Abstract
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 ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.
Highlights
A special feature of one-dimensional metric geometry is the compatibility of intrinsic and extrinsic measurements of the length of a curve
We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space
First we prove that every connected attractor of an IFS is ( /s)-Hölder path-connected, where s is the similarity dimension of the IFS
Summary
A special feature of one-dimensional metric geometry is the compatibility of intrinsic and extrinsic measurements of the length of a curve. If F is an IFS over a complete metric space, there exists a unique compact set KF in X (the attractor of F) such that. On the way to the proof of Theorem 1.3 (see §5), we rst parameterize IFS attractors without branching by ( /s)-Hölder arcs (see §4.1), where s is the similarity dimension. By Lemma 4.4, there exist two in nite words w , w ∈ AN such that for all n ∈ N, w (n) and w (n) are the unique vertices of valence 1 in Gn. Set. By (5.9), (5.10), and Corollary 2.8, FN ≡ FNN is a ( /s)-Hölder map with Hölder constant depending only on L , s, τ, C , C This completes the proof of Proposition 5.2 and Theorem 1.3, up to verifying. Suppose that there exists m ≤ N such that |a − b|∞ ≥ ( r)rm and |a − x|∞ ≤ ( /r)rm for all x ∈ R
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