HÖLDER PARAMETERIZATION OF ITERATED FUNCTION SYSTEMS AND A SELF-AFFINE PHENOMENON

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 is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in R may be strictly greater than n.

a continuous parameterization of finite total variation (intrinsic length) if and only if the set has finite one-dimensional Hausdorff measure H 1 (extrinsic length). In fact, any curve of finite length admits parameterizations f : [0, 1] → Γ, which are closed, Lipschitz, surjective, degree zero, constant speed, essentially two-to-one, and have total variation equal to 2H 1 (Γ); see Alberti and Ottolini [AO17,Theorem 4.4]. Unfortunately, this property-compatibility of intrinsic and extrinsic measurements of size-breaks down for higher-dimensional curves. While every curve parameterized by a continuous map of finite s-variation has finite s-dimensional Hausdorff measure H s , for each real-valued dimension s > 1 there exist curves with 0 < H s (Γ) < ∞ that cannot be parameterized by a continuous map of finite s-variation; e.g. see the "Cantor ladders" in [BNV19, §9.2]. Beyond a small zoo of examples, there does not yet exist a comprehensive theory of curves of dimension greater than one. Partial investigations on Hölder geometry of curves from a geometric measure theory perspective include [MM93], [MM00], [RZ16], [BV19], [BNV19], and [BZ20] (also see [Bad19]). For example, in [BNV19] with Naples, we established a Ważewski-type theorem for higher-dimensional curves under an additional geometric assumption (flatness), which is satisfied e.g. by von Koch snowflakes with small angles. The fundamental challenge is to develop robust methods to build good parameterizations.
Two well-known examples of higher-dimensional curves with Hölder parameterizations are the von Koch snowflake and the square (a space-filling curve). A common feature is that both examples can be viewed as the attractors of iterated function systems (IFS) in Euclidean space that satisfy the open set condition (OSC); for a quick review of the theory of IFS, see §2. Remes [Rem98] proved that this observation is generic in so far as every connected self-similar set in Euclidean space of Hausdorff dimension s ≥ 1 satisfying the OSC is a (1/s)-Hölder curve, i.e. the image of a continuous map f : [0, 1] → R n satisfying |f (x) − f (y)| ≤ H|x − y| 1/s for all x, y ∈ [0, 1] for some constant H < ∞. As an immediate consequence, for every integer n ≥ 2 and real number s ∈ (1, n], we can easily generate a plethora of examples of (1/s)-Hölder curves in R n with 0 < H s (Γ) < ∞ (see Figure 1). However, with the view of needing a better theory of curves of dimension greater than one, we may ask whether Remes' method is flexible enough to generate Hölder curves under less stringent requirements, e.g. can we parameterize self-similar sets in metric spaces or arbitrary connected IFS? The naive answer to this question is no, in part because measure-theoretic properties of IFS attractors in general metric or Banach spaces are less regular than in Euclidean space (see Schief [Sch96]). Nevertheless, combining ideas from Remes [Rem98] and Badger-Vellis [BV19] (or Badger-Schul [BS16]), we establish the following pair of results in the general metric setting. We emphasize that Theorems 1.1 and 1.2 do not require the IFS to be generated by similarities nor do they require the OSC. In the statement of the theorems, extending usual terminology for self-similar sets, we say that the similarity dimension of an IFS generated by contractions F is the unique number s such that where Lip φ = sup x =y dist(φ(x), φ(y))/ dist(x, y) is the Lipschitz constant of φ.
Theorem 1.1 (Hölder connectedness). Let F be an IFS over a complete metric space; let s be the similarity dimension of F. If the attractor K F is connected, then every pair of points is connected in K F by a (1/s)-Hölder curve.
Theorem 1.2 (Hölder parameterization). Let F be an IFS over a complete metric space; let s be the similarity dimension of F. If the attractor K F is connected, then K F is a (1/α)-Hölder curve for every α > s.
Early in the development of fractals, Hata [Hat85] proved that if the attractor K F of an IFS over a complete metric space X is connected, then K F is locally connected and path-connected. By the Hahn-Mazurkiewicz theorem, it follows that if K F is connected, then K F is a curve, i.e. K F the image of a continuous map from [0, 1] into X. Theorems 1.1 and 1.2, which are our main results, can be viewed as a quantitative strengthening of Hata's theorem. We prove the two theorems directly, in §3, without passing through Hata's theorem. A bi-Hölder variant of Theorem 1.1 appears in Iseli and Wildrick's study [IW17] of self-similar arcs with quasiconformal parameterizations.
Roughly speaking, to prove Theorem 1.1, we embed the attractor K F into ∞ and then construct a (1/s)-Hölder path between a given pair of points as the limit of a sequence of piecewise linear paths, mimicking the usual parameterization of the von Koch snowflake. Although the intermediate curves live in ∞ and not necessarily in K F , each successive approximation becomes closer to K F in the Hausdorff metric so that the final curve is entirely contained in the attractor. Building the sequence of intermediate piecewise linear paths is a straightforward application of connectedness of an abstract word space associated to the IFS. The essential point to ensure the limit map is Hölder is to estimate the growth of the Lipschitz constants of the intermediate maps (see §2.2 for an overview). Condition (1.1) gives us a natural way to control the growth of the Lipschitz constants, and thus, the similarity dimension determines the Hölder exponent of the limiting map (see §3). A similar technique allows us to parameterize the whole attractor of an IFS without branching by a (1/s)-Hölder arc (see §4).
To prove Theorem 1.2, we view the attractor K F as the limit of a sequence of metric trees T 1 ⊂ T 2 ⊂ · · · whose edges are (1/s)-Hölder curves. Using condition (1.1), one can easily show that (1.2) S α := sup n E∈Tn (diam E) α < ∞ for all α > s.
We then prove (generalizing a construction from [BV19,§2]) that (1.2) ensures K F is a (1/α)-Hölder curve for all α > s. Unfortunately, because the constants S α in (1.2) diverge as α ↓ s, we cannot use this method to obtain a Hölder parameterization at the endpoint. We leave the question of whether or not one can always take α = s in Theorem 1.2 as an open problem. The central issue is find a good way to control the growth of Lipschitz or Hölder constants of intermediate approximations for connected IFS with branching.
For self-similar sets with positive H s measure, we can build Hölder parameterizations at the endpoint in Theorem 1.2. The following theorem should be attributed to Remes [Rem98], who established the result for self-similar sets in Euclidean space, where the condition H s (K F ) > 0 is equivalent to the OSC (see Schief [Sch94]). In metric spaces, it is known that H s (K F ) > 0 implies the (strong) open set condition, but not conversely (see Schief [Sch96]). A key point is that self-similar sets K F with positive H s measure are necessarily Ahlfors s-regular, i.e. r s H s (K F ∩ B(x, r)) r s for all balls B(x, r) centered on K F with radius 0 < r diam K F . This fact is central to Remes' method for parameterizing self-similar sets with branching. See §5 for a details. Theorem 1.3 (Hölder parameterization for self-similar sets). Let F be an IFS over a complete metric space that is generated by similarities; let s be the similarity dimension of F. If the attractor K F is connected and H s (K F ) > 0, then K F is a (1/s)-Hölder curve.
As a case study, in §6, to further illustrate the results above, we determine the sharp Hölder exponents in parameterizations of connected self-affine Bedford-McMullen carpets. We also build parameterizations of connected self-affine sponges in R n (see Corollary 6.7).  • If Σ is a point, then Σ is (trivially) an α-Hölder curve for all α > 0.
• Otherwise, Σ is a (1/s)-Hölder curve, where s is the similarity dimension of Σ. The Hölder exponents above are sharp, i.e. they cannot be increased.
Of some note, the best Hölder exponent 1/s in parameterizations of a self-affine carpet can be strictly less than 1/2 (see Figure 2). A similar phenomenon occurs for self-affine arcs in R 2 (see §4.3). We interpret this as follows: If the supremum of all exponents appearing in the set of Hölder parameterizations of a curve Γ in a metric space is 1/s, then we may say that Γ has parameterization dimension s. (If Γ admits no Hölder parameterizations, then we say that Γ has infinite parameterization dimension.) Intuitively, the parameterization dimension is a rough gauge of how fast a denizen of a curve must walk to visit every point in the curve. Every non-degenerate rectifiable curve has parameterization dimension 1 and a square has parameterization dimension 2. More generally, every self-similar curve in R 2 that satisfies the OSC has parameterization dimension equal to its Hausdorff dimension. Theorem 1.4 implies that there exist self-affine curves in R 2 of arbitrarily large parameterization dimension.

Preliminaries
2.1. Iterated function systems. Let X be a complete metric space. A contraction in X is a Lipschitz map φ : X → X with Lipschitz constant Lip φ < 1, where An iterated function system (IFS) F is a finite collection of contractions in X. We say that F is trivial if Lip φ = 0 for every φ ∈ F; otherwise, we say that F is non-trivial.  [Hat85] are gems in geometric analysis and excellent introductions to the subject in their own right.
Theorem 2.1 (Hutchinson [Hut81]). If F is an IFS over a complete metric space, then there exists a unique compact set K F in X (the attractor of F) such that Above and below, the s-dimensional Hausdorff measure H s on a metric space is the Borel regular outer measure defined by The Hausdorff dimension dim H (E) of a set E in X is the unique number given by If there exists an open set U ⊂ X satisfying (2.6), and in addition, K F ∩ U = ∅, then we say that F satisfies the strong open set condition (SOSC). We say that the attractor K F of an IFS F over X is self-similar if each φ ∈ F is a similarity, i.e. there exists a constant 0 ≤ L φ < 1 such that Theorem 2.2 (Schief [Sch94], [Sch96]). Let K F be a self-similar set in X; let s = s-dim(F). If X is a complete metric space, then Moreover, the implications above are the best possible (unlisted arrows are false).
Given a metric space X, a set E ⊂ X, and radius ρ > 0, let N (E, ρ) denote the maximal number of disjoint closed balls with center in E and radius ρ. Following Larman [Lar67], X is called a β-space if for all 0 < β < 1 there exist constants 1 ≤ N β < ∞ and r β > 0 such that N (B, βr) ≤ N β for every open ball B of radius 0 < r ≤ r β . Theorem 2.3 (Stella [Ste92]). Let K F be a self-similar set in X; let s = s-dim(F). If X is a complete β-space, then The following pair of lemmas are easy exercises, whose proofs we leave for the reader.
Lemma 2.4. Let K F be a self-similar set in X; let s = s-dim(F). If H s (K F ) > 0, then K F is Ahlfors s-regular, i.e. there exists a constant 1 ≤ C < ∞ such that Lemma 2.5. Let F be an IFS over a complete metric space. If K F is connected, diam K F > 0, and φ ∈ F has Lip(φ) = 0, then K F agrees with the attractor of F \ {φ}.
2.2. Hölder parameterizations. Let s ≥ 1, let X be a metric space, and let f : [0, 1] → X. We define the s-variation of f (over [0, 1]) by where the supremum ranges over all finite interval partitions P of [0, 1]. Here and below a finite interval partition of an interval I is a collection of (possibly degenerate) intervals {J 1 , . . . , J k } that are mutually disjoint with I = k i=1 J i . We say that the map f is (1/s)-Hölder continuous provided that the associated (1/s)-Hölder constant By now, the following connection between continuous maps of finite s- The standard method to build a Hölder parameterization of a curve in a Banach space that we employ below is to exhibit the curve as the pointwise limit of a sequence of Lipschitz curves with controlled growth of Lipschitz constants. We will use this principle frequently, and also on one occasion in §3, the following extension where the intermediate maps are Hölder continuous.
Proof. Extend the sequence of functions f j 0 , . . . , f j 1 to an infinite sequence by setting f j ≡ f j 1 for all j > j 1 . Also choose any extension of the sequence of scales ρ j 0 , . . . , ρ j 1 satisfying (1). Then the full sequence (f j , ρ j ) ∞ j 0 satisfies the hypothesis of the lemma with A j ≡ A j 1 and B j ≡ 0 for all j > j 1 . Therefore, f j 1 ≡ lim j→∞ f j is (1/s)-Hölder with Höld 1/s f j 1 ≤ H.
By definition of the similarity dimension, we have L s 1 + · · · + L s k = 1. Define the alphabet A = {1, . . . , k}. Let A n = {i 1 · · · i n : i 1 , . . . , i n ∈ A} denote the set of words in A and of length n. Also let A 0 = { } denote the set containing the empty word of length 0. Let A * = n≥0 A n denote the set of finite words in A. Given any finite word w ∈ A * and length n ∈ N, we assign (2.16) A * w := {u ∈ A * : u = wv} and A n w = {wv ∈ A * w : |wv| = n}. The set A * w can be viewed in a natural way as a tree with root at w. We also let A N denote the set of infinite words in A. Given an infinite word w = i 1 i 2 · · · ∈ A N and integer n ≥ 0, we define the truncated word w(n) = i 1 · · · i n with the convention that w(0) = .
We now organize the set of finite words in A, according to the Lipschitz norms of the associated contractions! This will be used pervasively throughout the rest of the paper. For each word w = i 1 · · · i n ∈ A * , define the map (2.17) φ w := φ i 1 • · · · • φ in and the weight (2.18) L w := L i 1 · · · L in .
By convention, for the empty word, we assign φ := Id X and L := 1. For all w ∈ A * , define the cylinder K w to be the image of the attractor K := K F under φ w , Note that L uv = L u L v for every pair of words u and v, where uv denotes the concatenation of u followed by v. For each δ ∈ (0, 1), define (2.20) A * (δ) := {i 1 · · · i n ∈ A * : n ≥ 1 and L i 1 · · · L in < δ ≤ L i 1 · · · L i n−1 } with the convention L 1 · · · L i n−1 = 1 if n = 1. Also define A * (1) := { }. Finally, given any finite word w ∈ A * , set A * w (δ) := A * w ∩ A * (δ). Lemma 2.9. Given finite words w ∈ A * and w = wi 1 · · · i n and a number L w < δ ≤ L w , there exists a unique finite word u = wi 1 · · · i m (m ≤ n) such that u ∈ A * w (δ). Proof. Existence of u follows from the fact that the sequence a n = L wi 1 ···in is decreasing. Uniqueness of u follows from the fact that if wi 1 · · · i m ∈ A * w (δ), then for every l < m, L wi 1 ···i l ≥ δ, whence wi 1 · · · i l ∈ A * w (δ). Lemma 2.10. For every finite word w ∈ A * and number 0 < δ ≤ L w , Proof. By (2.15), we can choose N ∈ N sufficient large so that L u < L 1 δ for all words u ∈ A N (any integer N > log L k (L 1 δ) will suffice). In particular, if wv = wv 1 . . . v n ∈ A N w , then wv ∈ A * w (δ) (since L wv 1 ...v n−1 < δ) but wv has an ancestor wv 1 . . . v m ∈ A * w (δ) by Lemma 2.9. Hence the subtree T = N l=|w| A l w of A * w contains A * w (δ). To establish (2.21), we repeatedly use the defining condition L s 1 + · · · + L s k = 1 for the similarity dimension, first working "down" the tree T from each word w ∈ A * (δ) to its descendants in A N w and then working "up" the tree T level by level: Lemma 2.11. For all 0 < R ≤ 1, w ∈ A * (R), and 0 < r ≤ L w , Proof. Fix 0 < R ≤ 1, w ∈ A * (R), and 0 < r ≤ L w . Then L w < R ≤ L w /L 1 , and similarly, for all w ∈ A * w (r), we have L w < r ≤ L w /L 1 . By Lemma 2.10, Similarly, This establishes (2.22). To derive (2.23), simply take 0 < r ≤ 1 = R and w = .

Hölder connectedness of IFS attractors
In this section, we first prove Theorem 1.1, and afterwards, we derive Theorem 1.2 as a corollary. To that end, for the rest of this section, fix an IFS F = {φ 1 , . . . , φ k } over a complete metric space (X, d) whose attractor K := K F is connected and has positive diameter. Set s := s-dim(F), and for each i ∈ {1, . . . , k}, set L i := Lip(φ i ). By Lemma 2.5, we may assume without loss of generality that (3.1) 0 < L 1 ≤ · · · ≤ L k < 1.
In particular, we may adopt the notation, conventions, and lemmas in §2.3.
Proof. We first remark that K w = u∈A * w (δ) K u by Lemma 2.9. Define Then v ∈ E n for some 1 ≤ n ≤ N . Label v =: w n . By design of the sets E i , we can find a chain of distinct words w 1 , . . . , w n with K w i ∩ K w i+1 for all 1 ≤ i ≤ n − 1. Finally, x ∈ K w 1 , because w 1 ∈ E 1 . Theorem 1.1 is a special case of the following more precise result (take w to be the empty word). Recall that a metric space (X, d) is quasiconvex if any pair of points x and y can be joined by a Lipschitz curve f : [0, 1] → X with Lip(f ) X d(x, y). By analogy, the following proposition may be interpreted as saying that connected attractors of IFS are "(1/s)-Hölder quasiconvex".
Proposition 3.2. For any w ∈ A * and x, y ∈ K w , there exists a (1/s)-Hölder continuous Proof. By rescaling the metric on X, we may assume without loss of generality that diam K = 1. Furthermore, it suffices to prove the proposition for w = and K w = K.
For the general case, fix w ∈ A * and x, y ∈ K w . Choose x , y ∈ K such that φ w (x ) = x and φ w (y ) = y. Define If the proposition holds for w = , then there exists a (1/s)-Hölder map g : [0, 1] → K with g(0) = x , g(1) = y , and Höld 1/s g s,L 1 1.
To proceed, observe that by the Kuratowski embedding theorem, we may view K as a subset of ∞ , whose norm we denote by | · | ∞ . Fix any r > 0 with L 1 r ≤ L 1 (which ensures that r m+1 ≤ L 1 r m ≤ L w whenever w ∈ A * (r m )) and fix x, y ∈ K. The map f will be a limit of piecewise linear maps f n : [0, 1] → ∞ . In particular, for each m ∈ N, we will construct a subset W m ⊂ A * (r m ), a family of nondegenerate closed intervals E m , and a continuous map f m : [0, 1] → ∞ satisfying the following properties: Let us first see how to complete the proof, assuming the existence of family of such maps. On one hand, property (P3) gives On the other hand, by property (P2), |I| ≥ L s 1 r ms and diam f m (I) < r m for all I ∈ E m . Therefore, for all p, q ∈ [0, 1], By (3.2), (3.3), and Lemma 2.7, the sequence (f m ) ∞ m=1 converges uniformly to a (1/s)- Therefore, f ([0, 1]) ⊂ K and the proposition follows.
It remains to construct W m , E m , and f m satisfying properties (P1), (P2), and (P3). The construction is in an inductive manner.
By Lemma 3.1, there is a set W 1 = {w 1 , . . . , w n } of distinct words in A * (r), enumerated so that x ∈ K w 1 , y ∈ K wn , and To proceed, define E 1 = {I 1 , . . . , I n } to be closed intervals in [0, 1] with disjoint interiors, enumerated according to the orientation of [0, 1], whose union is [0, 1], and such that |I j | ≥ L s w i for all i ∈ {1, . . . , n}. We are able to find such intervals, since by Lemma 2.10, Next, define f 1 : [0, 1] → ∞ in a continuous fashion so that f 1 is linear on each I i and: (1) f 1 (0) = x and f 1 (I 1 ) is the segment that joins x with p 1 ; (2) f 1 (1) = y and f 1 (I n ) is the segment that joins p n−1 with y; and, Suppose that for some m ∈ N, we have defined W m ⊂ A * (r m ), a collection E m , and a piecewise linear map f m : [0, 1] → ∞ that satisfy (P1)-(P3). For each I ∈ E m , we will define a collection of intervals E m+1 (I) and a collection of words W m+1 (I) ⊂ A * (r m+1 ). We then set E m+1 = I∈Em E m+1 (I) and W m+1 = I∈Em W m+1 (I). In the process, we will also define f m+1 . To proceed, suppose that . . , I l } be closed intervals in I with mutually disjoint interiors, enumerated according to the orientation of I, whose union is I, and such that a ∈ I 1 , b ∈ I l and |I j | ≥ L s w j for all j ∈ {1, . . . , l}. We are able to find such intervals, since by our inductive hypothesis and Lemma 2.10, With the choices above, now define f m+1 |I : I → ∞ in a continuous fashion so that f m+1 |J is linear for each J ∈ E m+1 (I) and: (1) f m+1 (a) = f m (a) and f m+1 (I 1 ) is the segment that joins y with p 1 ; (2) f m+1 (b) = f m (b) and f m+1 (I l ) is the segment that joins p l−1 with f m (b); and, (3) for j ∈ {2, . . . , l − 1} (if any), f m+1 (I j ) is the segment that joins p j−1 with p j . Properties (P1), (P2), and the first claim of (P3) are immediate. To verify the second claim of (P3), fix z ∈ [0, 1]. By (P1), there exists I ∈ E m+1 such that z ∈ I. Let J be the unique element of E m such that I ⊂ J. Then there exists w ∈ A * (r m ) such that we have that f m+1 (∂I). Let y 1 ∈ ∂I and y 2 ∈ ∂J. We have 3.2. Hölder parameterization (Proof of Theorem 1.2). The proof of Theorem 1.2 is modeled after the proof of [BV19, Theorem 2.3], which gave a criterion for the set of leaves of a "tree of sets" in Euclidean space to be contained in a Hölder curve. Here we view the attractor K F as the set of leaves of a tree, whose edges are Hölder curves.
Proof of Theorem 1.2. Rescaling the metric d, we may assume for the rest of the proof that diam K = 1. Fix q ∈ K, and for each w ∈ A * , set q w := φ w (q) with the convention q = q. Fix α > s = s-dim F and fix L 1 r ≤ L 1 (once again ensuring that r m+1 ≤ L 1 r m ≤ L w for all w ∈ A * (r m )). By Lemma 2.11, for every integer m ≥ 0, the set A(r m ) has fewer than L −s 1 r −ms words, and moreover, for every w ∈ A * (r m ), the set A * w (r m+1 ) has at least 1 and fewer than L −s 1 r −s words. Since r L 1 , Below we call the elements of A * w (r m+1 ) the children of w ∈ A * (r m ), and we call w Hölder map with f w,u (0) = q w and f w,u (L s w ) = q u given by Proposition 3.2. Let also γ w,u be the image of f w,u . We can write K as the closure of the set For each integer m ≥ 0 and w ∈ A * (r m ) define where we sum over all descendants of w. Setting M := M , by (3.4), we have that M L 1 ,s,α 1. We will construct a (1/α)-Hölder continuous surjective map F : [0, M ] → K by defining a sequence F m : [0, M ] → K (m ∈ N) whose limit is F and whose image is the truncated tree . Conversely, if J ∈ N n , then there exist J 1 ∈ N m+1 and J 2 ∈ B m+1 such that J 1 ⊂ I and J 2 ⊂ I and card{I ∈ B m+1 ∪ N m+1 : I ⊂ J} ≤ L −s 1 r s . (P3) If I ∈ B m+1 , then either I ∈ B m or there exists J ∈ N m such that I ⊂ J.
(P4) For each I ∈ N m , |I| = M ηm(I) , F m |I is constant and equal to q η(I) and F m+1 |∂I = F m |∂I. (P5) For each I ∈ B m , there exists w ∈ A * (r m−1 ) and u ∈ A * w (r m ) such that |I| = L α w and F m |I = f w,u • ψ I where ψ I is (s/α)-Hölder with Höld s/α ψ I = 1. Conversely, for any w ∈ A * (r m−1 ) and u ∈ A * w (r m ) there exists I ∈ B m as above. Finally, We now complete the proof of Theorem 1.2, assuming Lemma 3.3. Let B m , N m , η m and F m be as in Lemma 3.3. Notice by (P2) that if I ∈ N m , then for all F n (I) ⊂ K ηm(I) . We claim that We now claim that for all m ∈ N and all x, y ∈ [0, 1], To prove (3.6) fix x, y ∈ [0, M ] and consider the following cases.
Case 2. Suppose that there exist I 1 , I 2 ∈ B m ∪ N m such that I 1 ∩ I 2 is a single point {z}, x ∈ I 1 and y ∈ I 2 . Then, by triangle inequality and Case 1, Case 3. Suppose that Case 1 and Case 2 do not hold. Let m 0 be the smallest positive integer m such that there exists I ∈ B m ∪ N m with x ≤ z ≤ y for all z ∈ I. In particular, suppose that Therefore, by Case 1 and the triangle inequality, By (3.5), (3.6) and Lemma 2.7, we have that F m converges pointwise to a (1/α)-Hölder This completes the proof of Theorem 1.2, assuming Lemma 3.3.
Proof of Lemma 3.3. We give the construction of B m , N m , η m and F m in an inductive manner. Suppose a union of closed intervals with mutually disjoint interiors, enumerated according to the orientation of [0, M ] such that |I j | = |I j | = 1 and |J j | = M w j . Set B 1 = {I 1 , I 1 , . . . , I n , I n }, ) be a (s/α)-Hölder orientation preserving (resp. orientation reversing) homeomorphism with Höld s/α ψ i = 1 (resp. Höld s/α ψ i = 1). Define now Suppose now that for some m ≥ 1, we have constructed B m , N m , η m and F m satisfying (P1)-(P5). For each I ∈ B m define F m+1 |I = F m |I. For each I ∈ N m we construct families B m+1 (I) and N m+1 (I) and then we set In the process we also define F m+1 and η m . Suppose that I ∈ N m and write I = [a, b]. By the inductive hypothesis (P3), there exists w ∈ A * (r m ) such that F m (I) = q w . Suppose that A * w (r) = {w 1 , . . . , w n }. Decompose I as a union of closed intervals with mutually disjoint interiors, enumerated according to the orientation of I such that |I j | = L α w and |J j | = M w j . Set B m+1 (I) = {I 1 , I 1 , . . . , I l , I l }, ) be a (s/α)-Hölder orientation preserving (resp. orientation reversing) homeomorphism with Höld s/α ψ i = 1 (resp. Höld s/α ψ i = 1). Define now

IFS without branching. Given an IFS
Proposition 4.1 (parameterization of connected IFS without branching). Let F be an IFS over a complete metric space; let s = s-dim(F). If K F is connected, diam K F > 0, and F has no branching, then there exists a (1/s)-Hölder homeomorphism f : For the rest of §4.1, fix an IFS F = {φ 1 , . . . , φ k } over a complete metric space (X, d) whose attractor K := K F is connected and has positive diameter. Adopt the notation and conventions set in the first paragraph of §3 as well as in §2.3. In addition, assume that F has no branching. Since diam K > 0, k ≥ 2. Replacing F with the iterated IFS F = {φ w : w ∈ A 2 } if needed, we may assume without loss generality that k ≥ 4. Finally, rescaling the metric d, we may assume without loss of generality that diam K = 1.
Given n ∈ N, we denote by G n the graph with vertices the set A n and (undirected) edges {{w, u} : w = u and K w ∩ K u = ∅}. For each n ∈ N and w ∈ A n , the valence val(u, G n ) of w in G n is the number of all edges of G n containing w.
Lemma 4.2. Each G n is a combinatorial arc. Moreover, there exist exactly two distinct i 0 , j 0 ∈ A such that for any n ∈ N the following properties hold.
(1) If w has valence 1 in G n , then there exists unique i ∈ {i 0 , j 0 } such that wi has valence 1 in G n+1 .
(2) If {w, u} is an edge of G n , then there exist unique i, j ∈ {i 0 , j 0 } such that {wi, uj} is an edge in G n+1 .
Proof. Because F has no branching, val(i, G 1 ) ∈ {1, 2} for all i ∈ A 1 . Therefore, either G 1 is a combinatorial circle or G 1 is a combinatorial arc. If G 1 is a combinatorial circle and {1, i} is any edge in G 1 , then there exists i 1 , i 2 , i 3 , j 1 ∈ A such that {1i 1 , 1i 2 }, {1i 2 , 1i 3 }, and {1i 2 , ij 1 } are edges in G 2 ; this implies val(1i 2 , G 2 ) ≥ 3 and we reach a contradiction. Thus, in fact, G 1 is a combinatorial arc. In particular, there exist exactly two words in A whose valence in G 1 is 1, say i 0 and j 0 . The rest of the proof follows from a simple induction, which we leave the reader.
From Lemma 4.2, we obtain two simple corollaries.
Lemma 4.3. For all n ∈ N and all w, u ∈ A n , K w ∩ K u is at most a point.
Proof. Fix w, u ∈ A n such that K w ∩ K u = ∅. We first claim that there exists unique i ∈ A and unique j ∈ A such that K wi ∩ K uj = ∅. Assuming the claim to be true, we have To prove the claim, fix i ∈ A such that K wi ∩ K u = ∅. By Lemma 4.2, we have that i ∈ {i 0 , j 0 } where {i 0 , j 0 } are the unique elements of A with valence 1 in G 1 ; say i = i 0 . If there exists w ∈ A \ {w, u} such that K w ∩ K w = ∅, then by Lemma 4.2 K w ∩K wj 0 = ∅ and K wj 0 ∩K u = ∅. If no such w exists, then val(w, G n ) = 1 which implies that val(wj 0 , G n+1 ) = 1 which also implies K wj 0 ∩ K u = ∅. In either case, K wj 0 ∩ K u = ∅ and i is unique.
Lemma 4.4. For all n ∈ N, there exist exactly two words w ∈ A n such that the set K w ∩ K \ K w contains only one point.
Proof. By Lemma 4.2, for each n ∈ N, there exist exactly two distinct words w, u ∈ A n that have valence 1 in G n . Fix one such word, say w. Then there exists a unique w ∈ A n \ {w} such that K w ∩ K w = K w ∩ K \ K w . By Lemma 4.3, the latter intersection is a single point.
We are ready to prove Proposition 4.1.
Proof of Proposition 4.1. By Lemma 4.4, there exist two infinite words w 0 , w 1 ∈ A N such that for all n ∈ N, w 0 (n) and w 1 (n) are the unique vertices of valence 1 in G n . Set It remains to show that f is a homeomorphism and suffices to show that f is injective. Recall the definitions of E m and f m from the proof of Proposition 3.2. By (P2) and (P3) therein, for each m ∈ N and I ∈ E m , there exists w I ∈ A * (r m ) such that f (I) ⊂ K w I . Moreover, w I = w J if I = J. In conjunction with the fact that f ([0, 1]) = K, we have that f (I) = K w I . By design of the map f , it is easy to see that K w I ∩ K w J if and only if I ∩ J. Assume x, y ∈ [0, 1] with x = y. Then there exists m ∈ N and disjoint I, J ∈ E m such that x ∈ I and y ∈ J.
4.2. Bounded turning and self-similar bi-Hölder arcs. With additional information on the contractions of F and how the components K i = φ i (K) of the attractor K intersect, the map f constructed in Proposition 4.1 is actually a (1/s)-bi-Hölder homeomorphism. We say that K has bounded turning if there exists C ≥ 1 such that for all In general, self-similar curves (even in R 2 ) do not have the bounded turning property; see [ATK03, Example 2.3] or [WX03, Theorem 2].
The following proposition follows from Theorem 1.5 and Lemma 3.2 of [IW17]. Proposition 4.5 (self-similar sets without branching and with bounded turning). Let F be an IFS over a complete metric space that is generated by similarities; let s = s-dim(F).
If K F is connected, diam K F > 0, F has no branching, and K F is bounded turning, then there exists a (1/s)-bi-Hölder homeomorphism f : [0, 1] → K.

4.3.
Sharp exponents for self-affine snowflake curves in the plane. For each line segment l ⊂ R 2 and α ∈ (0, 1), define the diamond D α (l) with axis l and aperture α, where p, q are the endpoints of l. We will build a family of self-affine snowflake curves as the IFS attractor of a chain of diamonds. Let l 0 := [0, 1] × {0} and let P = l 1 ∪ · · · ∪ l k , k ≥ 2, be a polygonal arc lying in {0, 1} ∪ int D 1/2 (l 0 ), enumerated so that Choose apertures α i ∈ (0, 1/2) small enough so that For each i ∈ {1, . . . , k}, fix an affine homeomorphism φ i : R 2 → R 2 such that φ i (l 0 ) = l i and φ i (D 1/2 (l 0 )) = D α i (l i ). Because each aperture α i < 1/2, where |l i | denotes the length of l i . In particular, F = {φ i : 1 ≤ i ≤ k} is an IFS over R 2 ; see Figure 3. Let s = s-dim(F) and let K = K F denote the attractor of F. Since Figure 3. Generators of an IFS for a self-affine snowflake curve.
F has no branching, the snowflake curve K is a (1/s)-Hölder arc by Proposition 4.1; the endpoints of K are p 0 = (0, 0) and p 1 = (1, 0). We now show that the exponent cannot be increased.

Hölder parameterization of self-similar sets (Remes' method)
Our goal in this section is to record a proof of Theorem 1.3 that combines original ideas of Remes [Rem98] with our style of Hölder parameterization from above. To aid the reader wishing to learn the proof, we have attempted to include a clear description of the key properties of parameterizations that approximate the final map (see Lemma 5.7), which are obscured in Remes' thesis.
As usual, we adopt the notation and conventions set in the first paragraph of §3 as well as in §2.3. Rescaling the metric, we may assume without loss of generality that diam K = 1. Since K is self-similar, it follows that If F has no branching (see §4.1 ), then a (1/s)-Hölder parameterization of K already exists by Proposition 4.1 . Thus, we shall assume F has branching, i.e. there exists m ∈ N and distinct words w 1 , . . . , w 4 ∈ A m such that K w 1 ∩K w i = ∅ for each i ∈ {2, 3, 4}. In the event that m ≥ 2 (see Example 5.1), we replace F with the self-similar IFS F = {φ w : w ∈ A m }. This causes no harm to the proof, because the attractors coincide, i.e. K F = K F , and s-dim(F ) = s-dim(F). Therefore, without loss of generality, we may assume that there exist distinct letters i 1 , i 2 , i 3 , i 4 ∈ A such that Example 5.1. Divide the unit square into 3 × 3 congruent subsquares with disjoint interiors S i (1 ≤ i ≤ 9). Let S 9 denote the central square and for each 1 ≤ i ≤ 8, let ψ i : R 2 → R 2 be the unique rotation-free and reflection-free similarity that maps [0, 1] 2 onto S i . The attractor of the IFS G = {ψ 1 , . . . , ψ 8 } is the Sierpiński carpet. Looking only at the intersection pattern of the first iterates ψ 1 (K G ),. . . ,ψ 8 (K G ), it appears that G has no branching. However, upon examining the intersections of the second iterates ψ i • ψ j (K G ) (1 ≤ i, j ≤ 8), it becomes apparent that G has branching.
To continue, use the Kuratowski embedding theorem to embed (K, d) into ( ∞ , | · | ∞ ). (If K already lies in some Euclidean or Banach space, or in a complete quasiconvex metric space, then the construction below can be carried out in that space instead.) Let d H denote the Hausdorff distance between compact sets in ∞ . By the Arzelá-Ascoli theorem, to complete the proof of Theorem 1.3, it suffices to establish the following claim. Remark 5.3. It is perhaps unfortunate that we have to invoke the Arzelá-Ascoli theorem to implement Remes' method. We leave as an open problem to find a proof of Theorem 1.3 that avoids taking a subsequential limit of the intermediate maps; cf. the proofs in §3 above or the proof of the Hölder traveling salesman theorem in [BNV19].
We devote the remainder of this section to proving Proposition 5.2.
and assign r := 1 4 L 1 τ . Then, since F consists of similarities, For all m ∈ N, define the set The separation condition (5.5) ensures that the words in A * (r m ) and points in Y m are in one-to-one correspondence. Unfortunately, the sets Y m are not necessarily nested. To proceed, fix an index N ∈ N. We will construct a map F N : [0, 1] → ∞ with Höld 1/s F N L 1 ,s,τ,C 1 ,C 2 1 and d H (F N ([0, 1]), K) L 1 ,τ r N .

Nets.
Following an idea of Remes [Rem98], starting from Y N and working backwards through Y 1 , we now produce a nested sequence of sets V 1 ⊂ · · · ⊂ V N recursively, as follows.
for each i ∈ {m, . . . , N } and each w ∈ A * (r i ), there exists a unique x ∈ K w ∩ V i . Replace each x ∈ Y m−1 by an element x ∈ V m ∩ K ux of shortest distance to x, where u x ∈ A * (r m−1 ) satisfies φ ux (v) = x. This produces the set V m−1 . See Figure 4.
Remark 5.4. The recursive definition of the sets V 1 ⊂ · · · ⊂ V N starting from a fixed level Y N is one obstacle to proving Theorem 1.3 without using the Arzelá-Ascoli theorem. (1) For each w ∈ A * (r m ), there exists a unique x ∈ V m ∩ K w .
Proof. The first claim and nesting property V m ⊂ V m+1 follow immediately by the design of the sets V m . Suppose that 1 ≤ m ≤ N − 1 and x ∈ V m+1 . By (1), there exists w ∈ A * (r m+1 ) such that x ∈ K w , say w = i 1 . . . i n ∈ A n . Set w = i 1 . . . i n−1 . Then L w < r m+1 ≤ L w ≤ L w /L 1 , since w ∈ A * (r m+1 ). If L w < r m ≤ L w , as well, then w ∈ A * (r m ), x ∈ V m , and we take x = x. Otherwise, L w < r m . Choose w = i 1 . . . i l , l ≤ n − 1 to be the shortest word such that L w < r m . Then w ∈ A * (r m ). By (1), there exists a unique x ∈ V m ∩ K w . Then |x − x | ∞ ≤ diam K w < r m by (5.3). This establishes the second claim.
For the third claim, we first prove that for all w ∈ A * (r m ) and x ∈ V m ∩ K w , by backwards induction on m. Equation (5.7) holds in the base case, because V N = Y N . Suppose for induction that we have established (5.7) for some 2 ≤ m + 1 ≤ N , and let w ∈ A * (r m ) and On the other hand, by the induction hypothesis, |y −φ wu (v)| ∞ ≤ r m+2 +· · ·+r N . Thus, since x is by definition a point in V m+1 that is nearest to φ w (v), Therefore, (5.7) holds for all m. Claim (3) follows, because where the last inequality holds since r < 1/2. Finally, for the last claim, if a, b ∈ V m are distinct, say with a ∈ K w ∩V m and b ∈ K u ∩V m for some w, u ∈ A * (r m ), then by (5.5),

5.3.
Trees. Next, we define a finite sequence of trees T m = (V m , E m ) m=1,...,N inductively, where the vertices V m were defined in the previous section and the edges E m will be specified below. By Lemma 5.5, for all m ∈ {1, . . . , N } and all x ∈ V m , there exists a unique w ∈ A * (r m ) such that x ∈ K w ; we denote this word w by x(m).
Let G 1 = (V 1 ,Ê 1 ) be the graph whose edge set is given bŷ The connectedness of K implies that G 1 is a connected graph, but not necessarily a tree. Now, removing some edges fromÊ 1 , we obtain a new set E 1 so that T 1 = (V 1 , E 1 ) is a connected tree. Because we assumed F has branching, see (5.4), we may assume that T 1 has at least one branch point, i.e. there exists x ∈ V 1 with valence in T 1 at least 3. Suppose that we have defined T m = (V m , E m ) for some m ∈ {1, . . . , N − 1}. For each x ∈ V m , let V m+1,x = V m+1 ∩ K x(m) and let T m+1,x = (V m+1,x , E m+1,x ) be a connected tree such that {y, z} ∈ E m+1,x only if y, z ∈ V m+1,x , y = z and K y(m+1) ∩ K z(m+1) = ∅. Moreover, since K x(m) is homothetic to K, we may require that T m+1,x has at least one branch point. Now, if {a, b} ∈ E m , there exists a ∈ V m+1,a and b ∈ V m+1,b such that K a (m+1) ∩ K b (m+1) = ∅. There is not a canonical choice, so we select one pair {a , b } for each pair {a, b} in an arbitrary fashion. Set   2r m by (5.3). The lower bound on the length is taken from (5.5).

Parameterization of T
It remains to show that F N is (1/s)-Hölder with Hölder constant independent of N .
To that purpose, we define an auxiliary sequence F 1 N , . . . , F N N ≡ F N to which we can apply Corollary 2.8. As already noted, we simply set F N N := F N . Next, suppose that 1 ≤ m ≤ N − 1. Let N m = {a 1 , a 2 , . . . , a l } denote the set of endpoints of intervals in E m , enumerated according to the orientation of [0, 1]. Letf j : [0, 1] → ∞ be defined by linear interpolation and the rulef j (a i ) = f j (a i ) for all i. We then let F j N be the unique map such thatf j = F j N • ψ = f j (i.e. F j N :=f j • ψ −1 ). By (P3), (P4) and (P5), for all 1 ≤ j ≤ N − 1, Next, we claim that for all 1 ≤ j ≤ N and all x, y ∈ [0, 1], Since each map F j N is continuous and linear on each interval ψ(I), I ∈ E j , the Lipschitz constant is given by by (P3). Fix I ∈ E j . To estimate |ψ(I)|, by (P6), (P7), and Lemma 2.11 we have Thus, we have established (5.10). Therefore, by (5.9), (5.10), and Corollary 2.8, F N ≡ F N N is a (1/s)-Hölder map with Hölder constant depending only on L 1 , s, τ, C 1 , C 2 . This completes the proof of Proposition 5.2 and Theorem 1.3, up to verifying Lemma 5.7. 5.5. Remes' Branching Lemma and the Proof of Lemma 5.7. We now recall a key lemma from Remes [Rem98], which lets us build the intermediate parameterizations in Lemma 5.7. In the remainder of this section, we frequently use the following notation and terminology. Given a, b ∈ V m with a = b, we let R m (a, b) denote the unique arc (the "road") in T m with endpoints a and b. A branch B of T m with respect to R m (a, b) is a maximal connected subtree of T m with at least two vertices such that B contains precisely one vertex x in R m (a, b) and x is terminal in B (i.e. x has valency 1 in B). See Figure  6. More generally, if T is a connected tree and S is a connected subtree of T , we define a branch B of T with respect to S to be a maximal connected subtree of T with at least two vertices such that B contains precisely one vertex x in S, and x is terminal in B. Figure 6. A road R 2 (a, b) (in red) and the 5 branches in T 2 with respect to R 2 (a, b) (in blue) for the IFS for the square in Fig. 3. Branches 2 and 3 contain a point in V 1 \ R 2 (a, b); branches 1, 4, and 5 do not.  N (a, b). Suppose that there exists m ≤ N such that |a − b| ∞ ≥ (2r)r m and |a − x| ∞ ≤ (4/r)r m for all x ∈ R.
(1) Control on number of branches from above: There exists C ≥ 1 depending only on L 1 , s, τ, C 1 , C 2 such that the number of the branches of T N with respect to R N (a, b) containing points in V m \ R is less than C.
(2) Control of the road length: There exists C ≥ 1 depending only on L 1 , s, τ, C 1 , C 2 such that if S = {z 1 , . . . , z l } is a subset of R, enumerated relative to the ordering induced by R N (a, b), and |z i − z i+1 | ∞ ≥ (2r)r m for all 1 ≤ i ≤ l − 1, then l ≤ C .
(3) Control on number of branches from below: There is t ∈ N depending only on L 1 , τ, s, C 1 , C 2 such that if m ≤ N − t, then the number of branches of T N with respect to R N (a, b) that contain some vertex in V m+t \ R is at least 2C + C + 2. Moreover, if c ∈ V m+t \ R is such a vertex and c ∈ K x t+m (c) , then c and c belong to the same branch of T N with respect to R N (a, b).
Proof. From the inductive construction, it is easy to see that the trees T 1 , . . . , T N satisfy the following property, which Remes calls the branch-preserving property: [Rem98, p. 23] Let 1 ≤ m ≤ n ≤ N , let x 1 , x 2 ∈ V m , let B be a branch of T m with respect to R m (x 1 , x 2 ), and let x 3 be a vertex of the branch B. Let x 1 , x 2 ∈ V n with x 1 ∈ K x 1 (m) and x 2 ∈ K x 2 (m) . Then all vertices in V n ∩ K x 3 (m) belong to the same branch of T n with respect to R n (x 1 , x 2 ).
Since we arranged for the attractor in our setting to satisfy (5.1), the proof of Lemma 5.8 follows exactly as the proof of [Rem98, Lemma 4.11] in Euclidean space. This is the only place in the proof of Theorem 1.3 where we use the assumption that H s (K) > 0.
(1) Denote by B the set of branches of T N with respect to R N (a, b) containing points in V m \ R. Let B ∈ B and let z B be the common vertex of the road and the branch B.

Among all vertices in
We claim that |x B − z B | ∞ ≤ 2r m . To prove the claim, note first that if |z B − y| ∞ > r m for any vertex y ∈ B ∩V N , then z B and y belong to two different sets K w , K u , respectively, with w, u ∈ A * (r m ). By design of T N and the branch-preserving property, we have that V N ∩K w ⊂ B, because the minimal connected subgraph containing those vertices contains no other vertices. Because one of those vertices belongs to V m , we get the claim.
(3) Set C = 2C + C + 1 and set t = log r (2r/C ) . Because |a − b| ∞ ≥ (2r)r m ≥ C r m+t , the road R N (a, b) contains at least C elements of V m+t . Since F has branching (recall (5.4)), there exist at least C branches of T N,m+t with respect to R N (a, b). By the branch-preserving property, for each such branch, there exists w ∈ A * (r m+t +1 ) such that the said branch contains all vertices in V N ∩K w . Thus, we may take t = log r (2r/C ) +1, which ultimately depends at most on L 1 , τ, s, C 1 , C 2 , and (3) holds.
(2) The map g 1 is a 2-to-1 piecewise linear tour of edges of T N,1 .
(3) For each I ∈ E 1 , g 1 maps the endpoints of I onto two vertices in V 1 and maps I piecewise linearly onto the road that joins the two vertices in T N,1 .
If N − t < 1, we simply set f 1 = g 1 and proceed to the inductive step. Otherwise, 1 ≤ N − t and to define f 1 , we modify the map g 1 on each interval in E 1 by inserting branches. Let {I 1 , . . . , I n } be an enumeration of E 1 . Let C be as in Lemma 5.8(1).
Lemma 5.9. Let I 1 = [x, y], a = g 1 (x) and b = g 1 (y). Let {B 1 , . . . , B p } be the branches of T N with respect to the road R N (a, b) that contain a set K w ∩ V N for some w ∈ A * (r t+1 ). There exist at most C indices j ∈ {1, . . . , p}, for which B j has parts that are traced by g 1 .
Proof. If B j is a branch as in the assumption of the lemma, then B j contains a point in V 1 . However, by Lemma 5.8(1), we know that no more than C such branches exist.
Writing I 1 = [x, y], since |g 1 (x) − g 1 (y)| ∞ > (2r)r and |g 1 (x) − z| ∞ ≤ (1/r)r for every vertex z of R N (g 1 (x), g 1 (y)) in T N , we can invoke Lemma 5.8(3). Thus, we can find a branch B of R N (g 1 (x), g 1 (y)) with respect to T N that contains all vertices of V N ∩ K w for some w ∈ A * (r t ) such that no part of it is traced by g 1 . We define f 1 |I 1 so that the following properties are satisfied.
(1) The map f 1 |I 1 is piecewise linear and traces all the edges of B ∪ g 1 (I 1 ) ⊂ T N .
Suppose that we have defined f 1 on I 1 , . . . , I i . To define f 1 |I i+1 , we first verify the following analogue of Lemma 5.9.
Lemma 5.10. Write I i+1 = [x, y], a = g 1 (x) and b = g 1 (y). Let {B 1 , . . . , B p } be the branches of T N with respect to the road R N (a, b) that contain a set K w ∩ V N for some w ∈ A * (r t+1 ). There exist at most 2C + 1 indices j ∈ {1, . . . , p} for which B j has been traced by f 1 |I 1 ∪ · · · ∪ I i .
Proof. There are two cases in which a branch B j has been traced by f 1 |I 1 ∪ · · · ∪ I i . The first case occurs when part of B j is already traced by g 1 (and hence by f 1 |I 1 ∪ · · · ∪ I i ). As in Lemma 5.9, at most C such branches B j exist. The second case occurs when we are traveling on the road R N (a, b) backwards. More specifically, the second case occurs when there exists i 1 ∈ {1, . . . , i} such that there is a part of g 1 (I 1 ) lying on R N (a, b) and part of B j is being traced by f 1 |I i 1 . In this situation, there are two possible subcases: (1) the right endpoint of I i 1 is mapped by g 1 into one of the branches of T N,1 with respect to R N (a, b) and by Lemma 5.8(1) at most C such branches exist; and, (2) f 1 |I i 1 contains a and since f 1 is essentially 2-1, at most one such interval exists.
For I i+1 , we now work exactly as with I 1 , but we choose a branch B j that has no edge being traced by f 1 |I 1 ∪ · · · ∪ I i . We can do so because by Lemma 5.8(3), there exist at least 2C + 2 branches of T N with respect to the road R N (a, b) that contain a set K w ∩ V N for some w ∈ A * (r t+1 ). Modifying g 1 on each I i completes the definition of f 1 .
We start by defining an auxiliary map g m+1 that visits the image of f m and T N,m+1 . In particular, define g m+1 : [0, 1] → T N and an auxiliary collection of intervals B m+1 of nondegenerate closed intervals in [0, 1] so that the following properties hold.
(1) The intervals in B m+1 have mutually disjoint interiors and collectively B m+1 = [0, 1]. Moreover, for any I ∈ B m+1 there exists unique J ∈ E m such that J ⊆ I. To define E m+1 , we will first identify the endpoints of its intervals. Towards this goal, let W m+1 denote the set of endpoints of the intervals in B m+1 and let P m denote the set of endpoints of the intervals in E m . By definition of B m+1 , we have P m ⊂ W m+1 .
Lemma 5.11. There exists a maximal set P m+1 contained in W m+1 with P m+1 ⊃ P m such that for any consecutive points x, y ∈ P m+1 , Proof. We start by making a simple remark. By design of B m+1 , for any two consecutive points x, y ∈ W m+1 , there exists w, u ∈ A * (r N ) such that g m+1 (x) ∈ K w , g m+1 (y) ∈ K u and K w ∩ K u = ∅. Hence To prove the lemma, it suffices (as W m+1 is finite) to construct a set P m+1 such that P m ⊂ P m+1 ⊂ W m+1 and P m+1 satisfies the conclusions of the lemma. The definition of P m+1 will be in an inductive manner. Set P (1) m+1 = P m . By the inductive hypothesis (P4), we have that |g m+1 (x)−g m+1 (y)| ∞ ≥ (2r)r m+1 for any two consecutive points x, y ∈ P (1) m+1 . Assume now that for some i ∈ N we have defined P (i) m+1 so that |g m+1 (x) − g m+1 (y)| ∞ ≥ (2r)r m+1 for any two consecutive points x, y ∈ P (i) m+1 . To define the next set P (i) m+1 , we consider two alternatives.
Define E m+1 to be the maximal collection of nondegenerate closed intervals in [0, 1] whose endpoints are consecutive points in the set P m+1 . If m+1 > N −t, set f m+1 := g m+1 . Otherwise, m + 1 ≤ N − t and to define f m+1 , we modify g m+1 on each I ∈ E m+1 like we did in the initial step.
Assume m + 1 ≤ N − t and let {I 1 , . . . , I q } be an enumeration of E m+1 . We start with I 1 . If g m+1 (I 1 ) traces a branch of T N with respect to R N (a, b) that contains all vertices of V N ∩ K w for some w ∈ A * (r m+t+1 ), then we set f m+1 |I 1 = g m+1 |I 1 . Suppose now that g m+1 (I 1 ) does not trace such a branch.
Lemma 5.12 (cf. Lemma 5.9). Let I 1 = [x, y], a = g m+1 (x) and b = g m+1 (y). Let {B 1 , . . . , B p } denote the branches of T N with respect to the road R N (a, b) that contain a set K w ∩ V N for some w ∈ A * (r m+t+1 ). Then there exist at most C indices j ∈ {1, . . . , p} for which B j has parts that are traced by g m+1 .
Proof. The branches of R N (a, b) with respect to g m+1 ([0, 1]) that are not in f m ([0, 1]) are branches that contain points in V m+1 . Therefore, by Lemma 5.8(1), there are at most C of them.
Since |a − b| ∞ > (2r)r m+1 and |a − z| ∞ ≤ (4/r)r m+1 for every vertex z of R N (a, b) in T N , we can invoke Lemma 5.8(3). In particular, there exist at least 2C + 2 branches of T N with respect to the road R N (a, b) such that for every branch there exists w ∈ A * (r m+t+1 ) such that all vertices of K w are in that branch. Fix such a branch B and define f m+1 |I 1 so that the following properties are satisfied.
(1) The map f m+1 |I 1 is piecewise linear and traces all the edges of B ∪ g m+1 (I 1 ) ⊂ T N . In fact, every edge of B is traced exactly twice. Moreover, for any edge e of B ∪ g m+1 (I 1 ) there exists J ⊂ I 1 such that f m+1 |I 1 maps J linearly onto e.
Lemma 5.13 (cf. Lemma 5.10). Let {B 1 , . . . , B p } be the branches of T N with respect to the road R N (a, b) that contain a set K w ∩ V N for some w ∈ A * (r t+m+1 ). There exist at most 2C + C + 1 indices j ∈ {1, . . . , p} for which B j has been traced by f m+1 |I 1 ∪ · · · ∪ I i .
Proof. There are two cases in which a branch B j has been traced by f 1 |I 1 ∪ · · · ∪ I i . The first case is when part of B j is already traced by by g m+1 (and hence f m+1 |I 1 ∪ · · · I i ). As in Lemma 5.12, at most C such branches exist.
The second case is when we are traveling on the road R N (a, b) backwards. Specifically, this case occurs when there exists i 1 ∈ {1, . . . , i} such that there is a part of g m+1 (I 1 ) lying on R N (a, b) and part of B j is being traced by f m+1 |I i 1 . There are three possible subcases: (1) the right endpoint of I i 1 is mapped by g m+1 into one of the branches of T N,1 with respect to R N (a, b) and by Lemma 5.8(1) at most C such branches exist; (2) the right endpoint of I i 1 is mapped onto the road R N (a, b) and by Lemma 5.8(2) at most C such points exist; and, (3) f m+1 |I i 1 contains a, and since f m+1 is essentially 2-to-1, at most one such interval exists.
For I i+1 we work exactly as with I 1 , but we choose a branch B that has not been traced by f m+1 |I 1 ∪ · · · ∪ I i . We can do so because by Lemma 5.8(3), there exist at least 2C + C + 2 such branches. Modifying g m+1 on each I i completes the definition of f m+1 .
With persistence, we have completed the proof of Lemma 5.7.

Bedford-McMullen carpets and self-affine sponges
Self-affine carpets were introduced and studied independently by Bedford [Bed84] and McMullen [McM84]. Fix integers 2 ≤ n 1 ≤ n 2 . For each pair of indices i ∈ {1, . . . , n 1 } and j ∈ {1, . . . , n 2 }, let φ i,j : R 2 → R 2 be the affine contraction given by For each nonempty set A ⊂ {1, . . . , n 1 } × {1, . . . , n 2 }, we associate the iterated function system F A = {φ i,j : (i, j) ∈ A} over R 2 and let S A denote the attractor of F A , called a Bedford-McMullen carpet. In general, we have S A ⊂ [0, 1] 2 . The following proposition serves as a brief overview of how the similarity dimension of F A compares to the Hausdorff, Minkowski, and Assouad dimensions of the carpet S A ; for definitions of these dimensions, we refer the reader to [McM84] and [Mac11].
(1) The similarity dimension is (2) [McM84] The Hausdorff dimension is The Minkowski dimension is dim M (S A ) = log n 1 r + log n 2 r −1 n 1 i=1 t i = log n 1 r + log n 2 (r −1 card A).
It is easy to see that for every Bedford-McMullen carpet, However, there is no universal comparison between the Assouad and similarity dimensions.  Our goal in this section is to establish the following statement, which encapsulates Theorem 1.4 from the introduction.
Theorem 6.2 (Hölder parameterization). Let 2 ≤ n 1 ≤ n 2 be integers and let A be as above. If S A is connected, then there exists a surjective (1/α)-Hölder map F : Furthermore, the exponent 1/α is sharp.
Note that the conclusion of Theorem 6.2 is trivial in the case that A ∈ {A 0 , . . . , A n 1 } or in the case that card A = 1. Below we give a proof of the sharpness of the exponent α, and in §6.2 we show why such a surjection exists. Lemma 6.3. If S A is connected and A ∈ {A 0 , . . . , A n 1 }, then there exists a pair of indices (i, j) ∈ A such that j < n 2 and (i, j + 1) ∈ A or such that j > 1 and (i, j − 1) ∈ A.
Proof. If card A ≥ 2, A ∈ {A 1 , . . . , A n 1 }, and the "first iteration" (i,j)∈A φ i,j ([0, 1] 2 ) does not touch the left or right edge, then the "second iteration" (i,j),(k,l)∈A φ i,j • φ j,k ([0, 1] 2 ) is disconnected. We leave the details as a useful exercise for the reader. It may help to visualize the diagrams in Figures 2 or 7. Corollary 6.5. Suppose that S A is connected, card A ≥ 2, and A ∈ {A 1 , . . . , A n 1 }. Then S A intersects both left and right edge of [0, 1] 2 .
We are ready to prove Theorem 6.2.
Proof of Theorem 6.2. With the conclusion being straightforward otherwise, let us assume that S A is a connected Bedford-McMullen carpet with card A ≥ 2 and A ∈ {A 0 , . . . , A n 1 }. Let s = s-dim F A . We defer the proof of existence of a (1/s)-Hölder parameterization of S A to §6.2, where we prove existence of Hölder parameterizations for self-affine sponges in R N (see Corollary 6.7). It remains to prove the sharpness of the exponent 1/s. Set k = card A and suppose that f : [0, 1] → S A is a (1/α)-Hölder surjection for some exponent α > 0. Since S A has positive diameter, the Hölder constant H := Höld 1/α f > 0. By Proposition 6.1, s-dim F A = log n 1 (k). Thus, we must show that α ≥ log n 1 k.
Fix m ∈ N and let A m , A * , and φ w be defined as in §2.3 relative to the alphabet {(i, j) : 1 ≤ i ≤ n 1 , 1 ≤ j ≤ n 2 }. For each m ∈ N and each word w = (i 1 , j 1 ) · · · (i m , j m ), set S w = φ w ([0, 1] 2 ). Let (i 0 , j 0 ) ∈ A be an index given by Lemma 6.3, i.e. an address in the first iterate such that the rectangle either immediately above or below is omitted from the carpet. Without loss of generality, we assume that j 0 < n 2 and (i 0 , j 0 + 1) ∈ A (there is no rectangle below (i 0 , j 0 )). Moreover, we assume that j 0 = min{j : (i 0 , j) ∈ A and (i 0 , j + 1) ∈ A}.
For each word w ∈ A m , we now define a "column of rectangles"S w , as follows.
Case 2.1. Suppose that u ∈ A m . Then, as in Case 1, setS w = j 0 j=0 S w(i 0 ,j) . Case 2.2. Suppose that u ∈ A and u(i 0 , n 2 ) ∈ A m+1 . Then we setS w = j 0 j=0 S w(i 0 ,j) . Case 2.3. Suppose that u ∈ A m and u(i 0 , n 2 ) ∈ A m+1 . Let j 1 = max{j : (i 0 , j −1) ∈ A}. Then we setS w = j 0 j=0 S w(i 0 ,j) ∪ n 2 j=j 1 S u(i 0 ,j) . In each case,S w ∩ S A is a connected set that intersects both the left and right edges of S w , but does not intersect the rectangles S u immediately above and belowS w . Moreover, the setsS w have mutually disjoint interiors. If τ w is the line segment joining the midpoints of upper and lower edges ofS w , then τ w contains a point of S A , which we denote by x w .
Consequently, there exists I w ⊂ [0, 1] such that f (I w ) is a curve inS w joining x w with one of the left/right edges ofS w . Clearly, the intervals I w are mutually disjoint and Since m is arbitrary, α ≥ log n 1 k.
Our strategy to parameterize a connected Bedford-McMullen carpet or self-affine sponge is to construct a Lipschitz lift of the set to a self-similar set in a metric space for which we can invoke Theorem 1.3. Then the Hölder parameterization of the self-similar set descends to a Hölder parameterization of the carpet or sponge.
Lemma 6.6 (Lipschitz lifts). Let N ≥ 2 be an integer, let 2 ≤ n 1 ≤ · · · ≤ n N be integers, and let A be a nonempty set as above. There exists a doubling metric d on R N such that if S A denotes the attractor of the IFS F A = {φ i : i ∈ A} over (R N , d), then (1) the identity map Id : S A → S A is a 1-Lipschitz homeomorphism; (2) s-dim F A = s-dim F A = log n 1 (card A) =: s, S A is self-similar, and H s ( S A ) > 0.
Proof. Consider the product metric d on R N given by d ((x 1 , . . . , x N ), (x 1 , . . . , x N )) = N i=1 |x i − x i | 2 log n i n 1 1/2 . In other words, d is a metric obtained by "snowflaking" the Euclidean metric separately in each coordinate. Note that if n 1 = · · · = n N , then d is the Euclidean metric. It is straightforward to check that (R N , d) is a doubling metric space and the identity map Id : (S A , d) → S A is a 1-Lipschitz homeomorphism; e.g. see Heinonen [Hei01]. We now claim that the affine contractions φ i generating the sponge S A become similarities in the metric space (R N , d). Indeed, let i = (i 1 , . . . , i N ) ∈ A. Then d(φ i (x 1 , . . . , x N ), φ i (x 1 , . . . , x N )) = N i=1 n −2 log n i n 1 i |x i − x i | 2 log n i n 1 1/2 = n −1 1 d((x 1 , . . . , x N ), (x 1 , . . . , x N )).
Since each of the similarities φ i have scaling factor n −1 1 , it follows that s-dim( F A ) = s-dim F A = log n 1 (card A) =: s Finally, F A satisfies the strong open set condition (SOSC) with U = (0, 1) N . Therefore, H s ( S A ) > 0 by Theorem 2.3, since doubling metric spaces are β-spaces.
Corollary 6.7. If S A is a connected self-affine sponge in R N , then S A is a (1/s)-Hölder curve, where s = log n 1 (card A) is the similarity dimension of F A .
Proof. Let S A denote the lift of the sponge S A in Euclidean space R N to the metric space (R N , d) given by Lemma 6.6. By Lemma 6.6 (2), the lifted sponge S A is a self-similar set and H s ( S A ) > 0, where s = s-dim F A = s-dim F A = log n 1 (card A). By Remes' theorem in metric spaces (Theorem 1.3), there exists a (1/s)-Hölder surjection F : [0, 1] → S A . By Lemma 6.6 (1), the identity map Id : S A → S A is a Lipschitz homeomorphism. Therefore, the composition G = [0, 1] → S A , G := Id • F is a (1/s)-Hölder surjection.