a parameterised family of iterated function systems by replacing each $t_i$ with a random vector in $\mathbb{R}^d$. In this paper we study whether a Lebesgue typical member of this family will satisfy the strong separation condition. Our main results show that if the similarity dimension of $Φ$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.

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Mathematics

Multivariate Random Variable

Combinatorics

Discrete Mathematics

Computer Science

Mathematical Analysis

Artificial Intelligence

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Iterated function Iterated function system Function (biology) Dimension (graph theory) Lebesgue integration Similarity (geometry) Mathematics Lebesgue measure Separation (statistics) Multivariate random variable Combinatorics Measurable function Pure mathematics Discrete mathematics Computer science Random variable Mathematical analysis Artificial intelligence Statistics Attractor Image (mathematics) Evolutionary biology Biology Bounded function

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2401.14175
PDF: https://arxiv.org/pdf/2401.14175
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Cited By: 1
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DOI: https://doi.org/10.48550/arxiv.2401.14175

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Title: On the strong separation condition for self-similar iterated function systems with random translations

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Language: en

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Publication year: 2024

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Publication date: 2024-01-25

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Authors: Simon Baker, Derong Kong, Zhiqiang Wang

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Concepts: Iterated function, Iterated function system, Function (biology), Dimension (graph theory), Lebesgue integration, Similarity (geometry), Mathematics, Lebesgue measure, Separation (statistics), Multivariate random variable, Combinatorics, Measurable function, Pure mathematics, Discrete mathematics, Computer science, Random variable, Mathematical analysis, Artificial intelligence, Statistics, Attractor, Image (mathematics), Evolutionary biology, Biology, Bounded function

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Cited by: 1

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