Hölder Parameterization of Iterated Function Systems and a Self-Aflne 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. more »
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;
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Publication Date:
NSF-PAR ID:
10280220
Journal Name:
Analysis and Geometry in Metric Spaces
Volume:
9
Issue:
1
Page Range or eLocation-ID:
90 to 119
ISSN:
2299-3274
Positive$$k\mathrm{th}$$$k\mathrm{th}$-intermediate Ricci curvature on a Riemanniann-manifold, to be denoted by$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when$$k =1$$$k=1$and$$k=n-1$$$k=n-1$respectively). In this work, we produce many examples of manifolds of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$withksmall by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension$$n\ge 7$$$n\ge 7$congruent to$$3\,{{\,\mathrm{mod}\,}}4$$$3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$for some$$k$k. We also prove that each dimension$$n\ge 4$$$n\ge 4$congruent to 0 or$$1\,{{\,\mathrm{mod}\,}}4$$$1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports closed manifolds which carry metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$with$$k\le n/2$k\le n/2$, but do not admit metrics of positive sectional curvature.