Hölder迭代函数系统的参数化与自仿射现象

IF 0.9 3区 数学 Q2 MATHEMATICS
Matthew Badger, Vyron Vellis
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引用次数: 3

摘要

摘要研究了完全度量空间中迭代函数系统(IFS)生成曲线的Hölder几何形状。Hata(1985)的一个定理断言IFS的每一个连通吸引子都是局部连通和路径连通的。我们给出了哈塔定理的一个定量强化。首先我们证明了IFS的每个连通吸引子都是(1/s)-Hölder路径连通的,其中s是IFS的相似维数。在端点α = s处,Remes(1998)的一个定理已经证明了欧氏空间中满足开集条件的连通自相似集是由(1/s)-Hölder曲线参数化的,并且证明了IFS的所有连通吸引子都是由(1/s)-Hölder曲线参数化的。在第二个结果中,我们展示了如何将Remes定理推广到完全度量空间中的自相似集,但在这种情况下,需要吸引子具有正的s维Hausdorff测度来代替开集条件。最后,我们确定了连通自仿射Bedford-McMullen地毯类参数化的明显Hölder指数,并建立了自仿射海绵的参数化。在自仿射设置中出现了一个有趣的现象。对于一个自相似曲线,其最优参数s总是不超过环境维数n,而对于一个自仿射曲线,其最优参数s可能严格大于n。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon
Abstract 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.
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来源期刊
Analysis and Geometry in Metric Spaces
Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology
CiteScore
1.80
自引率
0.00%
发文量
8
审稿时长
16 weeks
期刊介绍: Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.
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