The chaos game revisited: Yet another, but a trivial proof of the algorithm’s correctness

Martyn, Tomasz

doi:10.1016/j.aml.2011.08.020

Cited by 5 publications

(2 citation statements)

References 4 publications

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“…Then, (with a probability one) for appropriate big N < M, the set {x n : N ≤ n ≤ M } should approximate the attractor A F . The simple explanation (see [18]) is that with the probability one, we choose a sequence i = (i 1 , i 2 , . .…”

Section: Chaos Game and Disjunctive Sequencesmentioning

confidence: 99%

Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Strobin

2021

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Motivated by a recent paper of Leśniak and Snigireva [Iterated function systems enriched with symmetry, preprint], we investigate the properties of the semiattractor $$A_{\mathcal {F}\cup \mathcal {G}}^*$$ A F ∪ G ∗ of an IFS $$\mathcal {F}$$ F enriched by some other IFS $$\mathcal {G}$$ G . We show that in natural cases, the semiattractor $$A_{\mathcal {F}\cup \mathcal {G}}^*$$ A F ∪ G ∗ is in fact the attractor of certain IFSs related naturally with the IFSs $$\mathcal {F}$$ F and $$\mathcal {G}$$ G . We also give an example when $$A_{\mathcal {F}\cup \mathcal {G}}^*$$ A F ∪ G ∗ is not compact, yet still being the attractor of considered related IFSs. Finally, we use presented machinery to prove that the so called lower transition attractors due to Vince are semiattractors of enriched IFSs.

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Section: Chaos Game and Disjunctive Sequencesmentioning

confidence: 99%

Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Strobin

2021

Results Math

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“…Given that biological sequences are products of evolutionary processes, ergodic theory aids in modeling the evolutionary dynamics of sequences. This includes understanding how sequences evolve over time, the emergence of specific motifs, and the impact of mutations on sequence patterns [28, 29].…”

Section: Introductionmentioning

confidence: 99%

Complexity And Ergodicity In Chaos Game Representation Of Genomic Sequences

Mehrpooya,

Tavassoly

2023

Preprint

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The Chaos Game Representation (CGR) serves as a powerful graphical tool for transforming long one-dimensional sequences, such as genomic sequences, into visually insightful and complex graphical patterns. This algorithmic approach has revealed captivating fractal properties, providing a systems-level perspective on the underlying patterns inherent in specific genomic sequences. Each CGR image generated is distinctive and unique, capturing the individuality of different genomic sequences. We have investigated the nature of the fractal patterns discerned within CGR of genomic sequences, aiming to assess the ergodicity and identify potential ergodic patterns. Our findings present a mathematical proof establishing that CGRs are non-ergodic. This finding suggests that the distribution properties of nucleotide sequences play a pivotal role in shaping the emergent patterns within the CGR images.

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