Monatshefte für Mathematik

, Volume 180, Issue 4, pp 693–712 | Cite as

Peano curves with smooth footprints

Article

Abstract

We construct Peano curves \(\gamma : [0,\infty ) \rightarrow \mathbb {R}^2\) whose “footprints” \(\gamma ([0,t])\), \(t>0\), have \(C^\infty \) boundaries and are tangent to a common continuous line field on the punctured plane \(\mathbb {R}^2 {\backslash }\{\gamma (0)\}\). Moreover, these boundaries can be taken \(C^\infty \)-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.

Keywords

Space-filling curves Smoothness Convexity Line fields 

Mathematics Subject Classification

26A30 26E10 

1 Introduction

A continuous map \(\gamma : I \rightarrow \mathbb {R}^2\) defined on a nondegenerate interval \(I \subseteq \mathbb {R}\) is called a Peano curve if its image has nonempty interior. A lot of water has run under the bridge since Peano established the existence of such curves in 1890. Many interesting problems concerning such curves are discussed in the book [7] by Sagan.

By Sard’s theorem, Peano curves are non-differentiable. Nevertheless, they can have smooth “footprints”, as a consequence of our main result:

Theorem

There exist a Peano curve \(\gamma : [0,\infty ) \rightarrow \mathbb {R}^2\) and a continuous line field \(\Lambda \) on the punctured plane \(\mathbb {R}^2 {\backslash }\{\gamma (0)\}\) such that for every \(t>0\), the boundary of the set \(\gamma ([0,t])\) is a \(C^\infty \) curve \(C_t\) containing the point \(\gamma (t)\) and tangent to the line field \(\Lambda \) at each point.

Moreover, it is possible to choose the Peano curve \(\gamma \) so that each curve \(C_t\) is \(C^\infty \)-close to the circle \(x^2+y^2=t^2\).

Let us state the “moreover” part formally: Given any upper semicontinuous function \(k : (0, \infty ) \rightarrow \mathbb {N}\) and any lower semicontinuous function \(\varepsilon : (0,\infty ) \rightarrow (0,\infty )\), we can choose the Peano curve \(\gamma \) in the theorem with the following additional property: for each \(t>0\) the curve \(C_t = \partial \gamma ([0,t])\) is the image of a \(C^\infty \) embedding \(\beta _t\) of the circle \(\mathbb {T}:= \mathbb {R}/ 2\pi \mathbb {Z}\) into \(\mathbb {R}^2\) such that
$$\begin{aligned} \Vert \beta _t - \alpha _t \Vert _{k(t)} < \varepsilon (t), \end{aligned}$$
(1.1)
where \(\alpha _t : \mathbb {T}\rightarrow \mathbb {R}^2\) is the embedding \(\theta \mapsto (t \cos \theta , t \sin \theta )\) and \(\Vert \mathord {\cdot } \Vert _k\) is the usual \(C^k\) norm; see Sect. 2.1 for details.

Taking \(k \equiv 2\) and a sufficiently small function \(\varepsilon \), we can ensure that each curve \(C_t\) has everywhere nonzero curvature, and so we obtain:

Corollary

There exist Peano curves \(\gamma : [0,1] \rightarrow \mathbb {R}^2\) such that each set \(\gamma ([0,t])\) is convex.

This result was first obtained by Pach and Rogers1 in [6], and independently by Vince and Wilson in [10]. It is inspired by the following question, attributed by Pach and Rogers to Mihalik and Wieczorek (see also [3, Problem A.37]):

Question

Is there a Peano curve \(\gamma :I \rightarrow \mathbb {R}^2\) such that the image \(\gamma (J)\) of each subinterval \(J \subseteq I\) is a convex set?

To our knowledge, this question remains open. See [9] and [8, Chapter 6] for related information.

Coming back to our theorem, let us observe that the family of concentric circles can be replaced by an arbitrary smooth family of nested smooth Jordan curves contracting to a point. Indeed, it suffices to change coordinates by a suitable diffeomorphism of \(\mathbb {R}^2\).

Despite the fact that the boundaries of the “footprints” \(\gamma ([0,t])\) are smooth, the line field \(\Lambda \) is not. Indeed, \(\Lambda \) is not even locally Lipschitz, because otherwise it would be uniquely integrable. It is also known that generic (in the sense of Baire) continuous line fields are uniquely integrable (see [2, pp. 121–123]), which shows that \(\Lambda \) is quite pathological. Other highly non-uniquely integrable line fields are constructed in [1]; these are tangent to uncountably many \(C^k\)-foliations (where \(1 \leqslant k <\infty \)) and have the additional property of being Hölder continuous. It would be interesting to find what the optimal moduli of regularity of the line field \(\Lambda \) and of the Peano curve \(\gamma \) in our Theorem are—in particular, it is not clear whether they can be taken locally Hölder continuous on \(\mathbb {R}^2 {\backslash }\{\gamma (0)\}\) and \((0,\infty )\), respectively. Let us remark that the optimal Hölder coefficient of a general Peano curve is 1 / 2 (see e.g. [4, Prop. 2.3]).

It seems that it should be possible to extend the theorem to an arbitrary dimension \(n \geqslant 2\), so that \(\partial \gamma ([0,t])\) is a \(C^\infty \) hypersurface \(C^\infty \)-close to a sphere, and tangent to a continuous field of hyperplanes. Such a construction should follow the same ideas as the \(n=2\) case, but since it would be considerably more technical, we will not dwell on it.

This paper is divided into two parts: the longer part, Sect. 2, is devoted to the proof of a local, more flexible version of the theorem, namely Proposition 2.3. In the shorter part, Sect. 3, we “glue” these local constructions in order to prove the theorem.

This paper is based on the master’s dissertation [5] of the second named author, which is, in turn, inspired by ideas from [6, 10]. We thank the dissertation committee, especially Prof. Ricardo Sá Earp who posed questions that led to the improvement of the results of the dissertation, presented here. We also thank the referees for a number of valuable suggestions and corrections.

2 Local construction

2.1 Initial definitions and statement of the main proposition

The aim of this section is to prove Proposition 2.3 below, which constructs special Peano curves whose footprints are lunes, objects that are defined as follows:

Definition 2.1

Let \(f,g:[a,b]\rightarrow \mathbb {R}\) be \(C^\infty \) functions such that:
  1. (i)

    f and g and each of their derivatives coincide at a and at b, that is, \(f^{(k)}(a) = g^{(k)}(a)\) and \(f^{(k)}(b) = g^{(k)}(b)\) for all integers \(k \geqslant 0\);

     
  2. (ii)

    \(f(x) \leqslant g(x)\), for all \(x \in [a,b]\).

     
The plane region
$$\begin{aligned} L = L(f,g) = \{(x,y) \mid a \leqslant x \leqslant b, \ f(x) \leqslant y \leqslant g(x)\} \end{aligned}$$
is called a lune with domain [ab]. The support of a lune L(fg) is the (open) set
$$\begin{aligned} S_L = \{x \in [a,b] \mid f(x) < g(x)\}. \end{aligned}$$
A lune L is said to be simple if its support is a nonempty interval. The essential part of the lune L is defined as the closure of the interior of L:
$$\begin{aligned} L^{{\text {ess}}} = L^{{\text {ess}}}(f,g) = \overline{{{\mathrm{int}}}(L)}. \end{aligned}$$
Figure 1 shows an example of a simple lune.
Fig. 1

A typical lune in the construction

Lemma 2.2

For a simple lune L whose support is \(S_L = (c,d)\), we have
$$\begin{aligned} L^{{\text {ess}}} = \{(x,y) \in L \mid c \leqslant x \leqslant d \}. \end{aligned}$$
In particular, \(L^{{\text {ess}}}\) is itself a lune, and it is simple.

Proof

If \(L = L(f,g)\) has support \(S_L = (c,d)\) then its interior is clearly
$$\begin{aligned} \{(x,y) \mid c < x < d, f(x) < y < g(x)\}, \end{aligned}$$
so that \(\{(x,y) \in L \mid c \leqslant x \leqslant d \} = \overline{{{\mathrm{int}}}(L)} = L^{{\text {ess}}}\). \(\square \)
We let \(C^\infty ([a,b])\) denote the set of all \(C^\infty \) functions \(F: [a,b] \rightarrow \mathbb {R}\). The \(C^0\)norm of \(F \in C^\infty ([a,b])\) is \(\Vert F\Vert _0 = \sup _{x \in [a,b]} |F(x)|\). For \(k \in \mathbb {N}\), the \(C^k\)norm of F is
$$\begin{aligned} \Vert F\Vert _k = \max \left( \Vert F\Vert _0, \Vert F'\Vert _0, \ldots , \Vert F^{(k)}\Vert _0\right) . \end{aligned}$$
A basic neighborhood of F is a set of the form
$$\begin{aligned} N(F,k,\varepsilon ) = \big \{ G \in C^\infty ([a,b]) \mid \Vert G-F\Vert _k < \varepsilon \big \}. \end{aligned}$$
We endow the space \(C^\infty ([a,b])\) with the topology generated by the basic neighborhoods, called the \(C^\infty \)topology.

Later on we will work with the space \(C^\infty (\mathbb {T})\) of functions on the circle \(\mathbb {T}= \mathbb {R}/2\pi \mathbb {Z}\), which can be considered as \(2\pi \)-periodic functions on the line. The \(C^k\) norms and the \(C^\infty \) topology on this space are defined analogously.

If \(F,G \in C^\infty ([a,b])\) (or \(C^\infty (\mathbb {T})\)) are such that \(F(x) \leqslant G(x)\) for all x, we write \(F \leqslant G\).

We now state our main technical proposition:

Proposition 2.3

Let \(L = L(f,g)\) be a lune with domain [ab] and support (ab). Then there exist:
  • a Peano curve \(\gamma :[0,1] \rightarrow L\);

  • a continuous map \(t \in [0,1] \mapsto F_t \in C^\infty ([a,b])\);

  • a continuous function \(\psi : L \rightarrow \mathbb {R}\);

with the following properties:
  1. (i)

    \(F_0 = f\), \(F_1 = g\);

     
  2. (ii)

    If \(t \leqslant s\) then \(F_t \leqslant F_s\);

     
  3. (iii)

    Writing \(\gamma (t) = (x(t),y(t))\), we have \(y(t) = F_t(x(t))\).

     
  4. (iv)

    \(F'_t(x) = \psi (x,F_t(x))\);

     
  5. (v)

    for each \(t \in (0,1]\) we have \(\gamma ([0,t]) = L^{{\text {ess}}}(f,F_t)\);

     
  6. (vi)

    \(\gamma (0) = (a,f(a))\), \(\gamma (1) = (b, f(b))\).

     

Due to property (iii), the functions \(F_t\) are called ceiling functions. By property (iv), their graphs are tangent to the line field \(\Lambda (x,y)\) spanned by the vector field \((1,\psi (x,y))\). Note that for each \(t \in (0,1]\), the point \(\gamma (t)\) belongs to the boundary of \(\gamma ([0,t])\). Also, this boundary is everywhere tangent to the line field \(\Lambda \), except for the two extreme points where it is not differentiable. Finally, note that \(\gamma ([0,1]) = L^{{\text {ess}}}\).

Our construction actually yields simple lunes \(L^{{\text {ess}}}(f,F_t)\) for all \(t \in (0,1]\), but since this fact is not needed we will not justify it.

2.2 Lune subdivision processes

The proof of Proposition 2.3 involves a limiting process on a sequence of subdivisions of the original lune. The basic subdivision processes on a lune are described here.

Throughout the remainder of this section, fix a \(C^\infty \) function \(\varphi :\mathbb {R}\rightarrow \mathbb {R}\) with the following properties:
  1. (i)

    \(0 \leqslant \varphi (x) \leqslant 1\) for all \(x \in \mathbb {R}\);

     
  2. (ii)

    \(\varphi ^{-1}(0) = (-\infty , 1/3]\);

     
  3. (iii)

    \(\varphi ^{-1}(1) = [2/3,\infty ).\)

     
For any a, \(b \in \mathbb {R}, a< b\), let \(\varphi _{a,b}(x) = \varphi \left( \frac{x-a}{b-a}\right) \). Note that \(\varphi _{0,1} = \varphi \), \(\varphi _{a,b}^{-1}(0) = \left( -\infty ,\frac{2a+b}{3}\right] \) and \(\varphi _{a,b}^{-1}(1) = \left[ \frac{a+2b}{3}, +\infty \right) \). Also, clearly
$$\begin{aligned} \sup _{x \in \mathbb {R}} |\varphi _{a,b}^{(k)}(x)| = \sup _{x \in [a,b]} |\varphi _{a,b}^{(k)}(x)|, \end{aligned}$$
thus \(\Vert \varphi _{a,b}\Vert _k\) is defined for all k by taking the \(C^k\) norm of \(\varphi _{a,b}\) restricted to [ab]. Moreover, since \(\varphi _{a,b}^{(k)}(x) = \frac{1}{(b-a)^k} \cdot \varphi ^{(k)}\left( \frac{x-a}{b-a}\right) ,\) it follows that
$$\begin{aligned} \Vert \varphi _{a,b}\Vert _k = \max _{i \leqslant k}\frac{\Vert \varphi ^{(i)}\Vert _0}{(b-a)^i}. \end{aligned}$$
Thus,
$$\begin{aligned} \Vert \varphi _{a,b}\Vert _k \leqslant \max (1,(b-a)^{-k})\Vert \varphi \Vert _k. \end{aligned}$$
The \(C^k\) norm of a lune \(L = L(f,g)\) is defined as \(\Vert L\Vert _k = \Vert g - f\Vert _k\).

We have now set the stage for the definitions of the two basic subdivision processes:

Definition 2.4

(Slicing) Let \(L = L(f,g)\) be a simple lune, and let \(n \in \mathbb {N}\). For \(j = 0,1, \ldots , n\) set \(h_j = \left( 1 - \frac{j}{n}\right) f + \frac{j}{n} g\), and for \(i = 1,2, \ldots , n\) set \(L_i = L(h_{i-1}, h_i).\) The set of lunes \(\{L_1, L_2, \ldots ,L_n\}\) is called the n-slicing of L.

Proposition 2.5

The n-slicing \(\{L_1, \ldots , L_n\}\) of a simple lune L has the following properties:
  1. (a)

    \(S_{L_i} = S_L\) and, in particular, \(L_i\) is a simple lune, for each \(i \in \{1,2,\ldots , n\}\);

     
  2. (b)

    \(L = L_1 \cup L_2 \cup \cdots \cup L_{n}\) and \(L^{{\text {ess}}} = L_1^{{\text {ess}}} \cup L_2^{{\text {ess}}} \cup \cdots \cup L_{n}^{{\text {ess}}}\);

     
  3. (c)

    \(\Vert L_i\Vert _k = \frac{1}{n} \Vert L\Vert _k\) for each \(i \in \{1,\ldots , n\}\), and each \(k \in \mathbb {N}\).

     

Proof

Properties (a) and (b) are straightforward from the definition. Property (c) follows from the fact that \(\Vert L_i\Vert _k = \Vert h_{i+1} - h_i\Vert _k = \frac{1}{n}\Vert g - f\Vert _k\). \(\square \)

The second basic subdivision process is defined as follows:

Definition 2.6

(Bipartitioning) Let \(L=L(f,g)\) be a simple lune defined on [0, 1] with \(S_L = (a,b)\), and set \(h(x) = (1 - \varphi _{a,b}(x))g(x) + \varphi _{a,b}(x)f(x)\) for \(0 \leqslant x \le 1\). The pair of lunes \(\{L(f,h),L(h,g)\}\) is called the bipartition of L.

Proposition 2.7

The bipartition \(\{L_1, L_2\}\) of a lune L has the following properties:
  1. (a)

    \(S_{L_1} = \left( a, \frac{a+2b}{3}\right) \) and \(S_{L_2} = \left( \frac{2a+b}{3}, b\right) \); in particular, \(L_1\) and \(L_2\) are simple lunes;

     
  2. (b)

    \(L = L_1 \cup L_2\) and \(L^{{\text {ess}}} = L_1^{{\text {ess}}} \cup L_2^{{\text {ess}}}\);

     
  3. (c)

    \(\max (\Vert L_1\Vert _k,\Vert L_2\Vert _k) \leqslant 2^k\Vert \varphi _{a,b}\Vert _k\Vert L\Vert _k\), \(\forall k \in \mathbb {N}\).

     

Proof

Properties (a) and (b) are straightforward from the definition. As for (c), note that
$$\begin{aligned} h^{(k)}(x) = g^{(k)}(x) + \sum _{i=0}^k {\left( {\begin{array}{c}k\\ i\end{array}}\right) }\varphi _{a,b}^{(k-i)}(x)(f - g)^{(i)}(x), \end{aligned}$$
so that
$$\begin{aligned} \Vert h^{(k)} - g^{(k)}\Vert _{0} \leqslant 2^k \Vert \varphi _{a,b}\Vert _{k} \Vert f - g\Vert _{k} \end{aligned}$$
and
$$\begin{aligned} \Vert L_2\Vert _k \leqslant \max _{i \leqslant k} (2^i \Vert \varphi _{a,b}\Vert _{i} \Vert L\Vert _{i}) = 2^k \Vert \varphi _{a,b}\Vert _{k} \Vert L\Vert _{k} \, . \end{aligned}$$
The estimate of \(\Vert L_1\Vert _k\) is analogous. \(\square \)

Note that slicing decreases the norm while leaving the support unchanged, whereas bipartitioning reduces the size of the support while increasing the norm. By combining both processes, but slicing into many slices, we can make both the norms of the lunes and the sizes of their supports arbitrarily small. We formalize this idea in the next subsection.

2.3 A family of lunes

Let us begin the proof of Proposition 2.3. Let a simple lune \(L=L(f,g)\) with domain [ab] be given. Clearly it is sufficient to consider the case where \(S_L=(a,b)\). By rescaling if necessary, we can assume that \(a=0\) and \(b=1\). The functions f and g will be fixed for the remainder of this section.

Our recursive construction is based on the following definition:

Definition 2.8

Let \(\{m_j\}_{j \geqslant 1}\) be a sequence of positive integers. The set of words with respect to the sequence \(\{m_j\}\) is the set
$$\begin{aligned} \Omega = \{\omega = (i_1,i_2,\ldots , i_n) \mid n \in \mathbb {N}, \ i_j \in \{1,2, \ldots , m_j\}\quad \text { for } j=1,2,\ldots ,n\}, \end{aligned}$$
and its elements are called words with respect to\(\{m_j\}_{j \geqslant 1}\), or simply words. The length of a word \(\omega = (i_1, \ldots , i_n)\) is denoted as \(|\omega |\) and equals n.
The set of words with successor with respect to \(\{m_j\}\) is
$$\begin{aligned} \Omega ^*= \{\omega = (i_1,i_2,\ldots , i_n) \in \Omega \mid i_n < m_n \}. \end{aligned}$$
The successor of a word \(\omega = (i_1, \ldots , i_n) \in \Omega ^*\) is the word \(\omega ^+ = (i_1, \ldots , i_{n-1}, i_n+1)\).
Finally, if \(\omega _0 = (i_1, \ldots , i_k)\) and \(\omega _1 = (j_1, \ldots , j_n)\) are two words, the word
$$\begin{aligned} {\omega _0}*{\omega _1} = (i_1, \ldots , i_k, j_1, \ldots j_n) \end{aligned}$$
of length \(k + n\) is the concatenation of \(\omega _0\) and \(\omega _1\).

Our first goal is to recursively define both a sequence \(\{m_j\}_{j \geqslant 1}\) of integers and a family \(\{L_\omega \}\) of lunes indexed by words with respect to this sequence.

Let us fix a sequence of positive real numbers \(\{\varepsilon _n\}_{n \geqslant 2}\) such that
$$\begin{aligned} \sum \varepsilon _n < \infty \, . \end{aligned}$$
Let \(m_1 = 1\) and \(L_{(1)} = L\). For \(k \geqslant 2\):
Step 0.

Assume we know the values of \(m_1, m_2 \ldots , m_{k-1}\) and that \(L_\omega \) is defined for every word \(\omega \) of length \(k-1\) with respect to the sequence \(\{m_j\}_{j \geqslant 1}\) (although this sequence is not yet fully defined, the set of words of length \(k-1\) depends only on the \(k-1\) first integers in the sequence).

Step 1.
Pick \(n_k\) sufficiently large such that for every word \(\omega \) of length \(k-1\), the lunes generated in the \(n_k\)-slicing of \(L_\omega \), \(\left\{ L_\omega ^1, L_\omega ^2, \ldots , L_\omega ^{n_k}\right\} \), satisfy:
$$\begin{aligned} \Vert L_\omega ^j\Vert _k < \frac{\varepsilon _{k}}{2^{k} \Vert \varphi _{a,b}\Vert _{k}}, \quad j = 1, \ldots , n_k, \end{aligned}$$
(2.1)
where \(S_{L_\omega } = (a,b)\) [property (c) in Proposition 2.5 allows us to pick such an \(n_k\)]. Let \(m_k = 2 n_k\).
Step 2.
For every \(\omega \) of length \(k-1\), consider the lunes \(\left\{ L_\omega ^1, \ldots , L_\omega ^{n_k}\right\} \) generated in the \(n_k\)-slicing of \(L_\omega \). For \(j = 1, 2, \ldots , n_k\), let \(\{L_\omega ^{j,1},L_\omega ^{j,2}\}\) be the bipartition of \(L_\omega ^j\), and set
$$\begin{aligned} L_{{\omega }*{(2j-1)}} = L_\omega ^{j,1} \quad \text { and }\quad L_{{\omega }*{(2j)}} = L_\omega ^{j,2} . \end{aligned}$$
Thus we have defined \(L_{\omega '}\) for every word \(\omega '\) of length k. Moreover, by Proposition 2.7 and by inequality (2.1), \(\Vert L_{\omega '}\Vert _k \leqslant \varepsilon _k\) whenever \(|\omega '| = k\). Increment k and go to Step 0.
Fig. 2

Illustration of two recursion steps for the subdivision of a lune. At the end of the recursion step for \(k=2\), we have a total of 6 lunes, \(L_{(1,1)}, L_{(1,2)}, \ldots , L_{(1,6)}\). On the other hand, at the end of the recursion for \(k = 3\) we have a total of 24 lunes, ranging from \(L_{(1,1,1)}\) up to \(L_{(1,6,4)}\); note that \(L_{(1,j,k)}\) is always the result of a subdivision of \(L_{(1,j)}\). a A lune is first 3- sliced (here, schematically, \(n_{2}=3\)). b Each slice is bipartitioned, completing the recursion for \(k=2\). c Then, assuming \(n_{3}=2\), each of the previous lunes is 2-sliced. d Once again, the slices are bipartitioned, and the recursion with \(k=3\) is complete

This subdivision process is illustrated by Fig. 2. As a result of this construction, we obtain both a sequence \(\{m_j\}_{j \geqslant 1}\) and a family of lunes \(\{L_\omega \}_{\omega \in \Omega }\) indexed by words with respect to the aforementioned sequence. In what follows, let \(L_\omega = L(f_\omega , g_\omega )\) for all \(\omega \in \Omega \).

Remark 2.9

The following properties hold for the family \(\{L_\omega \}_{\omega \in \Omega }\) of lunes:
  1. (a)

    \(\Vert L_\omega \Vert _{|\omega |} \leqslant \varepsilon _{|\omega |}\);

     
  2. (b)

    \(f_{(1)} = f\), \(g_{(1)} = g\);

     
  3. (c)

    \(f_{{\omega }*{(1)}} = f_\omega \), \(g_{{\omega }*{(m_{|\omega |+1})}} = g_\omega \);

     
  4. (d)

    \(g_{{\omega }*{ (i)}} = f_{{\omega }*{ (i+1)}}\) for \(i=1,\ldots , m_{|\omega |+1} - 1\). In other words, \(g_\omega = f_{\omega ^+}\) for all \(\omega \in \Omega ^*\).

     

Lemma 2.10

If a word \(\omega \in \Omega \) has length n, then for each \(1 \leqslant \ell \leqslant m_{n}\),
$$\begin{aligned} \Vert f_{{\omega }*{(\ell )}} - f_\omega \Vert _{n} < \varepsilon _{n+1} + \varepsilon _n \end{aligned}$$

Proof

If \(\ell \) is odd, then \(f_{{\omega }*{(\ell )}}\) appeared after a slicing of \(L_\omega \), hence
$$\begin{aligned} \Vert f_{{\omega }*{(\ell )}} - f_\omega \Vert _n \leqslant \Vert L_\omega \Vert _n < \varepsilon _n. \end{aligned}$$
If \(\ell \) is even, then
$$\begin{aligned} \Vert f_{{\omega }*{(\ell )}} - f_\omega \Vert _n&\leqslant \Vert f_{{\omega }*{(\ell )}} - f_{{\omega }*{(\ell -1)}}\Vert _n + \Vert f_{{\omega }*{(\ell -1)}} - f_\omega \Vert _n \\&\leqslant \Vert g_{{\omega }*{(\ell -1)}} - f_{{\omega }*{(\ell -1)}}\Vert _n + \Vert f_{{\omega }*{(\ell -1)}} - f_\omega \Vert _n \\&\leqslant \Vert L_{{\omega }*{(\ell -1)}}\Vert _{n+1} + \Vert f_{{\omega }*{(\ell -1)}} - f_\omega \Vert _n \\&< \varepsilon _{n+1} + \varepsilon _n. \end{aligned}$$
\(\square \)

From this point on, we will assume that from an initial lune \(L = L(f,g)\) and a summable positive sequence \(\{\varepsilon _n\}_{n \geqslant 2}\), we have obtained, through the procedure described here, a set of words \(\Omega \) with respect to a sequence \(\{m_k\}_{k \geqslant 1}\), and a family of lunes \(\{L_\omega \}_{\omega \in \Omega }\), with all the properties that were mentioned.

2.4 A helpful Cantor set

Now that we have described the basic subdivision processes, we may begin to describe some auxiliary constructions that play an important part in the definition of a Peano curve (with some special properties) that will cover the initial lune L.

First, we will define a Cantor set K (Fig. 3) through a family of closed intervals \(\{J_\omega \}_{\omega \in \Omega }\) indexed by words with respect to \(\{m_k\}_{k \geqslant 1}\). The open intervals that will be removed from [0, 1], \(\{G_\omega \}_{\omega \in \Omega ^*}\), indexed by words with successor, will also play an important role.

First, set \(J_{(1)} = [0,1]\). For each \(k \geqslant 2\), assume that \(J_\omega \) has been defined for all \(\omega \in \Omega \) with \(|\omega | = k-1\). For each such \(\omega \), if \(J_\omega = [\alpha , \beta ]\) set
$$\begin{aligned} J_{{\omega }*{(\ell )}} = \left[ \alpha + \frac{2\ell -2}{2 m_k - 1}(\beta - \alpha ),\alpha + \frac{2\ell -1}{2 m_k - 1}(\beta - \alpha )\right]&, \quad 1 \leqslant \ell \leqslant m_k \, ,\\ G_{{\omega }*{(\ell )}} = \left( \alpha + \frac{2\ell -1}{2 m_k - 1}(\beta - \alpha ),\alpha + \frac{2\ell }{2 m_k - 1}(\beta - \alpha )\right)&, \quad 1 \leqslant \ell \leqslant m_k . \end{aligned}$$
Now set
$$\begin{aligned} K = \bigcap _{n \in \mathbb {N}} \, \bigcup _{\begin{array}{c} \omega \in \Omega ,\\ |\omega | = n \end{array}} J_\omega . \end{aligned}$$
Note that K is a Cantor set, and \([0,1] {\backslash }K = \bigcup _{\omega \in \Omega ^*} G_\omega \). The significance of this set K will become apparent later on, but the basic idea is as follows: the curve \(\gamma \) that we construct in this section will be such that \(\gamma (J_\omega ) = L_\omega ^{{\text {ess}}}\) (see Definition 2.1). However, usually \(\gamma (\sup J_\omega ) \ne \gamma (\inf J_{\omega ^+})\), so we connect these “subcurves” using \(G_\omega \).
Fig. 3

Illustration of the first few steps in the construction of K. Here \(m_2 = 4\) and \(m_3 = 2\)

2.5 The family of ceiling functions

A point \(t \in [0,1]\) belongs to the Cantor set K if and only if there exists a sequence \(\{\omega _n\}_{n \geqslant 1}\) of words with \(\omega _1 = (1)\), \(\omega _{n+1} = {\omega _n}*{(\ell )}\) for some \(\ell \in \mathbb {N}\) and such that \(t \in J_{\omega _n}\) for each n. Moreover, this sequence is unique, and we call it the defining sequence oftinK. For what follows, recall the notation that \(L_\omega = L(f_\omega , g_\omega )\).

Lemma 2.11

Let \(t \in K\) and let \(\{\omega _n\}_{n \geqslant 1}\) be the defining sequence of t in K. Then, for each \(k \geqslant 0\), the sequences of functions \(\{f_{\omega _n}^{(k)}\}_n\) and \(\{g_{\omega _n}^{(k)}\}_n\) are uniformly Cauchy and
$$\begin{aligned} \lim _{n \rightarrow \infty } f_{\omega _n}^{(k)} = \lim _{n \rightarrow \infty } g_{\omega _n}^{(k)}. \end{aligned}$$

Proof

Using Lemma 2.10 and noting that \(|\omega _n| = n\), we see that whenever \(n \geqslant k\) we have
$$\begin{aligned} \Vert f_{\omega _{n+1}} - f_{\omega _n}\Vert _k \leqslant \Vert f_{\omega _{n+1}} - f_{\omega _n}\Vert _n < \varepsilon _{n+1} + \varepsilon _n. \end{aligned}$$
If \(\varepsilon > 0\) is fixed and \(N \geqslant k\) is such that \(\sum _{n \geqslant N} \varepsilon _n < \frac{\varepsilon }{2}\), then if \(m > n > N\),
$$\begin{aligned} \Vert f_{\omega _{m}} - f_{\omega _n}\Vert _k \leqslant \sum _{j=n}^{m-1} \Vert f_{\omega _{j+1}} - f_{\omega _j}\Vert _k < 2 \sum _{j=n}^{m} \varepsilon _j < \varepsilon . \end{aligned}$$
Since \(\Vert h^{(k)}\Vert _0 \leqslant \Vert h\Vert _k\), it follows that \(\{f_{\omega _n}^{(k)}\}_n\) is a uniformly Cauchy sequence. Note that
$$\begin{aligned} \Vert g_{\omega _n}^{(k)} - f_{\omega _n}^{(k)}\Vert _0 \leqslant \Vert L_{\omega _n}\Vert _k < \varepsilon _n \, . \end{aligned}$$
Hence, the sequence \(\{g_{\omega _n}^{(k)}\}_n\) is also uniformly Cauchy, and both sequences have the same limit. \(\square \)

Definition 2.12

For \(t \in K\), let \(\{\omega _n\}_{n \geqslant 1}\) be the defining sequence of t in K. The ceiling function att is
$$\begin{aligned} F_t = \lim _{n \rightarrow \infty } f_{\omega _n} = \lim _{n \rightarrow \infty } g_{\omega _n}. \end{aligned}$$

A direct consequence of Lemma 2.11 is that, for each fixed t, \(F_t\) is a \(C^\infty \) function and \(F_t^{(k)} = \lim _{n \rightarrow \infty } f_{\omega _n}^{(k)} = \lim _{n \rightarrow \infty } g_{\omega _n}^{(k)}.\)

Lemma 2.13

If t, \(s \in K\), and \(t \leqslant s\), then \(F_t \leqslant F_s\).

Proof

Let \(\{\omega _{t,n}\}_{n \geqslant 1}\), \(\{\omega _{s,n}\}_{n \geqslant 1}\) be the defining sequences of t and s in K, respectively. Note that \(t < s\) if and only if for some \(N > 1\), \(\omega _{t,k} = \omega _{s,k}\) whenever \(k < N\) but \(\omega _{t,N} = {\omega _{t,N-1}}*{(j)}, \omega _{s,N} = {\omega _{s,N-1}}*{(\ell )}\) with \(j < \ell \). Now clearly, for each \(n > N\),
$$\begin{aligned} f_{\omega _{t,n}} \leqslant g_{\omega _{t,N}} \leqslant f_{\omega _{s,N}} \leqslant f_{\omega _{s,n}}, \end{aligned}$$
because \(L_{\omega _{t,n}}\) is a subdivision of \(L_{\omega _{t,N}}\). \(\square \)

Proposition 2.14

The function \(t \in K \mapsto F_t \in C^\infty ([0,1])\) is continuous.

Proof

Let \(k \in \mathbb {N}\), fix \(\varepsilon > 0\), and let \(N > k\) be such that
$$\begin{aligned} \sum _{n=N}^\infty \varepsilon _n < \frac{\varepsilon }{4}. \end{aligned}$$
Recall that the length of the gap \(G_\omega \) is the same for every word \(\omega \) such that \(|\omega | = N\); call this length \(\delta _N\). If t, \(s \in K\), with \(|t-s| < \delta _N\), then clearly t, \(s \in J_\omega \) for some \(\omega \) with \(|\omega | = N\). Let \(\omega _{t,n}\), \(\omega _{s,n}\) be the defining sequences of t and s in K (note that \(\omega _{t,N} = \omega _{s,N} = \omega \)). Since by Lemma 2.10
$$\begin{aligned} \Vert f_{\omega _{t,n}} - f_{\omega }\Vert _k \leqslant \sum _{j=N}^{n-1} \Vert f_{\omega _{t,j+1}} - f_{\omega _{t,j}}\Vert _k < 2 \sum _{j=N}^{n-1} \varepsilon _j \end{aligned}$$
whenever \(n > N\), it follows that by making \(n \rightarrow \infty \) we have
$$\begin{aligned} \Vert F_t - f_\omega \Vert _k \leqslant 2 \sum _{j=N}^\infty \varepsilon _j < \frac{\varepsilon }{2} \end{aligned}$$
and, analogously, \(\Vert F_s - f_\omega \Vert _k < \frac{\varepsilon }{2}.\) Therefore,
$$\begin{aligned} \Vert F_t - F_s\Vert _k \leqslant \Vert F_t - f_\omega \Vert _k + \Vert f_\omega - F_s \Vert _k < \varepsilon ,\end{aligned}$$
which completes the proof. \(\square \)

Lemma 2.15

Given \(\omega \in \Omega ^*\), suppose \(G_\omega = (\alpha ,\beta )\). Then \(\alpha \), \(\beta \in K\) and \(F_\alpha = F_\beta \).

Proof

Suppose \(|\omega | = N\). By the definition of K, it is clear that \(\alpha \in J_\omega \), \(\beta \in J_{\omega ^+}.\) In fact, for each \(n > N\) we have
$$\begin{aligned} \alpha \in J_{{\omega }*{(m_{N+1},m_{N+2}, \ldots , m_n)}}, \beta \in J_{{\omega ^+}*{\underbrace{\scriptstyle (0,0, \ldots , 0)}_{n-N}}}.\ \end{aligned}$$
Since
$$\begin{aligned} g_{{\omega }*{(m_{N+1},m_{N+2}, \ldots , m_n)}} = g_{\omega } = f_{\omega ^+} = f_{{\omega ^+}*{(0,0, \ldots , 0)}}, \end{aligned}$$
it follows that
$$\begin{aligned} F_\alpha = \lim _{n \rightarrow \infty } g_{\omega _{\alpha ,n}} = \lim _{n \rightarrow \infty } f_{\omega _{\beta ,n}} = F_\beta , \end{aligned}$$
where \(\omega _{\alpha ,n}\) and \(\omega _{\beta ,n}\) are the elements of the defining sequences of \(\alpha \) and \(\beta \) in K. \(\square \)

What Lemma 2.15 implies is that the function \(t \mapsto F_t\) can be extended to the interval [0, 1] in a natural way: for a point \(t \notin K\), there exists a unique \(G_\omega = (\alpha , \beta )\) such that \(t \in G_\omega \). Then \(F_t = F_\alpha = F_\beta \) is the ceiling function at t.

Proposition 2.16

The function \(t \in [0,1] \mapsto F_t \in C^\infty ([0,1])\) is continuous.

Proof

Given \(k \in \mathbb {N}\), fix \(\varepsilon > 0\) and let N and \(\delta _N\) be as in the proof of Proposition 2.14.

Suppose \(t < s\) and \(|t - s| < \delta _N\). If t, \(s \in K\), we already know that \(\Vert F_t - F_s\Vert _k < \varepsilon \). Otherwise, there exist \(\alpha \), \(\beta \in K\) such that \(t \leqslant \alpha \le \beta \leqslant s\) and \(F_t = F_\alpha , F_s = F_\beta \). Since \(|\alpha - \beta | \leqslant |t - s| < \delta _N\), we are done. \(\square \)

2.6 The function \(\psi \) and the associated line field

Having constructed the family \(\{F_t\}_{t \in [0,1]}\) of ceiling functions, our next task is to check that their graphs cover the lune L, and that the tangents to those graphs fit together to form a continuous line field \(\Lambda \) on L. The purpose of this subsection is to define this line field and show that it is continuous.

Lemma 2.17

Given \((x,y) \in L\), there exists \(t \in [0,1]\) (not necessarily unique) such that \(y = F_t(x)\). Moreover, if \(y = F_{t_1}(x) = F_{t_2}(x)\), then \(F_{t_1}'(x) = F_{t_2}'(x)\).

Proof

Take \((x,y) \in L\), i.e., such that \(x \in [0,1], f(x) \leqslant y \leqslant g(x)\). Since from the construction in Sect. 2.3 we know that for each n,
$$\begin{aligned} \bigcup _{1 \leqslant i \le m_n} L_{{\omega }*{(i)}} = L_\omega , \end{aligned}$$
it follows that there exists a sequence \(\{\omega _n\}_{n \geqslant 1}\) (not necessarily unique) such that \(\omega _1 = (1), \omega _{n} = {\omega _{n-1}}*{(i)}\) for some \(i \in \{1, 2, \ldots , m_n\}\) with \(f_{\omega _n}(x) \leqslant y \leqslant g_{\omega _n}(y)\). Let t be the unique element in the set \(\bigcap _{n \geqslant 1} J_{\omega _n} \subseteq K\); then we have
$$\begin{aligned} F_t(x) = \lim _{n \rightarrow \infty } f_{\omega _n}(x) = \lim _{n \rightarrow \infty } g_{\omega _n}(x) = y, \end{aligned}$$
proving the first part.

Now suppose \(t_1\), \(t_2 \in [0,1]\) are such that \(y = F_{t_1}(x) = F_{t_2}(x),\) and assume \(t_1 \leqslant t_2\). By Lemma 2.13, \(F_{t_1} \leqslant F_{t_2}\), so that x is a local maximum of the function \(F_{t_1} - F_{t_2}\). Therefore, \(F_{t_1}'(x) - F_{t_2}'(x) = 0\). \(\square \)

For each \((x,y) \in L\), take \(t \in [0,1]\) such that \(y = F_t(x)\), and set \(\psi (x,y) = F_t'(x)\). By Lemma 2.17, this is well defined (not depending on the choice of t).

Proposition 2.18

The function \(\psi : L \rightarrow \mathbb {R}\) is continuous.

Proof

Let \((x,y) \in L\) and let \((x_n,y_n) \in L\) be a sequence such that \((x_n, y_n) \rightarrow (x,y).\) By Lemma 2.17 and the previous paragraph, \(y_n = F_{t_n}(x_n)\) for some \(t_n\), and \(\psi (x_n,y_n) = F_{t_n}'(x_n)\).

Suppose, by contradiction, that \(\psi (x_n,y_n) \not \rightarrow \psi (x,y)\). In other words, \(F_{t_n}'(x_n)\not \rightarrow F_t'(x)\), where t is such that \(y = F_t(x)\). By passing to a subsequence if necessary, we may assume that \(|F_{t_n}'(x_n) - F_t'(x)| \geqslant \varepsilon \) for some \(\varepsilon > 0\).

Let \(\{t_{n_k}\}\) be a convergent subsequence of \(\{t_n\}\), such that \(t_{n_k} \rightarrow t^*\). By Proposition 2.16 and the continuity of \(F_{t^*}\), the distance
$$\begin{aligned} |F_{t_{n_k}}(x_{n_k}) - F_{t^*}(x)|&\leqslant |F_{t_{n_k}}(x_{n_k}) - F_{t^*}(x_{n_k})| + |F_{t^*}(x_{n_k}) - F_{t^*}(x)|\\&\leqslant \Vert F_{t_{n_k}} - F_{t^*}\Vert _0 + |F_{t^*}(x_{n_k}) - F_{t^*}(x)| \end{aligned}$$
can be made arbitrarily small as \(k \rightarrow \infty \), so that \(F_{t_{n_k}}(x_{n_k}) \rightarrow F_{t^*}(x)\) and \(F_{t^*}(x) = F_t(x)\). By Lemma 2.17, \(F_{t^*}'(x) = F_t'(x)\). If we pick k sufficiently large such that \(\Vert F_{t_{n_k}} - F_{t^*}\Vert _1 < \frac{\varepsilon }{2}\) and \(|F_{t^*}'(x_{n_k}) - F_{t^*}'(x)| < \frac{\varepsilon }{2}\), we obtain
$$\begin{aligned} |F_{t_{n_k}}'(x_{n_k}) - F_{t}'(x)| < \varepsilon , \end{aligned}$$
which is a contradiction. \(\square \)

We then set \(\Lambda (x,y)\) as the line whose direction vector is \((1,\psi (x,y))\), for each \((x,y) \in L\). By Proposition 2.18, this is a continuous line field.

2.7 A sequence of curves

We now proceed to the construction of a sequence \(\gamma _n\) of curves that converges uniformly to a Peano curve \(\gamma \), such that each \(\gamma _n\) is tangent to the line field \(\Lambda (x,y)\).

For an interval \(I=[\alpha ,\beta ]\) and for a, \(b \in \mathbb {R}\), let \(\zeta _{I,a,b}:I \rightarrow \mathbb {R}\) be a \(C^\infty \) strictly monotone function such that:
  1. (i)

    \(\zeta _{I,a,b}(\alpha ) = a\), \(\zeta _{I,a,b}(t) = b\);

     
  2. (ii)

    \(\zeta _{I,a,b}^{(k)}(\alpha ) = \zeta _{I,a,b}^{(k)}(\beta ) = 0\) for every \(k \geqslant 1\).

     
Additionally, for any \(h: [0,1] \rightarrow \mathbb {R}\) let \(\Gamma _h\) be the graph of h, parametrized in the obvious way, i.e., \(\Gamma _h(t) = (t,h(t))\) for \(t \in [0,1]\).
For each \(\omega \in \Omega \), write \(S_{L_\omega } = (a_{\omega }, b_{\omega }),\) and let
$$\begin{aligned} \gamma _1(t) = \Gamma _{f_{(1)}}\left( \zeta _{J_{(1)},a_{(1)},b_{(1)}}(t)\right) \end{aligned}$$
for \(t \in [0,1]\). Recursively, set for \(n \geqslant 2\) (Fig. 4):
$$\begin{aligned} \gamma _n(t) = {\left\{ \begin{array}{ll} \gamma _{n-1}(t), &{}\quad \text {if } t \in \bigcup _{\omega \in \Omega ^*, |\omega | < n}G_\omega , \\ \Gamma _{f_\omega }\left( \zeta _{J_\omega , a_\omega , b_\omega }(t)\right) , &{}\quad \text {if } t \in J_\omega \text { for some } \omega \in \Omega , |\omega | = n,\\ \Gamma _{g_\omega }\left( \zeta _{G_\omega , b_\omega , a_{\omega ^+}}(t)\right) , &{}\quad \text {if } t \in G_\omega \text { for some } \omega \in \Omega ^*, |\omega | = n. \end{array}\right. } \end{aligned}$$
Fig. 4

Schematic drawing of the curve \(\gamma _2\), assuming \(m_2 = 6\).

Lemma 2.19

If \(|\omega | = n\), then
  1. (i)

    \(\gamma _{n+1}(J_{{\omega }*{(i)}}) \subseteq L_{{{\omega }*{(i)}}}^{{\text {ess}}} \text { for any } 1 \leqslant i \le m_{n+1}\);

     
  2. (ii)

    \(\gamma _{n+1}(G_{{\omega }*{(i)}}) \subseteq L_{{{\omega }*{(i)}}}^{{\text {ess}}} \text { for odd } i, 1 \leqslant i < m_{n+1}\);

     
  3. (iii)

    \(\gamma _{n+1}(G_{{\omega }*{(i)}}) \subseteq L_{{{\omega }*{(i-1)}}}^{{\text {ess}}} \cup L_{{{\omega }*{(i)}}}^{{\text {ess}}} \text { for even } i, 1 \leqslant i < m_{n+1}\).

     
As a consequence, \(\gamma _{m}(J_\omega ) \subseteq L_\omega ^{{\text {ess}}}\) for all \(m \geqslant n\).

Proof

If \(t \in J_{{\omega }*{(i)}}\), then
$$\begin{aligned} \gamma _{n+1}(t) = \Gamma _{f_{{\omega }*{(i)}}}(\zeta _{J_{{\omega }*{(i)}}, a_{{\omega }*{(i)}}, b_{{\omega }*{(i)}}}(t)). \end{aligned}$$
Since \(a_{{\omega }*{(i)}} \leqslant \zeta _{J_{{\omega }*{(i)}}, a_{{\omega }*{(i)}}, b_{{\omega }*{(i)}}}(t) \leqslant b_{{\omega }*{(i)}},\) Lemma 2.2 implies property (i).
If \(t \in G_{{\omega }*{(i)}}\) with odd i, then \(L_{{\omega }*{(i)}}\) is the first lune in a bipartition of one of the slices of \(L_\omega \), hence \(b_{{\omega }*{(i)}} = (a_\omega + 2b_\omega )/3\) and \(a_{{\omega }*{(i+1)}} = (2a_\omega + b_\omega )/3\) (see Proposition 2.7). Since
$$\begin{aligned} \gamma _{n+1}(t) = \Gamma _{g_{{\omega }*{(i)}}}(\zeta _{G_{{\omega }*{(i)}}, b_{{\omega }*{(i)}}, a_{{\omega }*{(i+1)}}}(t)), \end{aligned}$$
property (ii) follows.

If \(t \in G_{{\omega }*{(i)}}\) with even i and \(i < m_{n+1}\), \(L_{{\omega }*{(i)}}\) is the second lune in a bipartition, hence \(b_{{\omega }*{(i)}} = b_\omega \) and \(a_{{\omega }*{(i+1)}} = a_\omega = a_{{\omega }*{(i-1)}}\). Therefore, if the value \(\zeta _{G_{{\omega }*{(i)}}, b_{{\omega }*{(i)}}, a_{{\omega }*{(i+1)}}}(t)\) is greater than or equal to \(a_{{\omega }*{(i)}}\) then \(\gamma _{n+1}(t) \in L_{{\omega }*{(i)}}\); on the other hand, if that value is less than or equal to \(a_{{\omega }*{(i)}} \leqslant b_{{\omega }*{(i-1)}}\) then \(\gamma _{n+1}(t) \in L_{{\omega }*{(i-1)}}\) (recall that if \(x \leqslant a_{{\omega }*{(i)}}\), \(g_{{\omega }*{(i)}}(x) = f_{{\omega }*{(i)}}(x) = g_{{\omega }*{(i-1)}}(x)\)), thus property (iii) follows.

To see the consequence, note that the three properties imply that for any \(\omega \in \Omega \) with \(|\omega | = n\), \(\gamma _{n+1}(J_{{\omega }*{(i)}}),\gamma _{n+1}(G_{{\omega }*{(i)}}) \subseteq L_\omega ^{{\text {ess}}}\) for each i. In particular, for any \(\omega ' \in \Omega \) with \(|\omega '| = k\), \(\gamma _{n+k+1}(J_{{{\omega }*{\omega '}}*{(i)}}), \gamma _{n+k+1}(G_{{{\omega }*{\omega '}}*{(i)}}) \subseteq L_{{\omega }*{\omega '}}^{{\text {ess}}} \subseteq L_\omega ^{{\text {ess}}}\). Since for each k we have
$$\begin{aligned} J_\omega = \bigcup _{\begin{array}{c} \omega ' \in \Omega \\ |\omega '| = k \end{array}} J_{{\omega }*{\omega '}} \cup \bigcup _{\begin{array}{c} \omega ' \in \Omega ^*\\ |\omega '| = k \end{array}} G_{{\omega }*{\omega '}}, \end{aligned}$$
it follows that \(\gamma _{n+k}(J_\omega ) \subseteq L_\omega ^{{\text {ess}}}\). \(\square \)

2.8 The Peano curve

We now wish to show that the curves \(\gamma _n\) defined above converge uniformly to some curve \(\gamma \).

Lemma 2.20

There exists a sequence \(D_n \rightarrow 0\) such that for each \(\omega \in \Omega \), the diameter \(D_\omega \) of \(L_\omega ^{{\text {ess}}}\) satisfies \(D_\omega ^2 \leqslant D_{|\omega |}\).

Proof

Let \((x_1,y_1), (x_2,y_2) \in L_\omega ^{{\text {ess}}}\), and assume without loss of generality that \(y_1 \leqslant y_2\). By Lemma 2.2, \(a_\omega \leqslant x_1,x_2 \leqslant b_\omega \) and \(f_\omega (x_1) \leqslant y_1 \leqslant y_2 \leqslant g_\omega (x_2)\). Hence,
$$\begin{aligned} {\text {dist}}((x_1,y_1),(x_2,y_2))^2&= (x_2 - x_1)^2 + (y_2- y_1)^2\\&\leqslant (b_\omega - a_\omega )^2 + (g_\omega (x_2) - f_\omega (x_1))^2\\&\leqslant (b_\omega - a_\omega )^2 + (g_\omega (x_2) - f_\omega (x_2) + f_\omega (x_2) - f_\omega (x_1))^2\\&\leqslant (b_\omega - a_\omega )^2 + \left( |g_\omega (x_2) - f_\omega (x_2)| + \left| \int _{x_1}^{x_2}f_\omega '(t) dt\right| \right) ^2\\&\leqslant (b_\omega - a_\omega )^2 + \left( \Vert L_\omega \Vert _0 + \Vert f_\omega \Vert _1 (b_\omega - a_\omega )\right) ^2. \end{aligned}$$
Suppose \(|\omega | = n\). Clearly \(\Vert L_\omega \Vert _0 \leqslant \Vert L_\omega \Vert _n < \varepsilon _n\), by the construction of the family of lunes. Moreover, also by construction (and by Proposition 2.7), \(b_\omega - a_\omega = (2/3)^n(b_{(1)} - a_{(1)}) \leqslant (2/3)^n\). Finally, let \(M = \sup _{t \in [0,1]} \Vert F_t\Vert _1\) (which is finite by Proposition 2.16). Since
$$\begin{aligned} f_\omega = \lim _{k \rightarrow \infty } f_{\omega * \underbrace{\scriptstyle (1,1, \ldots , 1)}_k } = F_t \end{aligned}$$
for some \(t \in [0,1]\), it follows that \(\Vert f_\omega \Vert _{1} \leqslant M\). In other words, if we take
$$\begin{aligned} D_n = \left( \frac{2}{3}\right) ^{2n} + \left( \varepsilon _n + M\left( \frac{2}{3}\right) ^n\right) ^{2} \rightarrow 0, \end{aligned}$$
we are done. \(\square \)

Lemma 2.21

The curves \(\gamma _n\) form a uniformly Cauchy sequence, and in particular converge uniformly to a continuous curve \(\gamma \).

Proof

Let \(\varepsilon > 0\) and take N such that \(D_N < \varepsilon .\) We wish to show that whenever \(m,n \geqslant N\), we have \(|\gamma _m(t) - \gamma _n(t)| < \varepsilon \) for each \(t \in [0,1]\).

In fact, if \(t \in \bigcup _{\omega \in \Omega ^*, |\omega | \leqslant N} G_\omega \) this is clear, because \(\gamma _k(t) = \gamma _N(t)\) for all \(k \geqslant N\). On the other hand, if \(t \in J_\omega \) for some \(\omega , |\omega | = N\), then Lemma 2.19 shows that \(\gamma _m(t)\), \(\gamma _n(t) \in L_\omega ^{{\text {ess}}}\), thus \(|\gamma _m(t) - \gamma _n(t)| \leqslant D_N < \varepsilon .\)\(\square \)

We now wish to relate the footprint \(\gamma ([0,t])\) with the ceiling function \(F_t\). We need a few results first:

Lemma 2.22

Let \(\omega \in \Omega \), \((x,y) \in L_\omega ^{{\text {ess}}}\). There exists a sequence \(\{\omega _n\}_{n \geqslant |\omega |}\) such that \(\omega _{|\omega |} = \omega \), \(\omega _{n+1} = {\omega _n}*{(\ell )}\) for some \(\ell \in \mathbb {N}\) and \((x,y) \in L_{\omega _n}^{{\text {ess}}}\) for each n.

Moreover, there exists \(t \in J_\omega \cap K\) such that the defining sequence \(\{\omega _{t,n}\}_{n \geqslant 1}\) of t satisfies \(\omega _{t,n} = \omega _n\) whenever \(n \geqslant |\omega |\).

Proof

The first part is immediate from the fact that
$$\begin{aligned} L_{\omega _n}^{{\text {ess}}} = \bigcup _{1 \leqslant i \leqslant m_{n+1}} L_{{\omega _n}*{(i)}}^{{\text {ess}}}. \end{aligned}$$
For the second part, we take t to be the single element of \(\bigcap _{n \geqslant |\omega |} J_{\omega _n}\). \(\square \)

The sequence in Lemma 2.22 is not necessarily unique, and we call every such sequence a defining sequence of (xy) in \(L_\omega \).

Lemma 2.23

For each \(\omega \in \Omega \), we have \(\gamma (J_\omega ) = \gamma (J_\omega \cap K) = L_\omega ^{{\text {ess}}}\).

Proof

Lemma 2.19 implies that \(\gamma (J_\omega ) \subseteq L_\omega ^{{\text {ess}}}\). For the other direction, take for each \((x,y) \in L_\omega ^{{\text {ess}}}\), \(\{\omega _n\}_{n \geqslant |\omega |}\) and \(t \in J_\omega \cap K\) as in Lemma 2.22. Since \(\gamma (t) \in L_{\omega _n}^{{\text {ess}}}\) for each n (and this is a sequence of nested compact sets), it follows that \(\gamma (t) = (x,y)\). Therefore, \(\gamma (J_\omega ) \subseteq L_\omega ^{{\text {ess}}} \subseteq \gamma (J_\omega \cap K) \subseteq \gamma (J_\omega )\) and we are done. \(\square \)

Lemma 2.24

For each \(t \in [0,1]\), we have \(\gamma ([0,t]) = \gamma (K \cap [0,t])\).

Proof

We need to show that for each \(t \notin K\), there exists \(s < t, s \in K\) with \(\gamma (s) = \gamma (t)\). Take \(t \notin K\), so that \(t \in G_{{\omega }*{(i)}}\) for some \(\omega \in \Omega ^*, |\omega | = n\), \(i < m_{n+1} - 1\). By Lemma 2.19, \(\gamma (t) = \gamma _{n+1}(t) \in L_{{{\omega }*{(i-1)}}}^{{\text {ess}}} \cup L_{{{\omega }*{(i)}}}^{{\text {ess}}}\) (or simply \(L_{{{\omega }*{(i)}}}^{{\text {ess}}}\) if \(i = 1\)). By Lemma 2.23, \( \gamma (J_{{\omega }*{(i-1)}} \cap K) \cup \gamma (J_{{\omega }*{(i)}} \cap K) = L_{{{\omega }*{(i-1)}}}^{{\text {ess}}} \cup L_{{{\omega }*{(i)}}}^{{\text {ess}}},\) and this yields the result. \(\square \)

Lemma 2.25

For \(t \in [0,1]\), if \(\gamma (t) = (x(t), y(t))\), then \(y(t) = F_t(x(t))\).

Proof

If \(t \notin K\), then \(t \in G_\omega = (\beta _{\omega }, \alpha _{\omega ^+})\) for some \(\omega \). By the definition of \(\gamma \), \(y(t) = g_\omega (x(t))\), because \(\gamma (t)\) is in the graph of \(g_\omega \). Since \(F_t = F_{\beta _\omega } = g_\omega = f_{\omega ^+} = F_{\alpha _{\omega ^+}}\), the result follows.

If \(t \in K\) and \(\omega _n\) is the defining sequence of t, we know by Lemma 2.23 that \((x(t),y(t)) \in L_{\omega _n}^{{\text {ess}}}\). By Lemma 2.2, \(f_{\omega _n}(x(t)) \leqslant y(t) \leqslant g_{\omega _n}(x(t))\). Since both bounding sequences converge to \(F_t(x(t))\), we are done. \(\square \)

Proposition 2.26

For each \(t \in (0,1]\), we have \(\gamma ([0,t]) = L^{{\text {ess}}}(f,F_t)\).

Proof

It suffices to show the result for \(t \in K\), for if \(t \notin K\) and \(t_K = \max \{s \in K \mid s \leqslant t\}\), then \(F_t = F_{t_K}\) and \(\gamma ([0,t]) = \gamma ([0,t] \cap K) = \gamma ([0,t_K])\).

Let \(t \in K\). We need to show that \(\gamma ([0,t]) = \gamma (K \cap [0,t]) = L^{{\text {ess}}}(f,F_t).\) To see that \(L^{{\text {ess}}}(f,F_t) \subseteq \gamma (K \cap [0,t]),\) take \((x,y) \in {{\mathrm{int}}}(L(f,F_t))\), so that \( f(x) < y < F_t(x)\). Let \(\{\omega _n\}_{n \geqslant 1}\) be a defining sequence of (xy) in \(L_{(1)} = L\), which is also the defining sequence of some \(s \in K\). Then \(\gamma (s) = (x,y)\). Since \(y = \lim f_{\omega _n}(x) = \lim g_{\omega _n}(x) = F_s(x)\), it follows that \(F_s(x) < F_t(x)\). Lemma 2.13 implies that we necessarily have \(F_s \leqslant F_t\) and thus \(s < t\). Consequently, \({{\mathrm{int}}}(L(f,F_t)) \subseteq \gamma (K \cap [0,t])\). Since \(\gamma (K \cap [0,t])\) is a compact set (thus closed), \(L^{{\text {ess}}}(f,F_t) = \overline{{{\mathrm{int}}}(L(f,F_t))} \subseteq \gamma (K \cap [0,t]).\)

To see that \(\gamma (K \cap [0,t]) \subseteq L^{{\text {ess}}}(f,F_t)\), write \(J_\omega = [\alpha _\omega , \beta _\omega ]\) for each \(\omega \in \Omega \). Take \(s \in K\), \(0 < s \leqslant t\), and suppose initially that \(s = \alpha _{\hat{\omega }}\) for some \(\hat{\omega }\). Since \(s > 0\), there exists a unique \(\omega \) such that \(s = \alpha _{\omega ^+}\). Since
$$\begin{aligned} \gamma (\alpha _{\omega ^+})&= \lim _{t \nearrow \alpha _{\omega ^+}} \gamma (t) \\&= \lim _{t \nearrow \alpha _{\omega ^+}} (\zeta _{G_\omega , b_\omega , a_{\omega ^+}}(t), g_\omega (\zeta _{G_\omega , b_\omega , a_{\omega ^+}}(t)))\\&= \lim _{t \nearrow \alpha _{\omega ^+}} (\zeta _{G_\omega , b_\omega , a_{\omega ^+}}(t), F_{\alpha _{\omega ^+}}(\zeta _{G_\omega , b_\omega , a_{\omega ^+}}(t))) \end{aligned}$$
is a limit point of the closed set \(L^{{\text {ess}}}(f,F_s)\), it follows that \(\gamma (s) \in L^{{\text {ess}}}(f,F_s) \subseteq L^{{\text {ess}}}(f,F_t)\).

If, on the other hand, \(s \notin \{\alpha _\omega \}_{\omega \in \Omega }\) and \(\{\omega _n\}\) is the defining sequence for s, then \(\alpha _{\omega _n} \nearrow s\), thus \(\gamma (\alpha _{\omega _n}) \rightarrow \gamma (s)\) and \(\gamma (\alpha _{\omega _n}) \in L^{{\text {ess}}}(f,F_{\alpha _{\omega _n}}) \subseteq L^{{\text {ess}}}(f,F_t)\). \(\square \)

We have thus concluded the proof of Proposition 2.3. To summarize, the continuity of \(\gamma \), \(F_t\), and \(\psi \) are established in Lemma 2.21, Propositions 2.16, and 2.18, respectively; property (i) comes directly from the definition; properties (ii) and (iii) are respectively Lemmas 2.13 and 2.25, property (iv) holds by definition, property (v) is Proposition 2.26, and property (vi) is also true by construction.

3 Proof of the Theorem

The first step is to obtain a cylindrical version of Proposition 2.3. Recall that \(\mathbb {T}= \mathbb {R}/ 2\pi \mathbb {Z}\), and that \(C^\infty (\mathbb {T})\) can be regarded as the space of \(C^\infty \)\(2\pi \)-periodic functions \(\mathbb {R}\rightarrow \mathbb {R}\), endowed with the \(C^\infty \) topology (see Sect. 2.1 for details).

Proposition 3.1

Given intervals \([t_0,t_1]\) and [cd], there exist:
  • a continuous map \(\gamma : [t_0,t_1] \rightarrow \mathbb {T}\times [c,d]\);

  • a continuous map \(t \in [t_0,t_1] \mapsto F_t \in C^\infty (\mathbb {T})\);

  • a continuous function \(\psi : \mathbb {T}\times [c,d] \rightarrow \mathbb {R}\);

with the following properties:
  1. (i)

    \(F_{t_0} \equiv c\), \(F_{t_1} \equiv d\);

     
  2. (ii)

    If \(t \leqslant s\) then \(F_t \leqslant F_s\);

     
  3. (iii)

    Writing \(\gamma (t) = (x(t),y(t))\), we have \(y(t) = F_t(x(t))\).

     
  4. (iv)

    \(F'_t(x) = \psi (x,F_t(x))\);

     
  5. (v)

    for each \(t \in (t_0, t_1]\), the image \(\gamma ([t_0,t])\) equals the closure of the interior of \(\{(x,y) \mid x\in \mathbb {T}, \ c \leqslant y \leqslant F_t(x)\}\);

     
  6. (vi)

    \(\gamma (t_0) = (0,c)\), \(\gamma (t_1) = (\pi ,d)\);

     

Proof

It is sufficient to consider the case \([t_0,t_1] = [c,d] = [0,1]\), since the general case follows by rescaling. Let \(g:\mathbb {R}\rightarrow [0,1]\) be a \(2\pi \)-periodic \(C^\infty \) function such that \(g(0)=0\), \(g(\pi )=1\), \(g^{(k)}(0) = g^{(k)}(\pi ) = 0\) for every \(k \geqslant 1\), and which is strictly monotone in each of the intervals \([0,\pi ]\) and \([\pi ,2\pi ]\). Consider the following four functions:
$$\begin{aligned} f_1, g_1&:[0,2\pi ]\rightarrow \mathbb {R}&\quad \text {given by}\quad f_1&\equiv 0,&\ g_1&= g|_{[0,2\pi ]}, \\ f_2, g_2&:[\pi ,3\pi ] \rightarrow \mathbb {R}&\quad \text {given by}\quad f_2&= g|_{[\pi ,3\pi ]},&\ g_2&\equiv 1, \end{aligned}$$
and the corresponding lunes \(L_1 = L(f_1,g_1)\) and \(L_2 = L(f_2,g_2)\). Applying Proposition 2.3 to the lune \(L_i\) we obtain a Peano curve \(\gamma _i\), a family of ceiling functions \(F_{i,t}\), and a function \(\psi _i\). Let \(p: L_1 \cup L_2 \rightarrow \mathbb {T}\times [0,1]\) be the bijective map defined by \(p(x,y) = (x \bmod 2\pi , y)\); set
$$\begin{aligned} \gamma ^*(t) = {\left\{ \begin{array}{ll} \gamma _1(3t), &{}\quad 0 \leqslant t < 1/3; \\ \left( 3\pi (1-t), g\left( 3\pi (1-t)\right) \right) , &{}\quad 1/3 \leqslant t \leqslant 2/3; \\ \gamma _2(3t-2), &{}\quad 2/3 < t \leqslant 1; \end{array}\right. } \end{aligned}$$
and \(\gamma = p \circ \gamma ^*\); set \(\psi = \psi ^* \circ p^{-1}\) for \(\psi ^*: L_1 \cup L_2 \rightarrow \mathbb {R}\) such that \(\psi ^*|{L_1} = \psi _1\) and \(\psi ^*|{L_2} = \psi _2\); finally, for \(x \in \mathbb {T}\), let
$$\begin{aligned} F_t(x) = {\left\{ \begin{array}{ll} F_{1,3t}(x), &{}\quad 0 \leqslant t < 1/3; \\ g(x), &{}\quad 1/3 \leqslant t \leqslant 2/3; \\ F_{2,3t-2}(x), &{}\quad 2/3 < t \leqslant 1. \end{array}\right. } \end{aligned}$$
Then \(\gamma \), \(\psi \) and \(F_t\) have the desired properties. \(\square \)

We improve the previous proposition by controlling the derivatives:

Proposition 3.2

Given intervals \([t_0,t_1]\), [cd] and numbers \(k_0 \in \mathbb {N}\), \(\delta _0 > 0\), there exist maps \(\gamma \), \(F_t\), and \(\psi \) satisfying properties (i)–(v) in Proposition 3.1 and, in addition, the following ones:
  1. (vi)

    \(\gamma (t_0) = (0,c)\), \(\gamma (t_1) = (0,d)\);

     
  2. (vii)

    \(\Vert F_t - c_t \Vert _{k_0} < \delta _0\) for every t, where \(c_t\) is the constant \(\frac{c(t_1-t) + d(t - t_0)}{t_1 - t_0}\).

     

Proof

Again, it is sufficient to consider the case \([t_0,t_1] = [c,d] = [0,1]\). Let \(\hat{\gamma }\), \(\hat{F}_t\), and \(\hat{\psi }\) be given by the previous proposition. By compactness, we have \(\Vert \hat{F}_t\Vert _{k_0} \leqslant C\) for some finite C independent of t. Fix an odd integer \(n > (C+1)/\delta _0\). By rescaling and translating we obtain \(\hat{\gamma }_j\), \(\hat{F}_{j,t}\), and \(\hat{\psi }_j\) for the cylinders \(\mathbb {T}\times [j/n, (j+1)/n]\), where \(j=0,1,\ldots n-1\), with the extra property that if \(t \in [j/n, (j+1)/n]\) then \(\Vert \hat{F}_{j,t} - j/n\Vert _{k_0} \leqslant C/n < \delta _0 - 1/n\). Finally, we rotate the cylinders so that everything glues: in other words, for each \(t \in [0,1]\) we let \(j = \lfloor {nt} \rfloor \) and define:
$$\begin{aligned} \gamma (t) = \hat{\gamma }_j(nt) + (j\pi ,0), \quad F_t(x) = \hat{F}_{j,nt-j}(x + j\pi ), \quad \psi (x,t) = \hat{\psi }_j(x + j\pi ,t). \end{aligned}$$
It is clear that these maps have the required properties (i)–(v) and (vi). To see property (vii), note that \(c_t = t\) so letting \(j = \lfloor {nt} \rfloor \) we have
$$\begin{aligned} \Vert F_t - c_t\Vert _{k_0} \leqslant \Vert F_t - j/n\Vert _{k_0} + 1/n < \delta _0 \, . \end{aligned}$$
\(\square \)

Finally, we explain how the previous proposition allows us to conclude:

Proof of Theorem

Let \(k :(0,\infty ) \rightarrow \mathbb {N}\) be upper semicontinuous and \(\varepsilon :(0,\infty ) \rightarrow (0,\infty )\) be lower semicontinuous. Then there exist two-sided sequences \(\{t_n\}_{n\in \mathbb {Z}}\), \(\{k_n\}_{n\in \mathbb {Z}}\), and \(\{\varepsilon _n\}_{n\in \mathbb {Z}}\) taking values in \((0,\infty )\), \(\mathbb {N}\), and \((0,\infty )\) respectively, such that \(\{t_n\}\) is monotonically increasing, \(\lim _{n \rightarrow -\infty } t_n = 0\), \(\lim _{n \rightarrow +\infty } t_n = \infty \), and
$$\begin{aligned} t\in [t_n, t_{n+1}] \ \Rightarrow \varepsilon (t) \geqslant \varepsilon _n \quad \text { and }\quad k(t) \leqslant k_n. \end{aligned}$$
For each n, let \(\delta _n = \varepsilon _n/2^{k_n}\), and apply Proposition 3.2 with both intervals equal to \(I_n = [t_n, t_{n+1}]\), thus obtaining a Peano curve \(\gamma _n: I_n \rightarrow \mathbb {T}\times I_n\), a family of ceiling functions \(F_{n,t}\) and a continuous function \(\psi _n\) defined on \(\mathbb {T}\times I_n\) satisfying all seven properties. Also, note that \(c_t = t\) in part (vii).
Define a diffeomorphism \(P :\mathbb {T}\times (0, \infty ) \rightarrow \mathbb {R}^2 {\backslash }\{(0,0)\}\) by \(P(\theta , r) = (r \cos \theta , r \sin \theta )\). Recall that \(\alpha _t(\theta ) = P(\theta ,t)\). We now construct maps \(\gamma ^*: (0,\infty ) \rightarrow \mathbb {T}\times (0,\infty )\), \(F_t^*: \mathbb {T}\rightarrow (0,\infty )\) and \(\psi ^*:\mathbb {T}\times (0,\infty ) \rightarrow \mathbb {R}\) by setting for \((x,t) \in \mathbb {T}\times [t_n, t_{n+1}]\):
$$\begin{aligned} \gamma ^*(t) = \gamma _n(t), \quad F_t^*(x) = F_{n,t}(x), \quad \psi (x,t) = \psi _n(x,t) \, . \end{aligned}$$
Next, we define the Peano curve \(\gamma \) by \(\gamma (0) = 0\) and \(\gamma (t) = P \circ \gamma ^*(t)\) for \(t > 0\). Let \(\Lambda ^*\) be the line field on \(\mathbb {T}\times (0,\infty )\) spanned by the vector field \(\frac{\partial }{\partial \theta } + \psi ^*(\theta ,r) \frac{\partial }{\partial r}\); by pushing it forward by the derivative of P, we obtain a line field \(\Lambda \) on \(\mathbb {R}^2 {\backslash }\{(0,0)\}\).

Note that, for each \(t > 0\), \(\gamma ^*([0,t]) =\{(x,y): x \in \mathbb {T}, 0 < y \leqslant F_t(x)\}\). It follows then that \(\beta _t: \theta \in \mathbb {T}\mapsto P(\theta ,F_t(\theta )) \in \mathbb {R}^2\) is a smooth embedding whose image is \(\partial \gamma ([0,t])\). By property (iv) of Proposition 3.1, \(\beta _t\) is tangent to the line field \(\Lambda \).

To conclude the proof we check that the proximity condition (1.1) is satisfied. Since \(\beta _t(\theta ) = F_t(\theta )\alpha _1(\theta )\) and \(\alpha _t(\theta ) = t \alpha _1(\theta )\), we have, for each \(k \in \mathbb {N}\),
$$\begin{aligned} \Vert \beta _t^{(k)}(\theta ) - \alpha _t^{(k)}(\theta )\Vert \leqslant \sum _{i=0}^k \left( {\begin{array}{c}k\\ i\end{array}}\right) \big |(F_t -t)^{(i)}(\theta )\big | \, \underbrace{\big |\alpha _1^{(k-i)}(\theta )\big |}_{1} \leqslant 2^k \Vert F_t - t\Vert _k \, . \end{aligned}$$
Given \(t>0\), let n be such that \(t \in I_n\). Then
$$\begin{aligned} \Vert \alpha _t - \beta _t\Vert _{k(t)} \leqslant \Vert \alpha _t - \beta _t\Vert _{k_n} \leqslant 2^{k_n} \Vert F_t - t \Vert _{k_n}< 2^{k_n} \delta _n = \varepsilon _n \leqslant \varepsilon (t). \end{aligned}$$
\(\square \)

Footnotes

  1. 1.

    Actually, Pach and Rogers obtained a curve with the additional properties that \(\gamma ([0,1]) = [0,1]^2\) and \(\gamma ([t,1])\) is convex for each t. These properties can easily be obtained by modifying our curve: the argument goes exactly as in the last paragraph of §3 in [6].

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

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