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Free, publiclyaccessible full text available June 1, 2023

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} .

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 pathconnected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )Hölder pathconnected, 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 selfsimilar 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 selfsimilar 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 selfaffine BedfordMcMullen carpets and build parameterizations of selfaffine sponges. An interesting phenomenon emerges in the selfaffine setting.more »