article · arXiv (Cornell University) · 2022

Transition Phenomena for the Attractor of an Iterated Function System

Krzysztof Leśniak, Nina Snigireva, Filip Strobin, Andrew Vince

Cited by 5Open access

Abstract

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and properties of two types of transition attractors, called the lower transition attractor $A_{\bullet}$ and the upper transition attractor $A^{\bullet}$, are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold $t_0$ such that the IFS $F_t$ has a unique attractor $A_t$ for $t<t_0$ and no attractor for $t>t_0$. At the threshold $t_0$, there is an $F_{t_0}$-invariant set $A^{\bullet}$ such that $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.

Cite this paper

APA 7

Leśniak, K., Snigireva, N., Strobin, F., & Vince, A. (2022). Transition Phenomena for the Attractor of an Iterated Function System. arXiv (Cornell University). https://doi.org/10.1088/1361-6544/ac8af1

MLA 9

Leśniak, Krzysztof, et al. “Transition Phenomena for the Attractor of an Iterated Function System.” arXiv (Cornell University), 2022, https://doi.org/10.1088/1361-6544/ac8af1.

Chicago

Leśniak, Krzysztof, Nina Snigireva, Filip Strobin, and Andrew Vince. 2022. “Transition Phenomena for the Attractor of an Iterated Function System.” arXiv (Cornell University). https://doi.org/10.1088/1361-6544/ac8af1.

Harvard

Leśniak, K. et al. (2022) ‘Transition Phenomena for the Attractor of an Iterated Function System’, arXiv (Cornell University). Available at: https://doi.org/10.1088/1361-6544/ac8af1.

BibTeX

@article{lesniak2022transition,
  title = {{Transition Phenomena for the Attractor of an Iterated Function System}},
  author = {Leśniak, Krzysztof and Snigireva, Nina and Strobin, Filip and Vince, Andrew},
  journal = {arXiv (Cornell University)},
  year = {2022},
  doi = {10.1088/1361-6544/ac8af1}
}

Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.