# Sewing lemma and knitting lemma for metric spaces

C Curry, D Manchon

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C Curry, D Manchon. Sewing lemma and knitting lemma for metric spaces. 2025. hal-04925197

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Distributed under a Creative Commons Attribution 4.0 International License
# SEWING LEMMA AND KNITTING LEMMA FOR METRIC SPACES

C. CURRY, D. MANCHON

**ABSTRACT.** We state and prove a sewing lemma in the general context of families of complete metric spaces indexed by an interval of the real line, encompassing the flow sewing lemma proved by I. Bailleul in 2015. A further generalisation to other metric parameter spaces $P$ than intervals is moreover proposed, leading to a representation of the groupoid of thin-equivalent Lipschitz paths on $P$. Under a stronger hypothesis, we finally prove a two-dimensional version, the knitting lemma, which gives rise to a representation of the Lipschitz homotopy groupoid of the parameter space, without thinness condition.

**Keywords:** sewing lemma, groupoids, paths, homotopy, Lipschitz norm.

MSC classification:18B40, 20L05, 41A99.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Reminders on groupoids</td><td>2</td></tr><tr><td>3. Extended metric spaces of continuous maps</td><td>4</td></tr><tr><td>3.1. Extended metric spaces</td><td>4</td></tr><tr><td>3.2. Extended metric spaces of continuous maps</td><td>5</td></tr><tr><td>3.3. Length spaces</td><td>6</td></tr><tr><td>4. Local flows, local approximate flows and the sewing lemma</td><td>6</td></tr><tr><td>5. Applications and generalisations</td><td>9</td></tr><tr><td>5.1. Applications</td><td>9</td></tr><tr><td>5.2. The groupoid of Lipschitz paths modulo thin Lipschitz equivalence</td><td>10</td></tr><tr><td>5.3. The Lipschitz homotopy groupoid</td><td>10</td></tr><tr><td>5.4. A generalisation of Theorem 6</td><td>11</td></tr><tr><td>5.5. The knitting lemma</td><td>11</td></tr><tr><td>References</td><td>14</td></tr></table>

## 1. INTRODUCTION

The sewing lemma first appeared in the work of Feyel and de la Pradelle [11], and was clarified by M. Gubinelli [15]. The statement is that, given a Banach space valued function $\mu(s, t)$ obeying

$$|\mu(s, t) - \mu(s, u) - \mu(u, t)| \le \sum_{i=1}^{n} C_i |t - u|^{a_i} |u - s|^{b_i} \quad \text{with } a_i + b_i = 1 + \varepsilon,$$

there exists a unique (up to additive constant) $\phi(t)$ such that $|\phi(t) - \phi(s) - \mu(s, t)| \le C|t - s|^{1+\varepsilon}$. The intended application was to show that integrals $I(s, t) = \phi(t) - \phi(s)$ could be defined through Riemann sums $\sum_i \mu(t_i, t_{i+1})$ in a limit where the partition of $[s, t]$ tends to zero. In this the inspiration was Lyons' theorem on almost multiplicative functionals, used to extend Young integration
to rough paths.

Shortly after, Feyel, de la Pradelle and Mokobodzki [12] proved a non-commutative sewing lemma for functions $\mu(s,t)$ taking values in an associative monoid equipped with a complete metric; a unique $\phi(t)$ exists under the hypothesis

$$d(\mu(s,t), \mu(s,u) \circ \mu(u,t)) \leq \sum_{i=1}^{n} C_i |t-u|^{a_i} |u-s|^{b_i} \text{ with } a_i + b_i = 1 + \varepsilon.$$

A similar result was proved by Bailleul [2] and used in the construction of flows driven by rough paths. Subsequently there have appeared several further generalizations [4, 5, 10] of non-commutative sewing lemmas.

Hitherto all sewing lemmas have been based on functions of two real variables $s, t \in \mathbb{R} \times \mathbb{R}$, indeed since the work of Gubinelli on rough paths [14] it has been possible to interpret such constructions in the language of cochain complexes. On examining the proofs of existing sewing lemmas, however, it becomes clear that it is not the order structure of $\mathbb{R}$ alone that facilitates the proof, but rather a groupoid structure that permits concatenation of paths and flows.

In the present work we

(1) Recall the definitions of groupoids (§2) and extended metric spaces (§3).

(2) State and prove a sewing lemma (Theorem 6) in the general context of complete metric spaces, encompassing all of the variants mentioned above.

(3) Propose in Paragraph 5.4 a generalization of the sewing lemma to more general parameter spaces $P$ than the real line, involving the groupoid of thin-equivalent Lipschitz paths on the metric space $P$.

(4) Prove in Paragraph 5.5 a *knitting lemma* for general metric parameter spaces $P$. This result, which holds true under a much more stronger condition than what is necessary for the sewing lemma, can be briefly described as a two-dimensional version of the latter, and leads to a representation of the Lipschitz homotopy groupoid (Theorem 15).

## 2. REMINDERS ON GROUPOIDS

Recall that a category is called small if all of its objects and morphisms both form sets [19]. A groupoid is a small category in which any morphism is invertible. In other words, a groupoid is given by a set $G$, a subset $G^{(0)}$, two surjective maps $r$ (the **range**) and $s$ (the **source**) from $G$ onto $G^{(0)}$, an involution

$$\iota: G \to G \\ \gamma \mapsto \gamma^{-1}$$

and a composition law

$$m: G^{(2)} \to G \\ (\gamma, \gamma') \mapsto \gamma\gamma'$$

where

$$G^{(2)} := \{(\gamma, \gamma') \in G \times G, s(\gamma) = r(\gamma')\},$$
subject to the following axioms:

$$r|_{G^{(0)}} = s|_{G^{(0)}} = \mathrm{Id}_{G^{(0)}},$$

$$\gamma s(\gamma) = r(\gamma)\gamma = \gamma,$$

$$r(\gamma\gamma') = r(\gamma), \quad s(\gamma\gamma') = s(\gamma'),$$

$$r \circ \iota = s, \quad s \circ \iota = r,$$

$$\gamma \circ \gamma^{-1} = r(\gamma), \quad \gamma^{-1} \circ \gamma = s(\gamma),$$

$$(\gamma\gamma')\gamma'' = \gamma(\gamma'\gamma''),$$

which are best visualised on the following diagrams:

<figure><img src="image_2.png" /></figure>

**Example 1: The pair groupoid of a set P.** It is defined by $G = P \times P$, with units $G^{(0)} = \{(x, x), x \in P\}$, source $s(x, y) = y$, range $r(x, y) = y$, inverse $(x, y)^{-1} = (y, x)$ and composition $(x, y)(y, z) = (x, z)$. The set of units is identified with P itself, through the diagonal embedding $x \mapsto (x, x)$. For any groupoid G with units $G^{(0)}$, the map $(r, s) : G \to G^{(0)} \times G^{(0)}$ is a groupoid morphism.

**Definition 1.** A path in a topological space P is a continuous map $\gamma : [0, 1] \to P$. The concatenation of two paths $\gamma$ and $\gamma'$ such that $\gamma(1) = \gamma'(0)$ is defined by

$$\gamma * \gamma'(t) = \gamma(2t) \text{ if } t \in [0, 1/2],$$

$$\gamma * \gamma'(t) = \gamma'(2t - 1) \text{ if } t \in [1/2, 1].$$

The reverse concatenation will be denoted by a simple dot, namely

$$\gamma \cdot \gamma' := \gamma' * \gamma.$$

It makes sense whenever $\gamma(0) = \gamma'(1)$. For any path $\gamma$ and for any $s, t \in [0, 1]$, the path $\gamma_{st}$ is defined by

$$\gamma_{st}(u) := \gamma(su + t(1-u)).$$

In particular, $\gamma_{1,0} = \gamma$. The path $\gamma_{0,1}$ is the path $\gamma$ reversed, denoted by $\gamma^{-1}$.

**Example 2. The fundamental groupoid of a topological space.** Recall that two paths $\gamma^0$ and $\gamma^1$ are homotopic if and only if

* they share the same starting and end points a and b, i.e. $\gamma^0(0) = \gamma^1(0) = a$ and $\gamma^0(1) = \gamma^1(1) = b$,

* there exists a homotopy between $\gamma^0$ and $\gamma^1$, i.e. a continuous map $H : [0, 1]^2 \to P$ such that $H(0, t) = \gamma^0(t)$ and $H(1, t) = \gamma^1(t)$ for any $t \in [0, 1]$, and moreover $H(s, 0) = a$, $H(s, 1) = b$ for any $s \in [0, 1]$.

We shall also use the convenient notation $\gamma^s(t) := H(s, t)$.

Homotopy is an equivalence relation denoted by $\sim$. Homotopy is compatible with concatenation: if $\gamma^0 \sim \gamma^1$ and $\gamma'^0 \sim \gamma'^1$, one has $\gamma^0\gamma'^0 \sim \gamma^1\gamma'^1$. The fundamental groupoid of P,
denoted by $\mathcal{G}_P$, is the set of homotopy classes of continuous paths in $P$ endowed with reversed concatenation. One can easily check the following chain rule:

$$ \gamma_{st} \sim \gamma_{su} \cdot \gamma_{ut}. $$

The (reversed) concatenation is not associative, but it is well-known that

$$ (1) \qquad (\gamma'' \cdot \gamma') \cdot \gamma \sim \gamma'' \cdot (\gamma' \cdot \gamma). $$

The homotopy between both sides of (1) is explicitly given by

$$ \begin{align*} H(s,t) &= \gamma(t) \text{ if } 0 \le t \le \frac{1}{4}(2-s), \\ \gamma'(t) &\text{ if } \frac{1}{4}(2-s) \le t \le \frac{1}{4}(3-s), \\ \gamma''(t) &\text{ if } \frac{1}{4}(3-s) \le t \le 1. \end{align*} $$

**Example 3.** The equivalence classes of paths in a topological space modulo thin equivalence ([3, Paragraph 2.2.1], see also [18]). A loop is a path $\gamma$ from $a$ to the same point $a$. A loop is *thin* if it is homotopic to the trivial loop with image $\{a\}$, by means of a homotopy $H : [0, 1]^2 \to P$ such that $H([0, 1]^2) \subseteq \gamma([0, 1])$. It is easily seen that any loop of the form $\gamma \cdot \gamma^{-1}$ is thin. Two paths $\gamma^0$ and $\gamma^1$ with starting point $a$ and endpoint $b$ are *thin equivalent* if the concatenation $(\gamma^1)^{-1} \cdot \gamma^0$ is thin.

Thin equivalence in compatible with (reverse) concatenation. It is easily seen that a path $\gamma^0$ is thin equivalent to any of its reparametrisations $\gamma^1 = \gamma^0 \circ \varphi$, where $\varphi : [0, 1] \to [0, 1]$ is a continuous map such that $\varphi(0) = 0$ and $\varphi(1) = 1$: an explicit homotopy is given by $\gamma^s(t) := \gamma^0 \circ \varphi_s(t)$ with $\varphi_s(t) := (1-s)t + s\varphi(t)$. The following homotopy between the constant loop $\gamma^0(0)$ and $(\gamma^0)^{-1}\gamma^1$, given by

$$ \begin{align*} K(s,t) &= \gamma^s(2t) \text{ if } 0 \le t \le \frac{s}{2}, \\ &= \gamma^s(s) \text{ if } \frac{s}{2} \le t \le 1 - \frac{\varphi_s(s)}{2}, \\ &= \gamma^0(2-2t) \text{ if } 1 - \frac{\varphi_s(s)}{2} \le t \le 1, \end{align*} $$

has therefore its image entirely contained in $\gamma^0([0, 1])$.

In particular, $(\gamma'' \cdot \gamma') \cdot \gamma$ is thin equivalent to $\gamma'' \cdot (\gamma' \cdot \gamma)$. The thin equivalence classes of paths therefore form a groupoid, denoted by $\mathcal{G}_P^{\text{th}}$. It is easily seen that two thin equivalent paths are homotopic. The inclusion of thin equivalence classes into homotopy classes induces a natural surjective groupoid morphism

$$ \pi : \mathcal{G}_P^{\text{th}} \longrightarrow \mathcal{G}_P. $$

**Remark 1.** The three groupoids above coincide for $P = \mathbb{R}$ endowed with its usual topology, i.e.

$$ \mathcal{G}_{\mathbb{R}}^{\text{th}} = \mathcal{G}_{\mathbb{R}} = \mathbb{R} \times \mathbb{R}. $$

**Definition 2.** Let $G$ be a groupoid, and let $E = (E_x)_{x \in G^{(0)}}$ be a family of sets parametrised by $G^{(0)}$. A left action of $G$ on $E$ is a family of maps $\varphi = (\varphi_\gamma)_{\gamma \in G}$ where:

$$ \bullet \quad \varphi_\gamma : E_{s(\gamma)} \to E_{r(\gamma)}, $$
• $\varphi_x = \mathrm{Id}_{E_x}$ for any $x \in G^{(0)}$,

• $\varphi_{\gamma\gamma'} = \varphi_{\gamma} \circ \varphi_{\gamma'}$ for any $(\gamma, \gamma') \in G^{(2)}$.

In particular, the $\varphi_\gamma$'s are bijective. The **action of the groupoid G on its set of units** is the particular case when $E_x$ is reduced to a single element for any $x \in G^{(0)}$.

**Definition 3.** A **topological groupoid** is both a groupoid and a topological space, such that all structure maps are continuous, the inverse being a homeomorphism. A left action of a topological groupoid G on a family $E = (E_x)_{x \in G^{(0)}}$ of topological spaces is **continuous** if all maps $\varphi_\gamma$ are continuous.

### 3. EXTENDED METRIC SPACES OF CONTINUOUS MAPS

#### 3.1. Extended metric spaces.

**Definition 4.** ([1, Pages 419–420], see also [4, Paragraph 1.2.3] and [6]) An **extended metric space** is a set $E$ endowed with a distance $d: E \times E \to \mathbb{R}^+ \cup \{\infty\}$, such that the usual axioms for a distance, i.e.

• $d(x, y) = 0$ if and only if $x = y$,

• $d(x, y) = d(y, x)$,

• triangle inequality $d(x, z) \le d(x, y) + d(y, z)$

are verified. The only difference with a usual metric space is that the infinite value for the distance is allowed. The **galaxy** of $x \in E$ is the subset of points at finite distance from $x$.

Any galaxy is a metric space in the usual sense, and any extended metric space is a disjoint (and disconnected) union of galaxies. The notion of completeness can be straightforwardly adapted to extended metric spaces, because all terms in a Cauchy sequence belong to the same galaxy, except perhaps a finite number of those.

#### 3.2. Extended metric spaces of continuous maps.
Let $X$ and $Y$ be two metric spaces (in the extended sense or not), where $Y$ is complete. The set $C_{YX}$ of continuous maps from $X$ to $Y$ carries a natural structure of complete extended topological space, given by the distance:

$$ (2) \qquad d_{YX}(f, g) := \sup_{x \in X} d_Y(f(x), g(x)). $$

The triangle inequality is obviously verified, and the completeness is verified as follows: if $(f_k)_{k \ge 1}$ is a Cauchy sequence in $C_{YX}$, then $(f_k(x))_{k \ge 1}$ is a Cauchy sequence in $Y$ for any $x \in X$, which thus converges to an element $f(x)$ of $Y$, which appears to define the evaluation at $x$ of the limit $f$ of the sequence $(f_k)$.

Recall that a map $f: X \to Y$ between two metric spaces is Lipschitz if $\mathrm{Lip}(f) < +\infty$, with

$$ (3) \qquad \mathrm{Lip}(f) := \sup_{x' \neq x''} \frac{d_Y(f(x'), f'x'')}{d_X(x', x'')} $$

In particular any Lipschitz map is continuous. If $Z$ is another metric space and $g: Y \to Z$ is another map, we obviously have

$$ \mathrm{Lip}(g \circ f) \le \mathrm{Lip}(g) \mathrm{Lip}(f). $$

The two following lemmas can be found in [10, Paragraph 2.1]. We simply rephrase them in the context of several metric spaces.
**Lemma 2.** Let X, Y, Z be three metric spaces. Let $f, f' \in C_{YX}$ and $g, g' \in C_{ZY}$. Then

$$ (4) \qquad d_{ZX}(g \circ f, g' \circ f') \le d_{ZY}(g, g') + \mathrm{Lip}(g')d_{YX}(f, f'). $$

*Proof.* From the triangle inequality in $C_{ZX}$ we have:

$$ \begin{aligned} d_{ZX}(g \circ f, g' \circ f') &\le d_{ZX}(g \circ f, g' \circ f) + d_{ZX}(g' \circ f, g' \circ f') \\ &\le \sup_{x \in X} d_Z(g \circ f(x), g' \circ f(x)) + \sup_{x \in X} d_Z(g' \circ f(x), g' \circ f'(x)) \\ &\le \sup_{y \in Y} d_Z(g(y), g'(y)) + \mathrm{Lip}(g') \sup_{x \in X} d_Y(f(x), f'(x)) \\ &\le d_{ZY}(g, g') + \mathrm{Lip}(g')d_{YX}(f, f'). \end{aligned} $$

□

Lemma 2 can be immediately generalised as follows:

**Lemma 3.** Let $X_0, \dots, X_r$ be a collection of metric spaces, with $X_j$ complete for $j = 0, \dots, n-1$. Let $f_j, f'_j \in C_{X_{j-1}X_j}$ for any $j = 1, \dots, r$. Then

$$ (5) \qquad d_{X_0 X_r} (f_1 \circ \dots \circ f_r, f'_1 \circ \dots \circ f'_r) \le \sum_{j=1}^{r} \left( \prod_{i=1}^{j-1} \mathrm{Lip}(f'_i) \right) d_{X_{j-1} X_j} (f_j, f'_j). $$

The various distances $d_X, d_Y, d_{YX}, \dots$ involved will occasionally be denoted by the same letter $d$ when the context clearly indicates which distance is to be considered.

3.3. **Length spaces.** Let $P$ be an extended metric space, and let $\gamma: I \to P$ a path, i.e. a continuous map from an interval $I$ of $\mathbb{R}$ into $P$. The length of the path $\gamma$ is defined by

$$ (6) \qquad \ell(\gamma) := \sup_{k \in \mathbb{N}, t_0 \le \dots \le t_k \in I} \sum_{i=1}^{k} d(\gamma(t_{i-1}), \gamma(t_i)). $$

The metric space $P$ is a length space if for any $x, y \in P$ the following equality holds:

$$ (7) \qquad d(x, y) = \inf_{\text{paths } \gamma: [0,1] \to P, \gamma(0)=x \text{ and } \gamma(1)=y} \ell(\gamma). $$

In other words, a length space is an extended metric space in which the distance is given by the minimum of the lengths of paths from $x$ to $y$. More details on length spaces can be found in [6, Paragraph 2.3].

## 4. LOCAL FLOWS, LOCAL APPROXIMATE FLOWS AND THE SEWING LEMMA

**Definition 5.** Let $(M_t, d_t)_{t \in \mathbb{R}}$ a one-parameter family of complete metric spaces, possibly in the extended sense. Let $C_{st}$ be the set of continuous maps from $M_t$ to $M_s$, endowed with the distance $d_{M_s M_t}$ that will be denoted by $d_{st}$. A **local flow** associated with these data is a continuous action of the pair groupoid $\mathbb{R} \times \mathbb{R}$ on the family $(M_t)$. It is therefore a two-parameter family $(\varphi_{st})_{s,t \in \mathbb{R}}$ of maps such that

* $\varphi_{st} \in C_{st}$,

* $\varphi_{ss} = \mathrm{Id}_{M_s}$ for any $s \in \mathbb{R}$,

* $\varphi_{st} = \varphi_{su} \circ \varphi_{ut}$ for any $s, t, u \in \mathbb{R}$.

In particular, if $(\varphi_{st})$ is a local flow, $\varphi_{st}: M_t \to M_s$ is a homeomorphism, with inverse $\varphi_{ts}$.
**Remark 4.** In view of our main result (Theorem 6 below), it is natural to ask for Lipschitz bounds on the homeomorphisms $\varphi_{st}$, namely

$$ (8) \qquad \mathrm{Lip}(\varphi_{st}) \le e^{L|t-s|} $$

for some positive constant $L$, thus taking the metric structure of the real line $\mathbb{R}$ into account.

**Remark 5.** Definition 5 and estimates (8) together make sense for any metric space $P$ instead of $\mathbb{R}$, we'll return to this point later.

**Definition 6.** Let $(M_t, d_t)_{t \in \mathbb{R}}$ a one-parameter family of complete metric spaces, possibly in the extended sense. A local approximate flow associated with these data is a two-parameter family $(\mu_{st})_{s,t \in \mathbb{R}}$ of maps such that

* $\mu_{st} \in C_{st}$,

* $\mu_{ss} = \mathrm{Id}_{M_s}$ for any $s \in \mathbb{R}$,

* There exists $L > 0$ such that $\mathrm{Lip}(\mu_{st}) \le 1 + L|t - s|$,

* There exists $\varepsilon > 0$, an integer $n \ge 1$ and two collections of positive constants $a_i, b_i, i = 1, \dots, n$ with $a_i + b_i = 1 + \varepsilon$, such that for any $s, t, u \in \mathbb{R}$ the following estimates hold:

$$ (9) \qquad d_{st}(\mu_{st}, \mu_{su} \circ \mu_{ut}) \le \sum_{i=1}^{n} C_i |t - u|^{a_i} |u - s|^{b_i}. $$

**Theorem 6.** Let $(\mu_{st})_{s,t \in \mathbb{R}}$ be a local approximate flow as in Definition 6. Then there exists a unique flow $(\varphi_{st})$ such that

$$ (10) \qquad d_{st}(\varphi_{st}, \mu_{st}) \le C'|t-s|^{1+\varepsilon}, $$

which is obtained as the limit in $C_{st}$ of the compositions

$$ \mu_{st_1} \circ \mu_{t_1 t_2} \circ \dots \circ \mu_{t_{k-1} t_k} $$

when the mesh of the subdivision $(s, t_1, \dots, t_{k-1}, t)$ of the interval between $s$ and $t$ tends to zero. One possible constant $C'$ is given by

$$ C' = 2^{1+\varepsilon} e^{L|t-s|} \left( \sum_{i=1}^{n} C_i \right) \zeta(1+\varepsilon), $$

where $\zeta$ is the Riemann zeta function. Moreover, the local flow $(\varphi_{st})$ satisfies the Lipschitz condition:

$$ (11) \qquad \mathrm{Lip}(\varphi_{st}) \le e^{L|t-s|}. $$

The proof of Theorem 6 requires several steps. Let $(\mu_{st})$ be a local approximate flow. Let $I = (t_0, \dots, t_k)$ be a subdivision of the interval between $s$ and $t$, arranged in increasing or decreasing order according to the sign of $t - s$, with $t_0 = s$ and $t_k = t$. The mesh of the subdivision is defined by:

$$ (12) \qquad \mathrm{mesh}(I) := \sup_{j=1,\dots,k} |t_j - t_{j-1}|. $$

Let us consider

$$ \mu_{st}^I := \mu_{t_0 t_1} \circ \mu_{t_1 t_2} \circ \dots \circ \mu_{t_{k-1} t_k}. $$

**Lemma 7.** There exists of a constant $K$ independent of the subdivision $I$ such that

$$ (13) \qquad d_{st}(\mu_{st}, \mu_{st}^I) \le K e^{L|t-s|} |t-s|^{1+\varepsilon}. $$
Proof. From I above we define $I_1, \dots, I_k$ above such that $I_j := [t_{j-1}, t_j]$ if $s < t$, and $I_j :=]t_j, t_{j-1}]$ if $s > t$. Denote by $|I_j| = |t_j - t_{j-1}|$ the length of the $j$th interval. We have then

$$ (14) \qquad \inf_{j=1,\dots,k-1} (|I_j| + |I_{j+1}|) \le \frac{2|t-s|}{k-1}, $$

as we can see straightforwardly by writing the whole interval between $s$ and $t$ (with $t$ excluded) as the disjoint union of the $I_j$'s, which in turn can be regrouped two by two, omitting $I_k$ if $k$ is odd. Let us consider the index $j$ realising the minimum in (14), and let $J$ be the subdivision obtained from $I$ by merging the two blocks $I_j$ and $I_{j+1}$, thus suppressing $t_j$. We have then

$$ \begin{aligned} d(\mu_{st}^I, \mu_{st}^J) &= d(\mu_{t_0t_1} \circ \dots \circ \mu_{t_{j-1}t_j} \circ \mu_{t_jt_{j+1}} \circ \dots \circ \mu_{t_{k-1}t_k}, \mu_{t_0t_1} \circ \dots \circ \mu_{t_{j-1}t_{j+1}} \circ \dots \circ \mu_{t_{k-1}t_k}) \\ &\le \prod_{r=1}^{j-1} \mathrm{Lip}(\mu_{t_{r-1}t_r}) \sum_{i=1}^n C_i |I_j|^{b_i} |I_{j+1}|^{a_i} \\ &\le e^{L|t-s|} \left( \sum_{i=1}^n C_i \right) \left( \frac{2}{k-1} \right)^{1+\varepsilon} |t-s|^{1+\varepsilon}. \end{aligned} $$

Iterating this process down to the trivial subdivision, we get (13), with

$$ (15) \qquad K = 2^{1+\varepsilon} \left( \sum_{i=1}^{n} C_i \right) \sum_{\ell=1}^{\infty} \left( \frac{1}{\ell} \right)^{1+\varepsilon} = 2^{1+\varepsilon} \left( \sum_{i=1}^{n} C_i \right) \zeta(1+\varepsilon). $$

**Lemma 8.** Let $(\mu_{st})$ be a local approximate flow, and let $I, J$ be two subdivisions of the interval between $s$ and $t$, with $J$ finer than $I$. Then

$$ (16) \qquad d(\mu_{st}^I, \mu_{st}^J) \le K e^{L|t-s|} e^{L \mathrm{mesh}(I)} \mathrm{mesh}(I)^{\varepsilon} |t-s|. $$

Proof. We denote by $J_j$ the subdivision of the block $I_j$ induced by $J$, and we use the shorthand $\mu_{t_{j-1}t_j}^J$ for $\mu_{t_{j-1}t_j}^{J_j}$. We can compute, using Lemma 7:

$$ \begin{aligned} d(\mu_{st}^I, \mu_{st}^J) &\le \sum_{j=0}^{r-1} d(\mu_{t_0t_1} \cdots \mu_{t_{j-1}t_j} \mu_{t_jt_{j+1}} \cdots \mu_{t_{k-1}t_k}, \mu_{t_0t_1} \cdots \mu_{t_{j-1}t_j} \mu_{t_jt_{j+1}}^J \cdots \mu_{t_{k-1}t_k}^J) \\ &\le e^{L|t-s|} \sum_{j=0}^{r-1} d(\mu_{t_jt_{j+1}}, \mu_{t_jt_{j+1}}^J) \\ &\le e^{L|t-s|} \sum_{j=0}^{r-1} K e^{L|t_{j+1}-t_j|} |t_{j+1} - t_j|^{1+\varepsilon} \\ &\le K e^{L|t-s|} e^{L \mathrm{mesh}(I)} \sum_{j=0}^{r-1} |t_{j+1} - t_j| |t_{j+1} - t_j|^{\varepsilon} \\ &\le K e^{L|t-s|} e^{L \mathrm{mesh}(I)} \sum_{j=0}^{r-1} |t_{j+1} - t_j| \mathrm{mesh}(I)^{\varepsilon} \\ &\le K e^{L|t-s|} e^{L \mathrm{mesh}(I)} \mathrm{mesh}(I)^{\varepsilon} |t-s|. \end{aligned} $$

<figure><img src="image_3.png" /></figure>
**Corollary 9.** For any pair $(I, I')$ of subdivisions we have:

$$ (17) \qquad d(\mu_{st}^I, \mu_{st}^{I'}) \le K e^{L|t-s|} (e^{L\operatorname{mesh}(I)} \operatorname{mesh}(I)^\varepsilon + e^{L\operatorname{mesh}(I')} \operatorname{mesh}(I')^\varepsilon) |t-s|. $$

*Proof.* It is an immediate consequence of the triangle inequality

$$ d(\mu_{st}^I, \mu_{st}^{I'}) \le d(\mu_{st}^I, \mu_{st}^J) + d(\mu_{st}^J, \mu_{st}^{I'}) $$

where $J$ is the joint subdivision of $I$ and $I'$, i.e. the coarsest subdivision which is both finer than $I$ and finer than $I'$. $\square$

**Corollary 10.** For any $s, t \in \mathbb{R}$, the family $(\mu_{st}^I)_I$ converges in $C_{st}$ to a limit $\varphi_{st}$ when $\operatorname{mesh}(I)$ tends to 0, and we have

$$ (18) \qquad d_{st}(\mu_{st}, \varphi_{st}) \le K e^{L|t-s|} |t-s|^{1+\varepsilon}. $$

where the constant $K$ is given by (15).

*Proof of Theorem 6.* Let $s, t, u \in \mathbb{R}$, with $u$ between $s$ and $t$. For any subdivision $I$ (resp. $I'$) of the interval between $s$ and $u$ (resp. between $u$ and $t$), the union of both subdivisions yields a subdivision $J$ of the interval between $s$ and $t$, with $\operatorname{mesh}(J) = \sup (\operatorname{mesh}(I), \operatorname{mesh}(I'))$, and we have

$$ \mu_{st}^J = \mu_{su}^I \circ \mu_{ut}^{I'}. $$

Letting the mesh of $I$ and $I'$ go to zero, Corollary 10 yields:

$$ (19) \qquad \varphi_{st} = \varphi_{su} \circ \varphi_{ut}. $$

It remains to show that $\varphi_{ts}$ is the inverse of $\varphi_{st}$. Choosing, for example, the regular subdivision $I_k = (t_0, \dots, t_k)$ with $t_j = s + j(t-s)/k$, and denoting by $\overline{I_k}$ the corresponding reverse subdivision, we have

$$ \mu_{st}^{I_k} \circ \mu_{ts}^{\overline{I_k}} = \mu_{t_0 t_1} \circ \dots \circ \mu_{t_{k-1} t_k} \circ \mu_{t_k t_{k-1}} \circ \dots \circ \mu_{t_1 t_0}. $$

Hence we get

$$ \begin{align*} d(\mu_{st}^{I_k} \circ \mu_{ts}^{\overline{I_k}}, \mu_{ss}) &\le \sum_{j=0}^{k-1} d(\mu_{t_0 t_1} \circ \dots \circ \mu_{t_{j-1} t_j} \circ \mu_{t_j t_{j-1}} \circ \dots \circ \mu_{t_1 t_0}, \mu_{t_0 t_1} \circ \dots \circ \mu_{t_j t_{j+1}} \circ \mu_{t_{j+1} t_j} \circ \dots \circ \mu_{t_1 t_0}) \\ &\le e^{L|t-s|} \sum_{j=0}^{r-1} d(\mu_{t_j t_j}, \mu_{t_j t_{j+1}} \circ \mu_{t_{j+1} t_j}) \\ &\le e^{L|t-s|} r \left( \sum_{i=1}^n C_i \right) \left( \frac{|t-s|}{r} \right)^{1+\varepsilon}, \end{align*} $$

which gives $\varphi_{st} \circ \varphi_{ts} = \varphi_{ss}$ by letting $r$ go to infinity. Finally, the Lipschitz condition follows from the Lipschitz condition for $\mu_{st}^{I_k}$:

$$ \mathrm{Lip}(\mu_{st}^{I_k}) \le \left( 1 + \frac{L|t-s|}{r} \right)^r, $$

letting $r$ go to infinity. $\square$
**Remark 11.** If $(\mu_s t)_{s,t \in \mathbb{R}}$ is an approximate flow in the sense of Definition 6, for any $s, t, u, v \in \mathbb{R}$ we have from Lemma 2 and (9):

$$ (20) \qquad \begin{aligned} d_{st}(\mu_{su} \circ \mu_{ut}, \mu_{sv} \circ \mu_{vt}) &\le d_{st}(\mu_{su} \circ \mu_{ut}, \mu_{su} \circ \mu_{uv} \circ \mu_{vt}) + d_{st}(\mu_{su} \circ \mu_{uv} \circ \mu_{vt}, \mu_{sv} \circ \mu_{vt}) \\ &\le (1 + |u - s|L) \sum_{i=1}^{n} C_i |t - v|^{a_i} |u - v|^{b_i} + \sum_{i=1}^{n} C_i |s - u|^{a_i} |u - v|^{b_i}. \end{aligned} $$

Applying the inequality (20) above for $v = t$ gives back (9). It gives moreover the optimal bound $0$ for $u = v$.

# 5. APPLICATIONS AND GENERALISATIONS

## 5.1. Applications.

**Example 1.** Consider the constant family $M_t = E$ where $E$ is a Banach space, and consider a local approximate flow defined by translations:

$$ \mu_{st}(x) := x + \tilde{\mu}(s, t) $$

for any $x \in E$. These are isometries, hence all Lipschitz constants involved are equal to one. The local approximate flow property is expressed as

* $\tilde{\mu}(t, t) = 0$ for any $t \in \mathbb{R}$,

* $|\tilde{\mu}(s, u) + \tilde{\mu}(u, t) - \tilde{\mu}(s, t)| \le \sum_{i=1}^{n} C_i |t - u|^{a_i} |u - s|^{b_i}$ for any $s, t, u \in \mathbb{R}$,

with $a_i + b_i = 1 + \varepsilon$. Theorem 6 for this example amounts to the existence of a unique map $\tilde{\varphi}: \mathbb{R} \times \mathbb{R} \to E$ such that $\tilde{\varphi}(s, t) = \tilde{\varphi}(s, u) + \tilde{\varphi}(u, t)$ and $|\tilde{\varphi}(s, t) - \tilde{\mu}(s, t)| \le C|t - s|^{1+\varepsilon}$. This is the original formulation of the sewing lemma [14, 11].

**Example 2.** A sewing lemma for maps with noncommuting values first appeared in [12]. Consider again the case of the constant family $M_t = E$ where $E$ is a Banach space, with a more general local approximate flow $(\mu_{st})$. Applying Theorem 6 in this case gives back Theorem 1 in [2]. Hypothesis (H3) therein, namely the uniform Lipschitz estimate (by the bound $e^{L|t-s|}$) for the maps $\mu_{st}^I$ when the subdivision $I$ varies, is a straightforward consequence of our Lemma 3.

## 5.2. The groupoid of Lipschitz paths modulo thin Lipschitz equivalence. 
Let $P$ be a metric space. Recall that a path $\gamma: [0, 1] \to P$ is Lipschitz if

$$ \mathrm{Lip}(\gamma) = \sup_{s,t \in [0,1], s \neq t} \frac{d_P(\gamma(s), \gamma(t))}{|s-t|} < +\infty. $$

**Definition 7.** Two Lipschitz paths $\gamma^0$ and $\gamma^1$ are Lipschitz-homotopic if there exists a Lipschitz homotopy $H: [0, 1]^2 \to P$ between both paths.

Two Lipschitz paths can be homotopic without being Lipschitz-homotopic. But any thin loop $\gamma$ is Lipschitz-thin, in the sense that the homotopy from $\gamma$ to the constant loop $\{\gamma(0)\}$ can be chosen as a Lipschitz homotopy. To see that, let $H: [0, 1]^2 \to P$ be a homotopy from $\gamma$ to the constant loop such that $H([0, 1]^2 \subset \gamma([0, 1])$. Recalling the notation $\gamma^s(t) = H(s, t)$, we choose the Lipschitz homotopy as follows:

$$ \begin{aligned} \bar{H}(s, t) &= \gamma(t) \text{ if } \gamma(t) \in \gamma^s([0, 1]), \\ &= \gamma(t_s) \text{ if not,} \end{aligned} $$

with $t_s := \inf\{t \in [0, 1], \gamma(t) \notin \gamma^s([0, 1])\}$. As a consequence, thin equivalence makes perfect sense in the subclass of Lipschitz paths.
**Proposition 12.** For any pair of paths $(\gamma, \gamma')$ in a metric space $P$, the following estimate holds:

$$ \mathrm{Lip}(\gamma \cdot \gamma') \le 2 \sup (\mathrm{Lip}(\gamma), \mathrm{Lip}(\gamma')). $$

Moreover, for any $s, t \in [0, 1]$ we have

$$ \mathrm{Lip}(\gamma_{st}) \le |t - s| \mathrm{Lip}(\gamma). $$

*Proof.* This is immediate. The factors 2 and $|t - s|$ come from rescaling. $\square$

Thin Lipschitz equivalence is compatible with concatenation, and $\gamma.(\gamma'.\gamma'')$ is Lipschitz-thin equivalent to $(\gamma.\gamma')\gamma''$ for any triple of Lipschitz paths composable that way. This therefore defines the **Lipschitz-thin equivalence groupoid** $G_P^{\mathrm{Lip-th}} \subseteq G_P^{\mathrm{th}}$ as the equivalence classes of Lipschitz paths modulo thin equivalence. The inclusion may be strict, because there can exist a path in $P$ which is not thin-homotopic to any Lipschitz path: an example of this situation is given by the image of a non-Lipschitz path (e.g. a sample path of a Brownian motion) in $\mathbb{R}^2$.

**5.3. The Lipschitz homotopy groupoid.** This notion is adapted from [8, Definition 4.1]. Let $P$ be a metric space, and let $x, y \in P$. Two Lipschitz paths with domain $[0, 1]$ from $x$ to $y$ are Lipschitz-homotopic if there exists a Lipschitz homotopy $H : [0, 1]^2 \to P$. This notion is obviously compatible with concatenation. The Lipschitz homotopy groupoid $G_P^{\mathrm{Lip}}$ is the set of Lipschitz paths $\gamma : [0, 1] \to P$ modulo Lipschitz homotopy.

Both groupoids $G_P$ and $G_P^{\mathrm{Lip}}$ often coincide, for example when $P$ is a smooth manifold: in this case, this is due to the fact that any continuous path is homotopic to a smooth (hence Lipschitz) one, and any two homotopic smooth paths can be interpolated by a smooth homotopy.

**5.4. A generalisation of Theorem 6.** Approximate local flows, in the sense of Definition 6, can be straightforwardly generalised as follows:

**Definition 8.** Let $P$ be a metric space, and let $(M_x, d_x)_{x \in P}$ be a family of complete metric spaces indexed by $P$, possibly in the extended sense. An **approximate action of the pair groupoid** $P \times P$ associated with these data is a two-parameter family $(\mu_{xy})_{x,y \in P}$ of maps such that

* $\mu_{xy} \in C_{xy}$,

* $\mu_{xx} = \mathrm{Id}_{M_x}$ for any $x \in \mathbb{R}$,

* There exists $L > 0$ such that $\mathrm{Lip}(\mu_{xy}) \le 1 + L \cdot d_P(x, y)$,

* There exists $\varepsilon > 0$, an integer $n \ge 1$ and two collections of positive constants $a_i, b_i, i = 1, \dots, n$ with $a_i + b_i = 1 + \varepsilon$, such that for any $x, y, z \in P$ the following estimates hold:

$$ (21) \qquad d_{xy}(\mu_{xy}, \mu_{xz} \circ \mu_{zy}) \le \sum_{i=1}^{n} C_i d_P(y, z)^{a_i} d_P(z, x)^{b_i}. $$

**Remark 13.** Estimates (21) are equivalent to the four-point estimates (22)

$$ d_{xy}(\mu_{xu} \circ \mu_{uy}, \mu_{xv} \circ \mu_{vy}) \le (1 + L d_P(u, x)) \sum_{i=1}^{n} C_i d_P(y, v)^{a_i} d_P(u, v)^{b_i} + \sum_{i=1}^{n} C_i d_P(x, u)^{a_i} d_P(u, v)^{b_i}. $$

the same way (9) are equivalent to (20) (see Remark 11).
**Theorem 14.** Let $P$ be a metric space, and let $((M_x), (\mu_{xy}))$ be an approximate action of the pair groupoid $P \times P$ in the sense of Definition 8. Then there exists a unique continuous action $\varphi$ of the Lipschitz thin equivalence groupoid $\mathcal{G}_P^{\text{Lip-th}}$ such that

$$ (23) \qquad d_{\gamma(s)\gamma(t)}(\varphi_{\gamma_{st}}, \mu_{\gamma(s)\gamma(t)}) \le C'|t-s|^{1+\varepsilon}, $$

which is obtained as the limit in $C_{\gamma(s)\gamma(t)}$ of the compositions

$$ \mu_{\gamma(s)\gamma(t_1)} \circ \mu_{\gamma(t_1)\gamma(t_2)} \circ \dots \circ \mu_{\gamma(t_{k-1})\gamma(t)} $$

when the mesh of the subdivision $(s, t_1, \dots, t_{k-1}, t)$ of the interval between $s$ and $t$ tends to zero. One possible constant $C'$ is given by

$$ C' = 2^{1+\varepsilon} e^{L \text{Lip}(\gamma)} |t-s| \left( \sum_{i=1}^{n} C_i \right) \zeta(1+\varepsilon). $$

Moreover, the continuous action $\varphi$ satisfies the Lipschitz condition:

$$ (24) \qquad \text{Lip}(\varphi_{\gamma_{st}}) \le e^{L \text{Lip}(\gamma)} |t-s|. $$

*Proof.* Theorem 14 is a direct consequence of Theorem 6 applied to the local approximate flow $(\mu_{st}^{\gamma}) := (\mu_{\gamma(s)\gamma(t)})$, where the path $\gamma: [0, 1] \to P$ is extended to $\mathbb{R}$ by $\gamma(u) := \gamma(0)$ for $u \le 0$ and $\gamma(u) = \gamma(1)$ for $u \ge 1$. $\square$

**5.5. The knitting lemma.** The tempting comparison with the higher-dimensional sewing lemma of [16] (see also [18], [17] and [9] in the two-dimensional case) is not straightforward. One however may look for a stronger notion of approximate action of the pair groupoid $P \times P$ so that the action of the groupoid $\mathcal{G}_P^{\text{Lip-th}}$ given by Theorem 14 factorises through an action of the fundamental groupoid $\mathcal{G}_P$, or at least the Lipschitz homotopy groupoid $\mathcal{G}_P^{\text{Lip}}$. A possible answer is given by the following theorem:

**Theorem 15 (knitting lemma).** Let $P$ be a metric space. Let $((M_x), (\mu_{xy}))$ be a strongly approximate action of the pair groupoid $P \times P$, in the sense that the three-point estimate (21) in Definition 8 (or, equivalently, the four-point estimate (22)) is replaced by the stronger four-point estimate

(25)

$$ d_{xy}(\mu_{xu} \circ \mu_{uy}, \mu_{xv} \circ \mu_{vy}) \le (1 + L d_P(u, x)) \sum_{i=1}^{n} C_i d_P(y, v)^{a_i} d_P(u, v)^{b_i} + \sum_{i=1}^{n} C_i d_P(x, u)^{b_i} d_P(u, v)^{a_i}. $$

for any $x, u, v, y \in P$, where $(a_i, b_i)_{i=1, \dots, n}$ is a collection of positive numbers such that

$$ a_i + b_i = 2 + \varepsilon $$

with $\varepsilon > 0$. Then there exists a unique continuous action $\varphi$ of the Lipschitz fundamental groupoid $\mathcal{G}_P^{\text{Lip}}$ such that

$$ (26) \qquad d_{\gamma(s)\gamma(t)}(\varphi_{\gamma_{st}}, \mu_{\gamma(s)\gamma(t)}) \le C'|t-s|^{1+\varepsilon}, $$

which is obtained as the limit in $C_{\gamma(s)\gamma(t)}$ of the compositions

$$ \mu_{\gamma(s)\gamma(t_1)} \circ \mu_{\gamma(t_1)\gamma(t_2)} \circ \dots \circ \mu_{\gamma(t_{k-1})\gamma(t)} $$
when the mesh of the subdivision $(s, t_1, \dots, t_{k-1}, t)$ of the interval between $s$ and $t$ tends to zero. One possible constant $C'$ is given by

$$ (27) \qquad C' = 2^{1+\varepsilon} e^{L \mathrm{Lip}(\gamma)|t-s|} \left( \sum_{i=1}^{n} C_i \right) \zeta(2+\varepsilon). $$

Moreover, the continuous action $\varphi$ satisfies the Lipschitz condition:

$$ (28) \qquad \mathrm{Lip}(\varphi_{\gamma_{st}}) \le e^{L \mathrm{Lip}(\gamma)|t-s|}. $$

*Proof.* The proof of Theorem 15 requires two steps.

**Step 1:** For any Lipschitz path $\gamma : [0, 1] \to P$, Equations (26) and (27) are obtained from Theorem 6 applied to the approximate flow $((\mu_\gamma)_{s,t})_{s,t \in [0,1]}$ given by $(\mu_\gamma)_{s,t} := \mu_{\gamma(s)\gamma(t)}$, with the modification $\zeta(2+\varepsilon)$ instead of $\zeta(1+\varepsilon)$ coming from the strong four-point estimate (25). The Lipschitz parameter from Definition 6 for the approximate flow $(\mu_\gamma)$ is indeed $L \mathrm{Lip}(\gamma)$. Details are left to the reader.

**Step 2:** It remains to show that $\varphi_\gamma$ does not depend on the choice of the path $\gamma$ in a given Lipschitz homotopy class. Let $x, y \in P$, let $\gamma^0$ and $\gamma^1$ be two Lipschitz continuous paths from $[0, 1]$ to $P$ with starting point $x$ and endpoint $y$. Let $H : [0, 1]^2 \to P$ be a Lipschitz homotopy from $\gamma^0$ to $\gamma^1$. We recall the convenient notation $\gamma^s(t) := H(s, t)$. Let $\ell$ be the Lipschitz norm of $H$, namely

$$ \ell := \sup_{t,s,t',s' \in [0,1], (s,t) \ne (s',t')} \frac{d_P(\gamma^s(t), \gamma^{s'}(t'))}{|t' - t| + |s' - s|}. $$

In particular, the Lipschitz norm of any path $\gamma^s$, $s \in [0, 1]$ is smaller or equal to $\ell$. As a consequence, $d_P(x, y) \le \ell$.

Let $k \ge 2$ be an integer, let $0 = t_0 \le t_1 \dots \le t_k = t$ be the regular subdivision of $[0, 1]$ with mesh $\delta = 1/k$, and let us denote $H(t_i, t_j) = \gamma^{t_i}(t_j)$ by $x_j^i$ for any $i, j \in \{0, \dots, k\}$. The mesh of the net $I = (x_j^i)_{i,j=0,\dots,k}$ is defined by

$$ (29) \qquad \mathrm{mesh}_H(I) := \sup \{ \sup_{i=0,\dots,k, j=1,\dots,k} d_P(x_{j-1}^i, x_j^i), \sup_{i=1,\dots,k, j=0,\dots,k} d_P(x_j^{i-1}, x_j^i) \}. $$

By definition we have $x_0^i = x$ and $x_k^i = y$ for any $i \in \{0, \dots, k\}$. The following estimate for the mesh of the net clearly holds true:

$$ (30) \qquad \mathrm{mesh}_H(I) \le \delta \ell. $$
<figure><img src="image_4.png" /></figure>

An example of net with $k = 6$.

Let $(\mu_{uv})$ be a strongly approximate action of the pair groupoid $P \times P$. Let us consider for any $i, j \in \{0, \dots, k-1\}$:

$$ (31) \qquad \mu_{xy}^{I,ij} := \mu_{x_0^i x_1^i} \circ \mu_{x_1^i x_2^i} \circ \dots \circ \mu_{x_{k-j-1}^i x_{k-j}^{i+1}} \circ \dots \circ \mu_{x_{k-1}^i x_k^{i+1}}. $$

In particular,

$$ (32) \qquad \mu_{xy}^{I,(i-1)(k-1)} = \mu_{xy}^{I,i0} = \mu_{x_0^i x_1^i} \circ \dots \circ \mu_{x_{k-1}^i x_k^i}. $$

We are now in position to compute:

$$
\begin{align*}
d_{xy}(\mu_{x_0^0 x_1^0} \circ \dots \circ \mu_{x_{k-1}^0 x_k^0}, \mu_{x_0^k x_1^k} \circ \dots \circ \mu_{x_{k-1}^k x_k^k})
& \\
&\le \sum_{i \in \{0, \dots, k-1\}} \left( \sum_{j \in \{0, \dots, k-2\}} d_{xy}(\mu_{xy}^{I,ij}, \mu_{xy}^{I,i(j+1)}) \right) + d(\mu_{xy}^{I,i(k-1)}, \mu_{xy}^{I,(i+1)0})
& \\
&\le \sum_{i \in \{0, \dots, k-1\}} \sum_{j \in \{0, \dots, k-2\}} d_{xy}(\mu_{xy}^{I,ij}, \mu_{xy}^{I,i(j+1)}) \quad \text{(from (32))}
& \\
&\le k^2 e^{\delta \ell L} (2 + \delta \ell L) \left( \sum_{r=1}^n C_i \right) (\delta \ell)^{2+\epsilon} \quad \text{(from (25) and (30))}
& \\
&\le e^{\delta \ell L} (2 + \delta \ell L) \left( \sum_{r=1}^n C_i \right) \ell^{2+\epsilon} \delta^{\epsilon}.
&
\end{align*}
$$

The right-hand side converges to zero as $\delta \to 0$ (in other words, for $k \to +\infty$), so that we get

$$ \varphi_{\gamma^0} = \varphi_{\gamma^1}, $$

which concludes the proof of Theorem 15.

<figure><img src="image_5.png" /></figure>

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C. CURRY: NTNU GJØVIK, CHARLES.CURRY@NTNU.NO

D. MANCHON: LABORATOIRE DE MATHÉMATIQUES BLAISE PASCAL (LMBP), CS 60026, 3 PLACE VASARÉLY, 63178 AUBIÈRE, FRANCE. DOMINIQUE.MANCHON@UCA.FR