Renormalization of the Hutchinson Operator
One of the easiest and common ways of generating fractal sets in ℝᴰ is as attractors of affine iterated function systems (IFS). The classic theory of IFS requires that they are made with contractive functions. In this paper, we relax this hypothesis, considering a new operator Hρ obtained by renorma...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209765 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Renormalization of the Hutchinson Operator / Y. Demichel // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859707034550665216 |
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| author | Demichel, Y. |
| author_facet | Demichel, Y. |
| citation_txt | Renormalization of the Hutchinson Operator / Y. Demichel // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | One of the easiest and common ways of generating fractal sets in ℝᴰ is as attractors of affine iterated function systems (IFS). The classic theory of IFS requires that they are made with contractive functions. In this paper, we relax this hypothesis, considering a new operator Hρ obtained by renormalizing the usual Hutchinson operator H. Namely, the Hρ-orbit of a given compact set K₀ is built from the original sequence (Hⁿ(K₀))ₙ by rescaling each set by its distance from 0. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for H. Our strategy to tackle the problem is to link these new sequences to some classic ones, but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations.
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| first_indexed | 2025-12-07T14:13:52Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209765 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:13:52Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Demichel, Y. 2025-11-26T11:24:55Z 2018 Renormalization of the Hutchinson Operator / Y. Demichel // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 28A80; 37C70; 47H10; 37C25; 37E05; 15A99 arXiv: 1803.06537 https://nasplib.isofts.kiev.ua/handle/123456789/209765 https://doi.org/10.3842/SIGMA.2018.085 One of the easiest and common ways of generating fractal sets in ℝᴰ is as attractors of affine iterated function systems (IFS). The classic theory of IFS requires that they are made with contractive functions. In this paper, we relax this hypothesis, considering a new operator Hρ obtained by renormalizing the usual Hutchinson operator H. Namely, the Hρ-orbit of a given compact set K₀ is built from the original sequence (Hⁿ(K₀))ₙ by rescaling each set by its distance from 0. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for H. Our strategy to tackle the problem is to link these new sequences to some classic ones, but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations. The present paper was completed during the thematic research semester Fractal Geometry and Dynamics, organized in the fall of 2017 at the Institut Mittag-Leffler, Stockholm, Sweden. The author is very grateful to the organizers for their warm welcome during their stay at the Institute. This work is partially supported by the French research group ‘Analyse Multifractale’ (CNRS-GDR3475). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Renormalization of the Hutchinson Operator Article published earlier |
| spellingShingle | Renormalization of the Hutchinson Operator Demichel, Y. |
| title | Renormalization of the Hutchinson Operator |
| title_full | Renormalization of the Hutchinson Operator |
| title_fullStr | Renormalization of the Hutchinson Operator |
| title_full_unstemmed | Renormalization of the Hutchinson Operator |
| title_short | Renormalization of the Hutchinson Operator |
| title_sort | renormalization of the hutchinson operator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209765 |
| work_keys_str_mv | AT demichely renormalizationofthehutchinsonoperator |