Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon

We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we...

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Autores principales:	Badger Matthew, Vellis Vyron
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	hölder curves parameterization iterated function systems self-affine sets primary 28a80 secondary 26a16, 28a75, 53a04 Analysis QA299.6-433
Acceso en línea:	https://doaj.org/article/0c87905a11d6469086eb3426c0600bc0
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spelling	oai:doaj.org-article:0c87905a11d6469086eb3426c0600bc02021-12-05T14:10:38ZHölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon2299-327410.1515/agms-2020-0125https://doaj.org/article/0c87905a11d6469086eb3426c0600bc02021-07-01T00:00:00Zhttps://doi.org/10.1515/agms-2020-0125https://doaj.org/toc/2299-3274We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝn is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝn may be strictly greater than n.Badger MatthewVellis VyronDe Gruyterarticlehölder curvesparameterizationiterated function systemsself-affine setsprimary 28a80secondary 26a16, 28a75, 53a04AnalysisQA299.6-433ENAnalysis and Geometry in Metric Spaces, Vol 9, Iss 1, Pp 90-119 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	hölder curves parameterization iterated function systems self-affine sets primary 28a80 secondary 26a16, 28a75, 53a04 Analysis QA299.6-433
spellingShingle	hölder curves parameterization iterated function systems self-affine sets primary 28a80 secondary 26a16, 28a75, 53a04 Analysis QA299.6-433 Badger Matthew Vellis Vyron Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
description	We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝn is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝn may be strictly greater than n.
format	article
author	Badger Matthew Vellis Vyron
author_facet	Badger Matthew Vellis Vyron
author_sort	Badger Matthew
title	Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
title_short	Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
title_full	Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
title_fullStr	Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
title_full_unstemmed	Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
title_sort	hölder parameterization of iterated function systems and a self-affine phenomenon
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/0c87905a11d6469086eb3426c0600bc0
work_keys_str_mv	AT badgermatthew holderparameterizationofiteratedfunctionsystemsandaselfaffinephenomenon AT vellisvyron holderparameterizationofiteratedfunctionsystemsandaselfaffinephenomenon
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Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon

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