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 $(1/\hat{I}\pm )$-Hölder curve for all $\hat{I}\pm >s$. At the endpoint, $\hat{I}\pm =s$, 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 ${\mathbb{R}}^n$ is always at most the ambient dimension $n$, the optimal parameter $s$ for a self-affine curve in ${\mathbb{R}}^n$ may be strictly greater than $n$.

37 pages, 7 figures. (v3: new title and abstract, new paragraph about parameterization dimension, added references and overhauled section 4)