R Ghezzi

F Jean

A new class of (H k , 1)-rectifiable subsets of metric spaces *

The main motivation of this paper arises from the study of Carnot-Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree k, which are Hölder but not Lipschitz continuous when k > 1. Replacing Lipschitz curves by this kind of curves we define (H k , 1)-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot-Carathéodory spaces.

Introduction

The main motivation of this paper arises from the study of Carnot-Carathéodory spaces. Recall that such a metric space (M, d) is defined by a sub-Riemannian manifold (M, D, g), where M is a smooth manifold, D a subbundle of T M and g a Riemannian metric on D. The absolutely continuous paths which are almost everywhere tangent to D are called horizontal and their length is obtained as in Riemannian geometry integrating the norm of their tangent vectors. The distance d is defined as the infimum of length of horizontal paths between two given points. By construction, only horizontal paths may have finite length and may be Lipschitz with respect to the distance. In contrast to the Euclidean case, both properties are independent on the regularity: all smooth non-horizontal paths have infinite length and are not Lipschitz. This gives rise to two kind of questions.

The first query concerns the measure of non-horizontal paths: what kind of notion is the best suited? One of our motivation is that, from an intrinsic point of view, computing measures of paths should allow to determine metric invariants of curves and thus recover metrically the structure of the manifold [START_REF] Gromov | Carnot-Carathéodory spaces seen from within[END_REF]. Since non-horizontal paths have a metric dimension greater than one (see [START_REF] Jean | Entropy and complexity of a path in sub-Riemannian geometry[END_REF]), Hausdorff measures are the most natural candidates. However they pose two problems: first they can hardly be computed (except for specific cases [START_REF] Agrachev | On the Hausdorff volume in sub-Riemannian geometry[END_REF]), second they do not appear as integrals along the path, which is what we expect for a measure generalizing the notion of length.

The second question comes from geometric measure theory. A typical problem in this field is whether it is possible to characterize the geometric structure of a set using only measures. This gave rise to the notion of rectifiable sets, which is based on Lipschitz functions, and to density results in Euclidean (see [START_REF] Besicovitch | On the fundamental geometrical properties of linearly measurable plane sets of points[END_REF][START_REF] Federer | The (ϕ, k) rectifiable subsets of n-space[END_REF][START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF] and [START_REF] Mattila | Geometry of sets and measures in Euclidean spaces: fractals and rectifiability[END_REF] for a complete presentation) and general metric spaces [START_REF] Kirchheim | Rectifiable metric spaces: local structure and regularity of the Hausdorff measure[END_REF]. In the context of Carnot-Carathéodory spaces rectifiable sets have been studied in Heisenberg groups (see [START_REF] Franchi | Rectifiability and perimeter in the Heisenberg group[END_REF][START_REF] Mattila | Characterizations of intrinsic rectifiability in Heisenberg groups[END_REF]) and a different notion of rectifiability was proposed in [START_REF] Magnani | Characteristic points, rectifiability and perimeter measure on stratified groups[END_REF]. However, in these spaces the class of Lipschitz paths is quite poor and does not include non-horizontal smooth curves which consequently are not rectifiable in the usual sense. To take into account the latter curves we need to define rectifiability through a larger class of paths, intrinsically characterized by the distance.

In this paper we address these issues in any metric space, not only in Carnot-Carathéodory ones, by defining a class of curves in the spirit of ( [START_REF] Ambrosio | Rectifiable sets in metric and Banach spaces[END_REF][START_REF] Kirchheim | Rectifiable metric spaces: local structure and regularity of the Hausdorff measure[END_REF]). Namely, we introduce curves on a metric space (M, d) that are continuously metric differentiable of degree k (m-C 1 k for short) as continuous curves γ : [a, b] → M such that the map

t → meas k t (γ) = lim s→0 d(γ(t + s), γ(t)) |s| 1/k
k is well-defined and continuous (see Definition 1). In an Euclidean space, this definition is useless since the class of m-C 1 k curves with non-zero measure is empty when k > 1 (see Proposition 2). However, in the sub-Riemannian context, for integer values of k this class of curves contains some smooth non-horizontal paths (see Proposition 1).

For m-C 1 k curves we can compute different kind of measures. First, we examine the Hausdorff measures: the usual ones H k and the spherical ones S k . Second, we study the k-dimensional length of a curve γ : [a, b] → M introduced in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF] and defined by

Length k (γ([a, b])) = ˆb a meas k t (γ)dt.
Third, we consider a measure based on approximations by finite sets called interpolation complexity (see [START_REF] Gauthier | On the one-step-bracket-generating motion planning problem[END_REF][START_REF] Jean | Paths in Sub-Riemannian Geometry[END_REF]). The first result of the paper (Theorem 1) states that for an injective m-

C 1 k curve γ : [a, b] → M we have H k (C ) = S k (C ) = Length k (C ),
where

C = γ([a, b]).
It also provides a relation between H k (C ) and the interpolation complexity. On the one hand, Theorem 1 gives an integral formula for the Hausdorff measure. On the other hand, it essentially implies that the considered measures are equivalent. Another interesting property of injective m-C 1 k curves with non-zero k-dimensional measure is that the k-dimensional density of H k ⌊ C exists and is constant along the curve (see Proposition 3).

We define (H k , 1)-rectifiable sets as sets that are covered, up to H k -null sets, by countable unions of m-C 1 k curves (see Definition 3). This notion is modeled on the definition of (H k , k)-rectifiable sets in R n , which are sets that are covered, up to H k -null sets, by countable unions of image of C 1 maps from R k to R n . Thanks to the properties of m-C 1 k curves, we show a density result for sets that are rectifiable according to our definition. Namely, the second main theorem of the paper (Theorem 2) states that if a set S is H k -measurable and satisfies H k (S) < +∞, then being (H k , 1)-rectifiable implies that the upper and lower densities of H k ⌊ S are bounded by positive constants.

Theorem 2 is inspired by the result proved in Federer [START_REF] Federer | Geometric measure theory[END_REF]Th. 3.2.19], which states that for a H k measurable subset E of the Euclidean n-space (H k , k)-rectifiability implies that the measure H k ⌊ E has k-dimensional density equal to 1 at H k -almost every point of E. The converse of this fact was proved for k = 1 and for a general measure µ in [START_REF] Moore | Density ratios and (φ, 1) rectifiability in n-space[END_REF]. Much later, Preiss showed not only that the converse holds true for any k, but also a stronger result: there exists a constant c > 1 (depending only on n and k) such that if [START_REF] Marstrand | The (ϕ, s) regular subsets of n-space[END_REF]Th. 1]) for (H k , 1)-rectifiable subsets, at least in Carnot-Carathéodory spaces. In Section 2.2 we construct m-C 1 k curves in sub-Riemannian manifolds having nonzero k-dimensional measure for integer values of k ≥ 1. When the curve is absolutely continuous, it is easy to see that being m-C 1 k with non-vanishing k-dimensional measure implies that k is an integer (see Corollary 1). The question is whether such result holds true without assuming absolute continuity.

0 < lim sup r→0 + µ(E ∩ B(x, r)) r k ≤ c lim inf r→0 + µ(E ∩ B(x, r)) r k < +∞,
The structure of the paper is the following. In Section 2 we give the definition of m-C 1 k curves in metric spaces and construct them in Carnot-Carathéodory spaces. We then study measures along curves. In Section 3.1 we recall different notions of measures. In Section 3.2 we show an auxiliary result for m-C 1 k injective curves with nonzero k-dimensional measure. In Section 3.3 we analyse m-C 1 k curves in (the Euclidean space or a) Riemannian manifold. The main theorem concerning injective m-C 1 k curves is proved in Section 3.4. Some possible generalizations to non m-C 1 k curves are discussed in Section 3.5. Finally, in Section 4 we define (H k , 1)-rectifiable sets and prove the density result.

m-C 1 k curves

Throughout the paper (M, d) denotes a metric space.

Definitions

Let γ : [a, b] → M be a continuous curve, where a, b ∈ R, and let k ≥ 1 be a real number.

Definition 1. We say that γ is m-differentiable of degree

k at t ∈ [a, b] if the limit lim s→0 t+s∈[a,b] d(γ(t + s), γ(t)) |s| 1/k (1)
exists and is finite. In this case, we call this limit the metric derivative of degree k of γ at t and we define moreover the k-dimensional infinitesimal measure of γ at t as

meas k t (γ) =   lim s→0 t+s∈[a,b] d(γ(t + s), γ(t)) |s| 1/k   k .
When γ is not m-differentiable of degree k at t we set meas k t (γ) = +∞. For the case k = 1, the notion of metric derivative is classical, see [START_REF] Ambrosio | Topics on analysis in metric spaces[END_REF]Def. 4.1.2]. The kdimensional infinitesimal measures of curves were introduced in the context of sub-Riemannian geometry in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF].

Note that if γ is m-differentiable of degree k at t then, for any k ′ , lim

s→0 t+s∈[a,b] d(γ(t + s), γ(t)) |s| 1/k ′ = lim s→0 t+s∈[a,b] 1 |s| 1/k ′ -1/k d(γ(t + s), γ(t)) |s| 1/k . Therefore, for any k ′ > k, meas k ′ t (γ) = 0. If moreover meas k t (γ) > 0, then for any k ′ < k meas k ′ t (γ) = +∞. Definition 2. Given k ≥ 1, we say that γ is differentiable of class m-C 1 k on [a, b] (m-C 1 k for short) if for every t ∈ [a, b] the curve is m-differentiable of degree k at t and the map t → meas k t (γ) is continuous.
Clearly, γ is m-C 1 k if and only if the limit in (1) exists and depends continuously on t. We shall see in the next section that when a smooth structure on M exists, m-C 1 k curves need not be differentiable in the usual sense. The following lemma states that they are Hölder continuous of exponent 1/k as functions from an interval to the metric space (M, d).

Lemma 1. Let γ : [a, b] → M be m-C 1 k on [a, b], k ≥ 1. For any t and t + s in [a, b], d(γ(t), γ(t + s)) = |s| 1/k (meas k t (γ) 1/k + ǫ t (s)), (2)
where ǫ t (s) tends to zero as s tends to zero uniformly with respect to t. This is a direct consequence of the continuity of t → meas k t (γ) on the compact interval [a, b].

Construction of m-C 1 k curves

In this section we consider a class of metric spaces which are also smooth manifolds and construct smooth m-C 1 k curves on them with non-vanishing metric derivative of degree k for some integer values of k. The analysis of this class of spaces is the main motivation of this paper.

Let (M, d) be a metric space defined by a sub-Riemannian manifold (M, D, g), i.e., M is a smooth manifold, D a subbundle of T M , g a Riemannian metric on D, and d is the associated sub-Riemannian distance. We assume that Chow's condition is satisfied: let D s denote the R-linear span of brackets of degree < s of vector fields tangent to D 1 = D; then, at every p ∈ M , there exists an integer r = r(p) such that D r(p) (p) = T p M , that is,

{0} ⊂ D 1 (p) ⊂ D 2 (p) ⊂ • • • ⊂ D r(p) (p) = T p M. (3)
Let A ⊂ M . A point p ∈ A is said A-regular if the sequence of dimensions n i (q) = dim D i (q), i = 1, . . . r(q) remains constant for q ∈ A near p, and A-singular otherwise. The set A is said

equiregular if every point of A is A-regular. A curve γ : [a, b] → M is equiregular if γ([a, b]) is equiregular. Proposition 1. Let γ : [a, b] → M be an equiregular curve of class C 1 and k ∈ N such that γ(t) ∈ D k (γ(t)) for every t ∈ [a, b]. Then γ is m-C 1 k on [a, b]. If moreover γ(t) / ∈ D k-1 (γ(t)) for a given t ∈ [a, b] then meas k t (γ) = 0.
The proof of this proposition is based on the notions of nilpotent approximation and privileged coordinates (see [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry[END_REF]) and some results in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF]. We do not give the complete argument, but only the underlying ideas. All the facts that here are simply claimed are already established and complete proofs can be found in the cited literature.

Sketch of the proof. Since γ([a, b]

) is equiregular, the integers w i defined by

w i = j, if n j-1 (γ(t)) < i ≤ n j (γ(t)), i = 1, . . . , n,
do not depend on t. We define for s ≥ 0 the dilation δ s : R n → R n by

δ s z = (s w 1 z 1 , . . . , s wn z n ).
Moreover, locally there exist n vector fields Y 1 , . . . , Y n whose values at each γ(t) form a basis of T γ(t) M adapted to the filtration (3) at γ(t), in the sense that, for every integer

i ≥ 1, Y 1 (γ(t)), . . . , Y n i (γ(t)) is a basis of D i (γ(t)). The local diffeomorphism x ∈ R n → exp(x n Y n ) • • • • • exp(x 1 Y 1 )(γ(t))
defines a system of coordinates φ t : q → x = (x 1 , . . . , x n ) on a neighborhood of γ(t), satisfying φ t (γ(t)) = 0. Following [START_REF] Bellaïche | The tangent space in sub-Riemannian geometry[END_REF]Sec. 5.3], there exists a sub-Riemannian distance d t on R n such that

• d t is homogeneous under the dilation δ s , i.e., d t (δ s x, δ s x ′ ) = s d t (x, x ′ ) for all s ≥ 0, x, x ′ ∈ R n ;
• when defined, the mapping t → d t (φ t (q), φ t (q ′ )) is continuous;

• for q in a neighborhood of γ(t), d(γ(t), q) = d t (0, φ t (q))(1 + ǫ t ( d t (0, φ t (q)))), where ǫ t (s) tends to zero as s tends to zero uniformly with respect to t.

The coordinates φ t are privileged at γ(t) and the distance d t is the sub-Riemannian distance associated with a nilpotent approximation at γ(t). Set φ t (γ(t)) = (γ 1 (t), . . . , γ n (t)). By the construction in the proof of [8, Le. 12], the limit

lim s→0 t+s∈[a,b] δ |s| -1/k φ t (γ(t + s))
exists at every t and is equal to x(t) = (x 1 (t), . . . , x n (t)), where

x j (t) = 0, w j = k γj (t), w j = k.
Using the properties of d t , we have

lim s→0 d(γ(t + s), γ(t)) |s| 1/k = lim s→0 d t (φ t (γ(t + s)), 0) |s| 1/k = lim s→0 d t (δ |s| -1/k φ t (γ(t + s)), 0) = d t (x(t), 0).
As a consequence, meas k t (γ) exists and is equal to d t (x(t), 0) k . Since the components of x(t) are continuous and the distance

d t depends continuously on t, γ is m-C 1 k . If moreover γ(t) / ∈ D k-1 (γ(t)) for a given t ∈ [a, b] then x(t) = 0, whence meas k t (γ) = 0.
Let us explain the construction in Proposition 1 through an example.

Example 1. Consider the Heisenberg group, that is, the sub-Riemannian manifold (R 3 , D, g) where D is the linear span of the vector fields

X 1 (x, y, z) = (1, 0, -y/2), X 2 (x, y, z) = (0, 1, x/2),
and g = dx 2 + dy 2 . Denote by d the Carnot-Carathéodory distance associated with the Heisenberg group. Recall that d is homogeneous with respect to the dilation

δ λ (x, y, z) = (λx, λy, λ 2 z), λ ≥ 0.
and it is invariant with respect to the group law

(x, y, z) * (x ′ , y ′ , z ′ ) = x + x ′ , y + y ′ , z + z ′ + 1 2 (xy ′ -x ′ y) .
Moreover, for each point (x, y, z

) ∈ R 3 , D 2 (x, y, z) = R 3 as [X 1 , X 2 ](x, y, z) = (0, 0, 1). Let γ(t) = (0, 0, t), for t ∈ R. Then γ is of class m-C 1 2 and meas 2 t (γ) is a positive constant. This is a consequence of Proposition 1 as γ is smooth and, for all t ∈ R, γ(t) ∈ D 2 (γ(t)). Let us compute explicitly meas 2 t (γ). Notice first that d(γ(t + s), γ(t)) = d((0, 0, s), (0, 0, 0)
), since d is invariant with respect to the group law. Hence, using the homogeneity of d and the fact d((0, 0, 1), 0) = d((0, 0, -1), 0),

lim s→0 d(γ(t + s), γ(t)) |s| 1/2 = lim s→0 d((0, 0, s), 0) |s| 1/2 = d((0, 0, 1), 0) = 2 √ π,
the value of the distance resulting from an isoperimetric problem. Note that such a computation can be generalized to any contact sub-Riemannian manifold, see [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF]Th. 22].

Remark 1. Note that the equiregularity assumption is essential to obtain the continuity of d t and φ t with respect to t. In particular the proof of [8, Le. 12] is not valid without this hypothesis 1 .

This assumption has also an intrinsic meaning. Indeed it is shown in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF] that the k-dimensional measure meas k t (γ) can actually be defined through the distance on the metric tangent space to (M, d) at γ(t). Since the metric tangent space does not vary continuously with respect to t around C -singular points, where C = γ([a, b]), in general non equiregular curves may not be m-C 1 k . Note that for every integer k ∈ {1, . . . , r(p)}, where p is regular, there exist C 1 equiregular curves with tangent vector belonging to D k \ D k-1 . As a consequence, for such integers k the class of m-C 1 k curves with non-vanishing metric derivative of degree k is not empty. For instance, this is the case in the Heisenberg group for k = 2, and in the Engel group (see below) for k = 2, 3. On the contrary, the next proposition states that in the Riemannian case, i.e., when D = T M , the class of m-C 1 k curves with non-vanishing derivative is empty except for k = 1 (the proof of Proposition 2 is postponed to Section 3.3).

Proposition 2. Let (M, g) be a Riemannian manifold. Let k ≥ 1 and assume that γ : [a, b] → M is a m-C 1 k curve such that t → meas k t (γ) does not vanish identically. Then k = 1. Let γ : [a, b] → M be of class m-C 1
1 and such that meas 1 t (γ) = 0 for every t ∈ [a, b]. Then γ is horizontal, i.e., it is absolutely continuous and γ(t) ∈ D 1 (γ(t)) almost everywhere on [a, b]. To see this, remark that by construction, γ is Lipschitz with respect to the sub-Riemannian distance. The metric g defined on D can be extended (at least in a tubular neighbourhood of γ([a, b])) to a Riemannian metric g on T M . In this way we obtain a Riemannian distance on M which is not greater than the sub-Riemannian distance. Hence γ is Lipschitz with respect to the chosen Riemannian distance which in turn implies that γ is absolutely continuous. Therefore, by [START_REF] Boscain | Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces[END_REF]Pr. 5]

γ is horizontal, i.e., γ(t) ∈ D 1 (γ(t)) almost everywhere on [a, b].
Using Proposition 1, this fact can be partially generalized to the case k > 1 under the following form. In particular Corollary 1 states that if meas k t (γ) ≡ 0 then k is an integer, provided that γ is absolutely continuous. An open question is whether the latter condition is necessary. If this were not the case then we would obtain a Marstrand's type Theorem [20, Th. 1] for m-C 1 k curves: indeed we shall see in Proposition 3 that along injective m-C 1 k curves with non-vanishing k-dimensional measure the density of H k exists and is constant.

Nevertheless, a m-C 1 k curve need not be C 1 in the usual sense as it is shown below. Example 2. Consider the Engel group, that is, the sub-Riemannian manifold (R 4 , D, g) where D is the linear span of the vector fields

X 1 (x, y, z, w) = (1, 0, 0, 0), X 2 (x, y, z, w) = (0, 1, x, x 2 /2),
and g = dx 2 + dy 2 . Let γ(t) = (0, 0, W (t), ϕ(t)), where ϕ ∈ C 1 and W is the Weierstrass function 1 The statement of Lemma 12 in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF] is incorrect. Indeed without equiregularity formula (3) therein does not hold.

W (t) = ∞ n=0 α n (cos(β n πt) -1), t ∈ R,
where 0 < α < 1, β > 1, and αβ > 1 see [START_REF] Weierstrass | On continuous functions of a real argument that do not have a well-defined differential quotient[END_REF]. It was proved in [START_REF] Hardy | Weierstrass's non-differentiable function[END_REF] that W (t) is continuous, nowhere differentiable on the real line, and satisfies

W (t + h) -W (t) = O(|h| ξ ), ξ = log(1/α) log β < 1, (4)
uniformly with respect to t ∈ R. Then, choosing α, β such that ξ > 2/3, γ is continuous and m-C 1 3 , but nowhere differentiable. Indeed, it is not hard to verify that the sub-Riemannian distance d satisfies the following homogeneity property λd((0, 0, z, w), (0, 0, z, w)) = d(0, (0, 0, λ 2 (z -z), λ 3 (w -w))), for every λ ≥ 0. Then we have

lim s→0 1 |s| 1/3 d((0, 0, W (t), ϕ(t)), (0, 0, W (t + s), ϕ(t + s))) = lim s→0 d 0, 0, 0, O(|s| ξ ) |s| 2/3 , ϕ ′ (t) .
Since ξ > 2/3, γ is m-differentiable of degree 3 at each t and meas 3 t (γ) = d(0, (0, 0, 0, ϕ ′ (t)) 3 , which is non-zero for a suitable choice of ϕ. Therefore, γ is m-C 1 3 and by the properties of

W (t), γ is nowhere differentiable. Notice that if γ is m-C 1 k and k ′ ≥ k, then γ is m-C 1 k ′ . Define k γ ≥ 1 as the infimum of k ≥ 1 such that γ is m-C 1 k .
Then k γ need not be an integer as it is shown in the next example. Moreover, γ is not necessarily m-C 1 kγ . Example 3. Consider the sub-Riemannian structure of Example 2 and the curve γ(t) = (0, 0, W (t), 0). Then k γ = 2/ξ may be any real number greater than 2 (see ( 4)), but γ is not m-C 1 kγ .

Measures along curves

This section is devoted to compute Hausdorff (and spherical Hausdorff) measures of continuous curves and to establish a relation with the k-dimensional length and with the complexity.

Different notions of measures

Denote by diam S the diameter of a set S ⊂ M . Let k ≥ 0 be a real number. For every set A ⊂ M , we define the k-dimensional Hausdorff measure

H k of A as H k (A) = lim ǫ→0 + H k ǫ (A)
, where

H k ǫ (A) = inf ∞ i=1 (diam S i ) k : A ⊂ ∞ i=1 S i , diam S i ≤ ǫ, S i closed set ,
and the k-dimensional spherical Hausdorff measure S k of A as S k (A) = lim ǫ→0 + S k ǫ (A), where

S k ǫ (A) = inf ∞ i=1 (diam S i ) k : A ⊂ ∞ i=1 S i , S i is a ball, diam S i ≤ ǫ .
In the Euclidean space R n , k-dimensional Hausdorff measures are often defined as 2 -k α(k)H k and 2 -k α(k)S k , where α(k) is defined from the usual gamma function as α(k) = Γ( 12

) k /Γ( k 2 + 1).
This normalization factor is necessary for the n-dimensional Hausdorff measure and the Lebesgue measure coincide on R n . For a given set A ⊂ M , H k (A) is a decreasing function of k, infinite when k is smaller than a certain value, and zero when k is greater than this value. We call Hausdorff dimension of A the real number dim H A = sup{k :

H k (A) = ∞} = inf{k : H k (A) = 0}.
Note that H k ≤ S k ≤ 2 k H k , so the Hausdorff dimension can be defined equally from Hausdorff or spherical Hausdorff measures. When the set A is a curve, another kind of dimensioned measures can be obtained from the integration of k-dimensional infinitesimal measures. Let γ : [a, b] → M be a continuous curve and

C = γ([a, b]). For k ≥ 1, we define the k-dimensional length of C as Length k (C ) = ˆb a meas k t (γ) dt. ( 5
)
where meas k t (γ) is as in Definition 1 (these lengths were introduced in [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF] in the sub-Riemannian context). Thanks to the properties of meas k t (γ), Length k (γ) is a decreasing function of k, infinite when k is smaller than a certain value, and zero when k is greater than this value. We call this value the length dimension of C .

Another way to measure the set C is to study its approximations by finite sets (see [START_REF] Jean | Paths in Sub-Riemannian Geometry[END_REF] and [14, p. 278]). Here we only consider approximations by ǫ-chains of C , i.e., sets of points q 1 = γ(a), . . . , q N = γ(b) in C such that d(q i , q i+1 ) ≤ ǫ. The interpolation complexity σ int (C , ǫ) is the minimal number of points in an ǫ-chain of C . This complexity has been computed in several cases in [START_REF] Gauthier | Motion planning and fastly oscillating controls[END_REF].

m-C 1

k curves with non-vanishing k-dimensional measure

In this section we prove the following proposition about m-C 1 k curves with non-vanishing k-dimensional measure. This result is the first step to prove Theorem 1.

Proposition 3. Let γ : [a, b] → M be an injective m-C 1 k curve and C = γ([a, b]). Assume meas k t (γ) = 0 for every t. Then H k (C ) = S k (C ) = Length k (C ) (6) lim ǫ→0 + ǫ k σ int (C , ǫ) = Length k (C ), (7)
and for every q ∈ C lim

r→0 + H k (C ∩ B(q, r)) 2r k = 1. ( 8
)
Remark 3. Equations ( 6), ( 8 If we drop the injectivity assumption we obtain the following weaker result.

Corollary 2. Let γ : [a, b] → M be a m-C 1 k curve and C = γ([a, b]). Assume meas k t (γ) = 0 for every t. Then H k (C ) = S k (C ) ≤ Length k (C ). ( 9
)
Proof of Corollary 2. Since meas k t (γ) = 0 for every t ∈ [a, b], γ is locally injective. Hence [a, b] is the disjoint union of a finite family of intervals I i such that γ| I i is injective. For each i there exists a measurable subset A i ⊂ I i such that C = ∪γ(A i ) and the sets γ(A i ) are pairwise disjoint. Using Remark 3, formula (6) applies to each γ(A i ). Since H k (C ) = i H k (γ(A i )) and S k (C ) = i S k (γ(A i )), we obtain [START_REF] Federer | The (ϕ, k) rectifiable subsets of n-space[END_REF].

The proof of Proposition 3 is based on the following result for bi-Hölder continuous curves.

Lemma 2. Let γ : [0, T ] → M be an injective curve and C = γ([0, T ]). Assume that there exist positive constants δ -, δ + , and η such that

δ -|s| 1/k ≤ d γ(t), γ(t + s) ≤ δ + |s| 1/k , ( 10
)
for every t, t + s ∈ [0, T ] with |s| < η. Then δ k -T ≤ H k (C ) ≤ δ k + T, (11)
δ k -T ≤ lim inf ǫ→0 + ǫ k σ int (C , ǫ) ≤ lim sup ǫ→0 + ǫ k σ int (C , ǫ) ≤ δ k + T, (12)
S k (C ) ≥ H k (C ) ≥ δ - δ + 2k S k (C ), (13)
and, for every t ∈ [0, T ] and r > 0 small enough,

δ - δ + k ≤ H k (C ∩ B(γ(t), r)) 2r k ≤ δ + δ - k . ( 14
)
Proof. Let ǫ > 0 be smaller than δ + η 1/k . We denote by N the smallest integer such that T ≤ N ( ǫ δ + ) k and define t 0 , . . . , t N by

t i = i ǫ δ + k for i = 0, . . . , N -1, t N = T. Set S i = γ([t i-1 , t i ]), i = 1, . . . , N . For t, t ′ in S i , one has |t -t ′ | ≤ ǫ k /δ k + ; it follows from (10) that d(γ(t), γ(t ′ )) ≤ δ + |t -t ′ | 1/k ≤ ǫ, (15)
which in turn implies diam

S i ≤ ǫ. Thus H k ǫ (C ) ≤ i (diam S i ) k ≤ N ǫ k . Using (N -1)ǫ k < T δ k + , we obtain H k ǫ (C ) ≤ δ k + T + ǫ k . It also results from inequality (15) that γ(t 0 ), . . . , γ(t N ) is an ǫ-chain of C which implies ǫ k σ int (C , ǫ) ≤ δ k + T + ǫ k ,
Taking the limit as ǫ → 0 in the preceding inequalities, we find

H k (C ) ≤ δ k + T and lim sup ǫ→0 + ǫ k σ int (C , ǫ) ≤ δ k + T.
We now prove converse inequalities for H k and σ int . Fix ǫ > 0 and consider a countable family S 1 , S 2 , . . . of closed subsets of M such that C ⊂ i S i and diam S i ≤ ǫ. For every i ∈ N, we set I i = γ -1 (S i ∩ C ). As γ is injective, if ǫ is small enough then it results from (10) that for any t, t ′ in I i there holds diam

S i ≥ d(γ(t), γ(t ′ )) ≥ δ -|t -t ′ | 1/k , which implies L 1 (I i ) ≤ (diam S i ) k /δ k -. Note that T ≤ i L 1 (I i ) since the sets I i cover [0, T ]. It follows that H k ǫ (C ) ≥ T δ k -, that is, H k ǫ (C ) ≥ δ k -T. (16)
In the same way, an ǫ-chain γ(t 0 ) = γ(0), . . . , γ(t

N ) = γ(T ) of C satisfies N ǫ k ≥ T δ k -since the injectivity of γ assures that ǫ ≥ d(γ(t i-1 ), γ(t i )) ≥ δ -|t i -t i-1 | 1/k . It follows that ǫ k σ int (C , ǫ) ≥ δ k -T .
Taking the limit as ǫ → 0 in this inequality and in [START_REF] Jean | Paths in Sub-Riemannian Geometry[END_REF], we find

H k (C ) ≥ δ k -T and lim inf ǫ→0 + ǫ k σ int (C , ǫ) ≥ δ k -T
, which completes the proof of ( 11) and ( 12). The first inequality in (13) always holds. Before proving the second one, let us recall a standard result in geometric measure theory (see for instance [10, 2.10.18, (1)]). Let X be a metric space and µ be a regular measure on X such that the closed balls in X are µ-measurable. If

lim sup y∈B(x,r) r→0 + µ(B(y, r)) (diam B(y, r)) k ≥ λ, for every point x ∈ X, then µ(X) ≥ λS k (X).
We will apply this result to the metric space (C , d| C ) and to the measure µ = H k ⌊ C .

If t ∈ [0, T ], then for r > 0 small enough, there holds

γ([t -r k δ k + , t + r k δ k + ]) ⊂ C ∩ B(γ(t), r) ⊂ γ([t -r k δ k - , t + r k δ k - ]). ( 17
)
The diameter of this set then satisfies

diam(C ∩ B(γ(t), r)) ≤ d γ t - r k δ k + , γ t + r k δ k + ≤ δ + 2 1 k r δ - .
Moreover, applying [START_REF] Franchi | Rectifiability and perimeter in the Heisenberg group[END_REF] to the curve

γ restricted to [t -r k δ k + , t + r k δ k + ],
we obtain

H k (C ∩ B(γ(t), r)) ≥ δ k - 2r k δ k + .
Thus we have, for every point γ(t ′ ) ∈ C , lim sup

r→0 + γ(t)∈B(γ(t ′ ),r) H k ⌊ C (B(γ(t), r)) (diam(C ∩ B(γ(t), r))) k ≥ δ - δ + 2k , which implies H k (C ) ≥ δ - δ + 2k S k (C ).
Finally, formula [START_REF] Gromov | Carnot-Carathéodory spaces seen from within[END_REF] results from [START_REF] Franchi | Rectifiability and perimeter in the Heisenberg group[END_REF] applied to the restrictions of γ in [START_REF] Jean | Entropy and complexity of a path in sub-Riemannian geometry[END_REF].

Proof of Proposition 3. By definition, Length

k (C ) = ˆb a meas k t (γ) dt.
Note that the k-dimensional length does not depend on the parameterization [START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF]Le. 16]. Thus, up to a reparameterization by the k-length, we assume that γ is defined on the interval [0, T ], with T = Length k (C ), and that meas k t (γ) ≡ 1. Fix δ > 0. Then, by Lemma 1, there exists η > 0 so that the hypothesis of Lemma 2 is satisfied with δ -= 1 -δ and δ + = 1 + δ. We let δ tends to zero in inequalities ( 11)-( 14) and the proposition follows. Remark 4. Another way to measure C using approximations by finite sets is to consider ǫ-nets, i.e., sets of points q 1 , . . . , q n ∈ M such that the union of closed balls B(q i , ǫ) covers C , and the metric entropy e(C , ǫ) which is the minimal number of points in an ǫ-net of C . Under the assumptions of Proposition 3, the following estimates can be deduced for a m-C 1 k curve:

S k (C ) 2 k ≤ lim inf ǫ→0 + ǫ k e(C , ǫ) ≤ lim sup ǫ→0 + ǫ k e(C , ǫ) ≤ S k (C ) 2 .

The Riemannian case

Let us come back to the case where (M, d) is a Carnot-Carathéodory space associated with a sub-Riemannian manifold (M, D, g). A consequence of Proposition 3 is that if the structure is Riemannian, i.e., D = T M , then the class of m-C 1 k curves having non-zero metric derivative of degree k is empty if k > 1, as stated in Proposition 2.

Proof of Proposition 2. Let s ∈ [a, b] such that meas k s (γ) = 0. Thus, restricted to a small enough neighbourhood I = [s -δ, s + δ] of s, the curve γ is injective and meas k t (γ) = 0 on I. Up to reparameterizing γ| I , we may assume moreover meas k t (γ) ≡ 1 on I. Also, it is sufficient to consider the case M = R n and d is the Euclidean distance on R n . Denote by C the set γ(I). By Proposition 3, we have 0 < H k (C ) < +∞ and for every t ∈ I

lim r→0 H k (C ∩ B(γ(t), r)) 2r k = 1.
Moreover, by Lemma 1, there exist 0 < ρ < 1 such that

(1 -ρ)|t -t ′ | 1/k ≤ ||γ(t) -γ(t ′ )|| ≤ (1 + ρ)|t -t ′ | 1/k , ∀ t, t ′ ∈ I.
The proposition then results from the lemma below.

Lemma 3. Let k ≥ 1 and let γ : [a, b] → R n be a bi-Hölder curve of exponent 1/k, i.e., δ -|t -t ′ | 1/k ≤ ||γ(t) -γ(t ′ )|| ≤ δ + |t -t ′ | 1/k , ∀ t, t ′ ∈ [a, b], (18)
with δ -, δ

+ = 0. Set C = γ([a, b]). If 0 < H k (C ) < +∞
lim r→0 H k (C ∩ B(γ(t), r)) r k = c, then k = 1.
Proof. Under the assumptions of the lemma, Marstrand's Theorem [START_REF] Marstrand | The (ϕ, s) regular subsets of n-space[END_REF]Th. 1] assures that k ∈ N and k ∈ {1, 2, . . . , n}. Applying Preiss' result [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF], there exists a countable family of k-dimensional

submanifolds N i ⊂ R n such that H k (C \ ∪ i N i ) = 0. Since H k (C ) > 0, there exists i such that H k (C ∩ N i ) > 0. Let us rename N i by N . Then L k N (C ∩ N ) = H k ⌊ N (C ) > 0, where L k N is the k-dimensional Lebesgue measure on N . Hence there exists a density point γ(t 0 ) ∈ C ∩ N , that is, a point such that lim r→0 L k N (C ∩ N ∩ B N (γ(t 0 ), r)) L k N (B N (γ(t 0 ), r)) = 1,
where

B N (γ(t 0 ), r) is the open ball in N centered at γ(t 0 ) of radius r. Let us identify N with {(x 1 , . . . , x n ) ∈ R n : x k+1 = • • • = x n = 0}
by choosing local coordinates around γ(t 0 ). Using the inequalities [START_REF] Kirchheim | Rectifiable metric spaces: local structure and regularity of the Hausdorff measure[END_REF] and the density point t 0 it is not hard to prove that there exists a bi-Lipschitz homeomorphism from B N (γ(t 0 ), δ -) endowed with the Euclidean distance to (-1, 1) endowed with the distance | • | 1/k (see for instance the argument in the proof of [START_REF] Assouad | Plongements lipschitziens dans R n[END_REF]Pr. 4.12]). Since the topological dimension of B N (γ(t 0 ), δ -) is k, then k must be equal to 1.

Comparison of measures for m-C 1 k curves

Next theorem generalizes the first part of Proposition 3 to the case when the metric derivative may vanish. Namely, it compares the H k measure and the S k measure of sets that are images of m-C 1 k curves. Also, it provides a relation among such measures, the k-length and the behaviour of the complexity of the curve.

Theorem 1. Let γ : [a, b] → M be an injective m-C 1 k curve and C = γ([a, b]). Then H k (C ) = S k (C ) = Length k (C ) = lim ǫ→0 ǫ k σ int (C , ǫ).
If moreover H k (C ) > 0 or Length k (C ) > 0, then for every k ′ ≥ 1

H k ′ (C ) = S k ′ (C ) = Length k ′ (C ) = lim ǫ→0 ǫ k ′ σ int (C , ǫ).
Remark 5. When M is a sub-Riemannian manifold, in many cases Gauthier and coauthors (see [START_REF] Gauthier | Motion planning and fastly oscillating controls[END_REF] and references therein) computed the interpolation complexity of curves as integral of some geometric invariants. Jointly with Theorem 1, these results provide a way of computing Hausdorff measures of curves as well as a geometric interpretation of Hausdorff and infinitesimal measures.

Corollary 3. Let γ : [a, b] → M be a m-C 1 k curve. Then, for every measurable set A ⊂ [a, b], S k (γ(A)) = H k (γ(A)) and H k (γ(A)) ≤ Length k (γ(A)).
If moreover γ is injective then

H k (γ(A)) = Length k (γ(A)).
Remark 6. Let us consider the case where γ is an injective m-C 1 k curve. Recalling the definition of Length k , we have

H k (C ) = ˆb a meas k t (γ) dt,
that is, we have an integral formula for the k-dimensional Hausdorff measure. Moreover, Theorem 1 implies that the Hausdorff dimension k H of C coincides with the length dimension of C . If in addition H k H (C ) (or Length k H (C )) is finite, then Corollary 3 implies that H k H ⌊ C is absolutely continuous with respect to the push-forward measure2 γ * L 1 and that its Radon-Nikodym derivative is meas k t (γ).

I = {t ∈ [a, b] : meas k t (γ) = 0}, (19)
and its complementary

I c = [a, b] \ I. The set I is the union of a disjointed countable family of open subintervals I i of [a, b]. Note that, since meas k t (γ) = 0 for all t ∈ I c , one has Length k (C ) = ˆI meas k t (γ)dt = i ˆIi meas k t (γ)dt.
By Remark 3, we have the equality

H k (γ(I i )) = ˆIi meas k t (γ)dt, ∀i. (20)
Since γ is injective, we have

H k (C ) ≥ i H k (γ(I i )),
whence we obtain

H k (C ) ≥ Length k (C ).
The next step is to prove the converse inequality. Let δ > 0. Since the function t → meas k t (γ) 1/k is uniformly continuous on [a, b], there exists η > 0 such that, if t, t ′ ∈ [a, b] and |t -t ′ | < η, then |meas k t (γ) 1/k -meas k t ′ (γ) 1/k | < δ. In the covering I = i I i , only a finite number N δ of subintervals I i may have a Lebesgue measure greater than η. Up to reordering, we assume L 1 (I i ) < η if i > N δ . Set J = I c ∪ i>N δ I i . Since the restriction of meas k t (γ) to I c is identically zero, there holds meas k t (γ) 1/k < δ for every t ∈ J. The k-dimensional Hausdorff measure of C satisfies

H k (C ) ≤ i≤N δ H k (γ(I i )) + H k (γ(J)) = i≤N δ ˆIi meas k t (γ)dt + H k (γ(J)), (21)
in view of [START_REF] Marstrand | The (ϕ, s) regular subsets of n-space[END_REF]. It remains to compute H k (γ(J)). Being the complementary of

i≤N δ I i in [a, b], J is the disjointed union of N δ + 1 closed subintervals J i = [a i , b i ] of [a, b].
For each one of these intervals we will proceed as in the proof of Proposition 3. Let ǫ > 0 and i ∈ {1, . . . , N δ + 1}. We denote by N ′ the smallest integer such that b i -a i ≤ N ′ ( ǫ 2δ ) k and define t 0 , . . . , t N ′ by

t j = a i + j ǫ 2δ k for j = 0, . . . , N ′ -1, t N ′ = b i .
We then set S j = γ([t j , t j-1 ]). Applying Lemma 1, we get, for any t, t ′ ∈ [t j , t j-1 ],

d γ(t), γ(t ′ ) = |t -t ′ | 1/k (meas k t (γ) 1/k + ǫ t (t -t ′ )).
Note that meas k t (γ) 1/k < δ since t ∈ J. Note also that, if ǫ is small enough, then ǫ t (|t -t ′ |) is smaller than δ. Therefore d γ(t), γ(t ′ ) < 2δ|t -t ′ | 1/k ≤ ǫ and diam S j ≤ ǫ. As a consequence

H k ǫ (γ(J i )) ≤ N ′ ǫ k ≤ (2δ) k (b i -a i ) + ǫ k ,
and

H k (γ(J i )) ≤ (2δ) k (b i -a i ). It follows that H k (γ(J)) ≤ i≤N δ +1 (2δ) k (b i -a i ) ≤ (2δ) k (b -a).
Finally, formula [START_REF] Mattila | Geometry of sets and measures in Euclidean spaces: fractals and rectifiability[END_REF] yields

H k (C ) ≤ i≤N δ ˆIi meas k t (γ)dt + (2δ) k (b -a).
Letting δ → 0, we get H k (C ) ≤ ´I meas k t (γ)dt = Length k (C ), and thus H k (C ) = Length k (C ). Similarly we can show that S k (C ) and the limit of ǫ k σ int (C , ǫ) are equal to Length k (C ).

Proof of Corollary 3. When γ is injective, the conclusions follow from Theorem 1 and from the regularity of L 1 and H k measures (see Remark 3).

Assume now that γ is not injective. We slightly modify the second part of the proof of Theorem 1 replacing the equality in [START_REF] Mattila | Geometry of sets and measures in Euclidean spaces: fractals and rectifiability[END_REF] by

H k (C ) ≤ i≤N δ H k (γ(I i )) + H k (γ(J)) ≤ i≤N δ ˆIi meas k t (γ)dt + H k (γ(J)),
which is a consequence of Corollary 2. This shows that H k (C ) ≤ Length k (C ) and therefore

H k (γ(A)) ≤ Length k (γ(A))
. Moreover, we have H k (γ(I c )) = 0, where I c is the complementary of the set I defined in [START_REF] Magnani | Characteristic points, rectifiability and perimeter measure on stratified groups[END_REF], which in turn implies S k (γ(I c )) = 0. Thus

S k (C ) = S k (γ(I)) = H k (γ(I)) = H k (C ),
where the second equality results from Corollary 2.

Generalization to non m-C 1 k curves

In this section we present some possible generalizations of the preceding results (in particular Theorem 1) to non m-C 1 k curves. Consider first the case of a continuous curve γ : [a, b] → M , C = γ([a, b]). For k ≥ 1, we define I k to be the set of points t ∈ [a, b] such that meas k t (γ) is not continuous at t (that is, such that γ is not m-C 1 k at t). A standard argument of measure theory allows to show the following fact. Assume that [a, b] \ I k is an open subset of [a, b] of full L 1 measure and that H k (γ(I k )) = 0. Then the conclusions of Corollary 3 still hold. Moreover, if γ is injective, then the equalities between H k (C ), S k (C ), and Length k (C ) as in Theorem 1 hold true. The result on the complexity is not valid anymore, since lim ǫ→0 ǫ k σ int (C , ǫ) is not a measure. A curve C satisfying the properties above actually appears as a particular case of (H k , 1)-rectifiable set, which will be studied in the next section.

It is however worth to mention a consequence of the result claimed above (and of Proposition 1) in the context of Carnot-Carathéodory spaces. Let (M, D, g) be a sub-Riemannian manifold, γ : [a, b] → M be an absolutely continuous injective curve, and C = γ([a, b]). Let m C ≥ 1 be the smallest integer such that γ(t) ∈ D m C (γ(t)) almost everywhere. We denote by

I C the set of points t ∈ [a, b] such that either γ is not C 1 at t or γ(t) is C -singular. Corollary 4. Assume that [a, b]\I C is an open subset of [a, b] of full L 1 measure and H m C (γ(I C )) = 0. Then, for any k ≥ 1, H k (C ) = S k (C ) = Length k (C ),
and

dim H C = m C .
When the sub-Riemannian manifold is equiregular, it is already known [14, p. 104] that the Hausdorff dimension of a one-dimensional submanifold C is the smallest integer k such that T q C ⊂ D k (q) for every q ∈ C. Corollary 4 generalizes this fact.

Any injective m-C 1

k curve being bi-Hölder (see Lemma 1), it is also natural to generalize our results to such curves. Consider then a bi-Hölder curve of exponent 1/k, i.e. a curve γ : [a, b] → M , i.e., there exist positive constants δ -and δ + such that, for every t, t

+ s ∈ [a, b], δ -|s| 1/k ≤ d γ(t), γ(t + s) ≤ δ + |s| 1/k .
For such a curve C = γ([a, b]), the k-dimensional length does not always exists but Lemma 2 gives estimates of H k (C ) and S k (C ) in function of T = b -a, and the following result for upper and lower density:

δ - δ + k ≤ lim inf r→0 + H k (C ∩ B(q, r)) 2r k ≤ lim sup r→0 + H k (C ∩ B(q, r)) 2r k ≤ δ + δ - k . (22)
Let us remark that there is no hope to obtain a density result such as (8) for bi-Hölder curves. Indeed Assouad proved in [START_REF] Assouad | Plongements lipschitziens dans R n[END_REF] that, for any k < n, there exist bi-Hölder curves of exponent 1/k from (-1, 1) to R n (both endowed with a Euclidean metric). When k > 1 the density of these curves cannot be constant, for otherwise Lemma 3 would yield a contradiction. This strongly hints that there is not a Rademacher's-type result in this context, that is, being bi-Hölder of exponent 1/k does not imply being m-C 1 k almost everywhere. In what follows, in particular in the definition of (H k , 1)-rectifiability, we will work with m-C 1 k curves and not with bi-Hölder curves. The drawback is that our definitions will not be invariant under bi-Lipschitz equivalence of metric spaces since the m-C 1 k property is not invariant under such equivalence, contrarily to the bi-Hölder property. However we think that rectifiable sets should be defined as sets which admit almost everywhere a metric derivative. As noticed above, the use of bi-Hölder curves would not guarantee such a property.

(H k , 1)-rectifiable sets and a density result

In this section we use m-C 1 k curves to define a new class of (H k , 1)-rectifiable subsets of metric spaces.

Consider the Euclidean space R n . Recall that, given a positive measure µ on Borel subsets of R n , a subset S ⊂ R n is (µ, k)-rectifiable if there exists a countable family of Lipschitz functions [10, 3.2.14]). Considering µ = H k ′ on R n (with the Euclidean structure), one has that if k ′ > k then there are no (H k ′ , k)-rectifiable sets of positive H k ′ measure. This follows from the requirement of γ i being Lipschitz. On the other hand, if one consider images under Hölder continuous functions γ i then the case k ′ > k becomes of interest. This suggests the next definition.

γ i : V i → R n , i ∈ N, where V i is a bounded subset of R k , such that µ(S \ ∪ i∈N γ i (V i )) = 0 (see
Consider a metric space (M, d).

Definition 3. A subset S ⊂ M is (H k , 1)-rectifiable if there exists a countable family of m-C 1 k curves γ i : I i → M , i ∈ N, I i closed interval in R such that H k (S \ ∪ i∈N γ i (I i )) = 0. Remark 7.
If M is a manifold and d is the distance associated with a Riemannian structure on M , the class of (H k , 1)-rectifiable sets with positive and finite H k measure is empty unless k = 1 (see Proposition 2). Since m-C 1 1 curves are Lipschitz, in this case Definition 3 coincides with the usual definition of (H 1 , 1)-rectifiable sets. Conversely, when (M, d) is the Carnot-Carathéodory space associated with a genuine sub-Riemannian manifold, there exist (H k , 1)-rectifiable sets (of positive and finite H k measure) for some integers k > 1 (see Section 2.2). When a subset is H k -measurable and has finite H k measure, being (H k , 1)-rectifiable implies boundedness for the lower and upper densities of the measure H k ⌊ S .

Theorem 2. Assume S ⊂ M is a (H k , 1)-rectifiable and H k -measurable set such that H k (S) < +∞. Then for H k -almost every q ∈ S 2 ≤ lim inf r→0 + H k (S ∩ B(q, r)) r k ≤ lim sup

r→0 + H k (S ∩ B(q, r)) r k ≤ 2 k . (23)
Recall that in [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF]Co. 5.5] it was proved that, in the Euclidean case, there exists a constant c > 0 such that if a µ-measurable subset E ⊂ R n with finite µ measure satisfies 0 < lim sup

r→0 + µ(E ∩ B(x, r)) r k ≤ c lim inf r→0 + µ(E ∩ B(x, r)) r k < +∞, (24)
for µ-almost every x ∈ E, then E is (µ, k)-rectifiable. This result provides a characterization of rectifiable sets as the converse is also true (see [START_REF] Federer | Geometric measure theory[END_REF]Th. 3.2.19]). Theorem 2 implies that if S ⊂ M is (H k , 1)-rectifiable in the sense of Definition 3 then 0 < lim sup

r→0 + H k (S ∩ B(q, r)) r k ≤ 2 k-1 lim inf r→0 + H k (S ∩ B(q, r)) r k < +∞, (25)
for H k -almost every q ∈ S. The last estimate is, mutatis mutandis, the assumption [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF] in the result by Preiss. An open question is whether the same conclusion of [24, Cor. 5.5] holds with our notion of (H k , 1)-rectifiable sets. Namely, is condition [START_REF] Weierstrass | On continuous functions of a real argument that do not have a well-defined differential quotient[END_REF] for H k -almost every q ∈ S ⊂ M sufficient to show that a H k -measurable set S of finite H k measure is (H k , 1)-rectifiable in the sense of Definition 3?

Proof of Theorem 2. By assumption, there exists a countable family of m-C 1 k curves γ i : I i → M such that I i is a closed interval and H k (S \ ∪ i γ i (I i )) = 0. Since by Corollary 3, for every i, H k (γ i ({t | meas k t (γ i ) = 0})) = 0, we may assume meas k t (γ i ) = 0 for every t ∈ I i and then, by a reparameterization, meas k t (γ i ) ≡ 1. This implies that every γ i is locally injective. Hence, without loss of generality, we may assume that every γ i is injective and moreover that the sets γ i (I i ) are pairwise disjoint.

Since H k (S) < +∞, to prove the upper bound lim sup

r→0 + H k (S ∩ B(q, r)) r k ≤ 2 k ,
for H k -almost every q ∈ M , it suffices to use [10, 2.10.19 (5)].

Let us show the lower bound in [START_REF] Moore | Density ratios and (φ, 1) rectifiability in n-space[END_REF], namely, that lim inf

r→0 + H k (S ∩ B(q, r)) r k ≥ 2, ( 26
)
for H k -almost every q ∈ S. Let Ĩi = γ -1 i (γ i (I i )∩S). Then ∪ i∈N γ i ( Ĩi ) ⊂ S and H k (S \∪ i∈N γ i ( Ĩi )) = 0. We may assume H k (γ i ( Ĩi )) > 0 for each i. Then, by Corollary 3, since meas k t (γ i ) ≡ 1, L 1 ( Ĩi ) = H k (γ i ( Ĩi )) > 0. Therefore almost every t ∈ Ĩi is a density point for the Lebesgue measure on Ĩi , i.e., lim r→0 L 1 ( Ĩi ∩ B(t, r)) 2r = 1, where B(t, r) = (t -r, t + r). Hence, for H k -almost every q ∈ S there exist a unique i and a unique t ∈ Ĩi such that q = γ i (t) and t is a density point for L 1 ⌊ Ĩi . Since γ i ( Ĩi ) ⊂ S, we deduce H k (S ∩ B(q, r)) r k ≥ H k (γ i ( Ĩi ) ∩ B(q, r)) r k = L 1 ( Ĩi ∩ γ -1 i (B(q, r))) r k , the last equality following by Corollary 3. Now, for any δ > 0, from Lemma 1, for |t -s| ≤ r k (1+δ) k we have d γ(t), γ(s) ≤ |t -s| 1/k (1 + δ) ≤ r.

This implies B(t, r k /(1 + δ) k ) ⊂ γ -1 i (B(q, r)). Therefore

L 1 ( Ĩi ∩ γ -1 i (B(q, r))) r k ≥ L 1 ( Ĩi ∩ B(t, r k /(1 + δ) k ))) r k .
The right-hand side of the inequality above tends to 2/(1 + δ) k , as r goes to 0, since t is a density point for L 1 ⌊ Ĩi . Letting δ go to 0, we conclude lim inf r→0 L 1 ( Ĩi ∩ γ -1 i (B(q, r))) r k ≥ 2, which shows (26).

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Construction of m-C 1 k curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Measures along curves 3.1 Different notions of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 m-C 1 k curves with non-vanishing k-dimensional measure . . . . . . . . . . . . . . . . . 3.3 The Riemannian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Comparison of measures for m-C 1 k curves . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Generalization to non m-C 1 k curves . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Corollary 1 .

 1 Let k ≥ 1 and let γ : [a, b] → M be equiregular and of class m-C 1 k , with meas k t (γ) ≡ 0. If γ is absolutely continuous, then k is the smallest integer m such that γ(t) ∈ D m (γ(t)) almost everywhere.

Remark 2 .

 2 Notice that for any injective m-C 1 1 curve the equality H 1 (C ) = Length 1 (C ) holds (see[START_REF] Ambrosio | Topics on analysis in metric spaces[END_REF] Th. 4.1.6, 4.4.2]).

  ) hold when we replace[a, b] by the open interval (a, b). Also, they hold for unbounded intervals. Therefore, thanks to the regularity of L 1 and H k measures, equations (6),[START_REF] Falbel | Measures of transverse paths in sub-Riemannian geometry[END_REF] are still verified when we replace C by γ(A), for any measurable set A ⊂ [a, b].

Proof of Theorem 1 .

 1 Clearly it suffices to prove the first statement of the theorem. Consider the (possibly empty) open subset of [a, b]

  and if there exits a positive constant c such that for every t ∈ [a, b]

Given a Borel set E ⊂ M the push-forward measure γ * L 1 is defined byγ * L 1 (E) = L 1 (γ -1 (E ∩ C )).

* This work was supported by the Digiteo grant Congeo and by the ANR project GCM, program "Blanche", project number NT09 504490.


Ilaria Brunetti

Eitan Altman

Majed Haddad

Pricing access to the Internet with partial information

We shall consider two competition problems between service providers with asymmetric information. The utility of each one of them depends on the demand it gets and in its price. The demand itself is also a function of the prices of the providers. In both problems there is one provider (player 1) that has more information than the other (player 2) on the demand function. The more informed provider plays first, and then the second observes the move of the first provider and chooses accordingly its own action: it determines its price per unit demand. In the first problem that we consider, the first provider does not control its price (it has a fixed price known to the other provider which does not depend on the information that is unknown to provider 2). Before player 2 takes its action it receives a signal (or a recommendation) from the more informed player, i.e. from provider 1. The pure actions of provider 1 are thus the possible choices of what signal to send. The second problem that we consider is the same as the first one except that the actions of provider 1 is to choose its price. Since player 2 observes the choice of price of player 1 before it takes its own pricing decision, we can consider the choice of price by player 1 has also a role of signalling. We reduce each one of the problem to an equivalent four by four matrix game.

Introduction and Model

This note uses Bayesian game theory in order to study pricing issues in competitive environment. We model two problems related to pricing under the assumption that only one mobile fully knows the demand function. In the first part we study the signalling problem in which the informed player can signal information to the other one and has to decide on a signalling rule such that, given an optimal reaction of the other player reacts to the signal, the utility of the informed player would be maximized. In a second sceario that we study, each of the two players determine their prices. Yet the order of the decisions is such that the player that is not informed of the demand function that is used, determines its own price after observing the choice of the first player. We formulate both games as Bayesian games and show how to transform them into equivalent matrix games.

Our starting point is to define two service providers. The demand d i that provider i receives is a function of both its own price as well as of its competitor. We thus write it as d i (p 1 , p 2 ). We assume that the utility for provider i is given by

U i = d i p i , i = 1, 2
Each provider wishes to maximize the expectation of its own utility.

The Signalling Game

Let us first consider the following scenario, which is common knowledge for both providers (which we call players).

• The price p 1 of player 1 is fixed and known.

• Player 2 can choose one of two possible levels of pricing: ζ l and ζ h .

• The demand d i for each provider is a function of the prices of both providers. The demand function of player 2 is known to both players.

• We assume that the demand of player 1 depends on some parameter θ. θ is known to player 1 but not to player 2. It can take one of two values: θ 1 and θ 2 .

The game is then played as follows.

• The value of θ is chosen by nature according to some distribution π. It equals θ r with probability π r .

• Player 1 observes θ and then sends a signal to player 2. It has four pure strategies for signalling: q 1 q 1 , q 1 q 2 , q 2 q 1 and q 2 q 2 . Here, q i q j is the strategy according to which player 1 signals q i if it observes θ 1 and signals q j if it observes θ 2 .

• Player 2 has also four pure strategies. We write them as

ζ l ζ l , ζ l ζ h , ζ h ζ l and ζ h ζ h .
Here, a strategy of the form ζ i ζ j means choosing ζ i if player 2 observes a signal q 1 by player 1 and choosing ζ j otherwise.

The game that we study is known as a game with "cheap talk"; this is a signaling game where the sender incurs no cost for his signals [START_REF] Kreps | signaling[END_REF]Sec. 7].

Transformation into a matrix game

We can now represent the game as a 4 × 4 matrix game.

Fix i ∈ (1, 2), j ∈ (1, 2), m ∈ (l, h), n ∈ (l, h). Assume that player 1 chooses q i q j and that player 2 chooses ζ m ζ n .

For r = 1, 2, define ξ(r, i, j) to be i if r = 1 and j otherwise. Player 1 sends signal q ξ(r,i,j) to player 2 if θ = θ r . Define φ(r, i, j, m, n) to be the price chosen by player 2 as a result of the signal ξ(r, i, j) when it uses the strategy

ζ m ζ n . φ(r, i, j, m, n) equals ζ m if ξ(r, i, j) = 1 and is otherwise ζ n . Then U 1 (q i q j ; ζ m ζ n ) = E q i q j ;ζmζn [p 1 d 1 (p 1 , p 2 )] (1) = r=1,2 π r E q i q j ;ζmζn [p 1 d 1 (p 1 , p 2 )|θ = θ r ] (2)
where E q i q j ;ζmζn [p

1 d 1 (p 1 , p 2 )|θ = θ r ] = p 1 d 1 (p 1 , φ(r, i, j, m, n).
Thus we can rewrite:

U 1 (q i q j ; ζ m ζ n ) = p 1 r=1,2 π r d 1 (p 1 , φ(r, i, j, m, n)
For player 2 we obtain:

U 2 (q i q j ; ζ m ζ n ) = E q i q j ;ζmζn [p 2 d 2 (p 1 , p 2 )] (3) = r=1,2 π r • φ(r, i, j, m, n) • d 2 (p 1 , φ(r, i, j, m, n) (4)

The Linear Case

We now consider a linear demand function:

d i = d 0 i -A ii p i + A ji p j (5)
Assume that A is known but that d 0 2 is unknown. It can be written as a function of a parameter θ, i.e., d 0 2 = d 0 2 (θ). It follows from the above considerations that the utility functions of player 1 and player 2 can be expressed as:

U 1 (q i q j ; ζ m ζ n ) = p 1 r=1,2 π r θ r -A 11 p 1 + A 21 φ(r, i, j, m, n) (6) U 2 (q i q j ; ζ m ζ n ) = r=1,2 π r • φ(r, i, j, m, n) d 0 2 (θ r ) -A 22 φ(r, i, j, m, n) + A 12 p 1 (7)
3 Prices as signals

Now consider the following variant of the problem. Instead of a fixed price p 1 , the first player can choose one of two values as its own price instead of just a signle. Yet the price itself serves also as a signal that player 2 can see. q i , i = 1, 2 now stands for the price chosen by player 1.

In this case assuming that q i stands for the price chosen by player 1 and following the previous definitions, we have that q ξ(r,i,j) corresponds to the price chosen by player 1 if he plays strategy q i q j and θ = θ r . The utility functions we obtain are:

U 1 (q i q j ; ζ m ζ n ) = r=1,2 π r • q ξ(r,i,j) (d 1 (q ξ(r,i,j) , φ(r, i, j, m, n))) U 2 (q i q j ; ζ m ζ n ) = r=1,2 π r • φ(r, i, j, m, n)(d 2 (q ξ(r,i,j) , φ(r, i, j, m, n)))
The Linear Case If we consider again the linear demand function defined in (5), the utility functions of player 1 and player 2 can be expressed as:

U 1 (q i q j ; ζ m ζ n ) = r=1,2 π r • q ξ(r,i,j) (θ r -A 11 q ξ(r,i,j) + A 21 φ(r, i, j, m, n)) U 2 (q i q j ; ζ m ζ n ) = r=1,2 π r • φ(r, i, j, m, n)(d 0 2 (θ r ) -A 22 φ(r, i, j, m, n) + A 12 q ξ(r,i,j) )
It follows from the above considerations that p 1 = q ξ(r,i,j) , yielding the following expressions for the utility functions of player 1 and player 2

U 1 (q i q j ; ζ m ζ n ) = r=1,2
π r • q ξ(r,i,j) (θ r -A 11 q ξ(r,i,j) + A 21 φ(r, i, j, m, n)) (8)

U 2 (q i q j ; ζ m ζ n ) = r=1,2 π r • φ(r, i, j, m, n)(d 0 2 (θ r ) -A 22 φ(r, i, j, m, n) + A 12 q ξ(r,i,j) ) (9)

Discussion

In this paper, we studied a model of pricing access to the Internet and investigated the role of signalling (or recommendations) in identifying side payments (price per unit demand). We considered the case of two Internet service providers aiming at maximizing their own expected utility. Under this setting, we assume that that the amount of information made available to both ISPs on the demand function is not the same: the more informed provider plays first, and then the second observes the move of the first provider and chooses accordingly its own action.

We studied two configurations of the proposed problem. In the first configuration, one service provider (say provider 1) has a fixed price known to the other provider (say provider 2). The pure actions of provider 1 are thus the possible choices of what signal to send. In the second configuration, provider 1 is allowed to choose his prices which serves as a signal to provider 2. We then reduced each one of the configuration to an equivalent four by four matrix game and studied the equilibria Our model is, needless to say, a mere caricature that captures certain types of interactions between ISPs. The biggest benefit is that it is tractable, as evidenced by the obtained expressions in this paper. The litmus test of its usefulness will be its ability, or otherwise, to explain some observed behavior, even if only qualitatively. Studies in this direction are ongoing. Finally, an extension of our model to account for aspects of quality of service (QoS) should be brought in to enrich the model. We hope to pursue some of these in future works.


Parikshit Dutta

Mathieu Saujot

Elise Arnaud

Benoit Lefèvre

Emmanuel Prados

Uncertainty Propagation and Sensitivity Analysis During Calibration of TRANUS, an Integrated Land Use and Transport Model

Keywords: Uncertainty propagation, sensitivity analysis, calibration under uncertainty, hypothesis testing, integrated land use and transport models, TRANUS, Grenoble

In this work, propagation of uncertainty during calibration process of TRANUS, an integrated land use and transport model (ILUTM), has been investigated. It has also been examined, through a sensitivity analysis, which input parameters affect the variation of the outputs the most. Moreover, a probabilistic verification methodology of calibration process, which equates the observed and calculated production, has been proposed. The model chosen as an application is the model of the city of Grenoble, France. For sensitivity analysis and uncertainty propagation, Monte Carlo method was employed, and a statistical hypothesis test was used for verification. The parameters of the induced demand function in TRANUS, were assumed as uncertain in the present case. It was found that, if during calibration, TRANUS converges, then with a high probability the calibration process is verified. Moreover, a weak correlation was found between the inputs and the outputs of the calibration process. The total effect of the inputs on outputs was investigated, and the output variation was found to be dictated by only a few input parameters.

I. INTRODUCTION

Integrated land use and transport modeling has attracted the attention of researchers, recently. In this regard, the main thrust has been to develop models which are generic, user friendly and robust. Over the years, a large number of such models have come into existence [START_REF] Wegener | Overview of land use transport models[END_REF], [START_REF] Echenique | An integrated land use and transport model[END_REF]. It is well known that integration of land use and transport system creates a complex nonlinear system, which evolves in different scales [START_REF] Echenique | An integrated land use and transport model[END_REF]. Analyzing these complex systems is typically a hard problem, especially in the presence of uncertainty, whose effects may be difficult to assess [START_REF] Krishnamurthy | Propagation of uncertainty in transportation land use models: Investigation of dram-empal and utpp predictions in austin, texas[END_REF]. In such cases, calibration plays a central role, as it helps us determine optimal parameters, creating a robust model [START_REF] Iacono | Models of transportation and land use change: a guide to the territory[END_REF], [START_REF] Sullivan | Integrated land-use, transportation and environmental modeling: Validation case studies[END_REF]. Propagation of uncertainty and assessing sensitivity of the input parameters on the outputs during the calibration process, is significant for proper tuning of the model and ensuring better predicting capabilities. Moreover, if a large number of parameters are involved, such an analysis can help to significantly reduce the dimension of the whole calibration problem, which in turn helps parameter estimation [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF], [START_REF] Campolongo | From screening to quantitative sensitivity analysis. a unified approach[END_REF]. On the other hand, it is necessary that the calibrated model converges and the outputs produced by the calibrated model match actual observed data [START_REF] Martens | Validation and verification of regression in small data sets[END_REF]. The goal here is to create a model that is robust to uncertainties, while verifying its consistency with observations. Uncertainty can come from various sources in an integrated land use and transportation model (ILUTM), and is well documented [START_REF] Mahmassani | Uncertainty in transportation systems evaluation: issues and approaches[END_REF], [START_REF] Krishnamurthy | Propagation of uncertainty in transportation land use models: Investigation of dram-empal and utpp predictions in austin, texas[END_REF]. Moreover, while developing such a model, little can be done regarding lack of data and mismeasurement thereof, biased sampling and mis-specification [START_REF] Zhao | The propagation of uncertainty through travel demand models: An exploratory analysis[END_REF]. Hence, in practice, quantifying uncertainty for these models is challenging [START_REF] Waddell | Integrated land use and transportation planning and modelling: addressing challenges in research and practice[END_REF]. Regarding evolution of uncertainty, scientists have investigated the impact of uncertainty in land use part of UrbanSim [START_REF] Pradhan | Uncertainty propagation in an integrated land use-transportation modeling framework: output variation via urbansim[END_REF], [START_REF] Waddell | Modeling urban development for land use, transportation, and environmental planning[END_REF], and several other ILUTMs like combination of DRAM-EMPAL and UTPP [START_REF] Krishnamurthy | Propagation of uncertainty in transportation land use models: Investigation of dram-empal and utpp predictions in austin, texas[END_REF]. Researchers have also investigated the role of uncertainty in decision making and use parameter estimation in this case [START_REF] Duthie | Highway improvement project rankings due to uncertain model inputs: Application of traditional transportation and land use models[END_REF]. Sevcíková et al. have used Bayesian inference for estimation of optimal parameters in UrbanSim [START_REF] Sevcíková | Assessing uncertainty in urban simulations using bayesian melding[END_REF], [START_REF] Sevcíková | Uncertain benefits: Application of bayesian melding to the alaskan way viaduct in seattle[END_REF]. Regarding model calibration, Clay et al. investigates multivariate uncertainty analysis along with validation exercises in MEPLAN, another ILUTM [START_REF] Hunt | Theory and application of an integrated land use and transport modelling framework[END_REF], [START_REF] Clay | Multivariate uncertainty analysis of an integrated land use and transportation model: Meplan[END_REF]. But an intensive analysis, focusing on uncertainty propagation and sensitivity analysis of the calibration process of ILUTMs has largely been ignored. In this work, we have investigated effect of uncertainty during model calibration process of TRANUS, an ILUTM [START_REF] De La Barra | Integrated land use and transport modeling: the tranus experience[END_REF], when applied to model the region of Grenoble in France. In particular, we analyze the propagation of the uncertainty and we perform a sensitivity analysis of the calibration process developed in TRANUS. Moreover, a probabilistic verification methodology for TRANUS calibration is proposed to check if the calibration task has been achieved.

TRANUS has been used by city planners and modelers to simulate land use and transport structures in prospective studies in many cities [START_REF] De La Barra | Tranus-j: putting large models into small computers[END_REF], [START_REF] Anez | Development of tranus: An integrated land use and transport modeling system[END_REF]. It predicts the land use and travel demand of a model and can help to evaluate some optimal parameter values for sustainable development [START_REF] Vichiensan | Evaluation system of policy measure alternatives for a metropolis based on tranus from the view point of sustainability[END_REF]. Calibration in TRANUS is important as it estimates price adjustment factors which are then used for future predictions [START_REF] De La Barra | Integrated land use and transport modeling: the tranus experience[END_REF]. Uncertainty and sensitivity analysis during calibration of TRANUS is significant as it helps ranking input parameters according to their impact on variability of the adjustment factors.

The rest of the paper is organized as follows. First, TRANUS and its modeling philosophy is described in brief. Next, TRANUS calibration methodology is presented, along with method used to perform sensitivity analysis and uncertainty propagation, and then the proposed probabilistic verification methodology is formulated. Afterward, we describe the Grenoble model and the parameters assumed as uncertain during calibration process. Finally, the results achieved from calibration of Grenoble model are presented, and an analysis of the results obtained from uncertainty propagation and sensitivity analysis are carried out.

II. TRANUS DESCRIPTION

TRANUS provides a generic framework to model land use and transportation in an integrated manner, both in urban and regional levels. The region of interest is divided into economic sectors and spatial zones. Then TRANUS combines two modules: the land use and activity module which simulates a spatial economic system by assessing the activity locations and economic sector interactions; and a transportation module, which estimates the use of the transport network and the associated disutility.

The land use and activity module estimates the productions and the consumptions for a zone at a given period, and the demands of flows that this activity generates. These demands of flows are then fed to the transportation module. In this way the movements of people or freight are explained as the results of the economic and spatial interaction between activities, the transport system and the real estate market. Then from the transportation network, once the corresponding trips are generated, travel flows are allocated to the network according to travel demand. In turn, the accessibility that results from the transport system influences the location and interaction between activities through transport disutilities, also affecting land rent.

The two modules in the system use discrete choice logit models, linked together in a consistent way. This includes activity-location, land-choice, and multi-modal path choice and assignment. The modules are then run iteratively, such that production and consumption demands for each area are met and equilibrium is achieved.

For the sake of brevity, an in-depth analysis using mathematical equations has not been presented to describe TRANUS land use and transportation modules. Interested readers may consult [START_REF] De La Barra | The mathematical and algorithmic structure of tranus[END_REF] for detailed mathematical treatment of the subject.

III. CALIBRATION OF THE LAND USE MODULE

The current work focuses primarily on the land use module. TRANUS land use algorithm consists in the resolution of a system of around 20 deterministic nonlinear equations and inequalities. The solution of such a system represents an economic equilibrium between supply and demand. This system of equations contains a number of economic parameters (e.g. demand elasticity parameters, location dispersion parameters, etc.) which can be a priori roughly estimated with a more or less important degree of uncertainty. In practice, it is crucial to precisely calibrate these parameters in order to get satisfactory simulations. In TRANUS, this calibration is not automatic. It is done via a series of trials and errors and requires lots of expertise. In addition to these economic parameters, TRANUS contains a set of intrinsic parameters called "adjustment factors" which allow the adjustment of the prices in order to correct the modeling errors. By adding these parameters, the system of equations is then relaxed. This increases the set of solutions and thus, in a sense, the system then becomes better-posed. Contrary to the economic parameters, TRANUS contains a procedure enabling the automatic assessment of these adjustment factors. Hence, the goal of this paper is to specifically work on this adjustment factor estimation procedure. We analyze its relevance and its sensibility with respect to economic parameters.

TRANUS calibration is generally is done with respect to a "base year", for which the production values, real estate prices and transport flows are available as observed data for the region of interest. Let us consider a region, divided into N sectors and M zones. Productions and prices are available as observed data for a given base year. We denote by X obs ∈ R N ×M and P obs ∈ R N ×M the set of observed production data and observed data prices respectively. The data and economic parameters serve as inputs to the calibration process, and adjustment factors serve as outputs. The calibration process can be mathematically described with the following inputoutput model,

[X, H] ⊤ = f (X obs , P obs , ρ), (1)
where ρ = {ρ i } ∈ R P is the vector of economic parameters, which are uncertain. X = {X nm } ∈ R N ×M is the matrix of computed productions with X nm being the production for sector n in zone m. The adjustment factors H ∈ R N ×M allow to correct the primary given prices by assigning proper values of unit production prices P = {P nm },

P = H + P obs , (2)
in such a way that the computed productions X match the observed productions X obs . In other words, H are introduced to correct the model in order to get a solution which verifies

X ∼ = X obs . (3)
Prices are in effect often quite volatile and, generally, the price data is the least accurate among all data. The adjustment factors H allow thus to compensate for this and also for the various other imperfections of the model. The approach used to estimate the adjustment factors H and the productions X is iterative. The process loops until convergence is reached for the prices P and productions X, or until a maximum given number of iterations if not the case. The process is explained through a flow diagram in fig. 1.

A. Uncertainty and Sensitivity Analysis

Given ρ admits a PDF P(ρ), the PDF of H, P(H), and the sensitivity of P(H) to ρ, are to be determined. In this work, Monte Carlo approach has been used, both for uncertainty propagation and sensitivity analysis [START_REF] Saltelli | Sensitivity analysis[END_REF].

Let ρ (k) , k = 1, 2, . . . , K be an unbiased sample of P(ρ), then, ∀ρ (k) we can compute H (k) from eqn. [START_REF] Wegener | Overview of land use transport models[END_REF]. The moments of H then can be approximated using Monte Carlo (MC) approximations, i.e.,

E [H] = 1 K K i=1 H (k) , E H 2 = 1 K K i=1 H (k) 2 , . . . (4)
P(H) can be then estimated using histogram approximation [START_REF] Magnello | Karl pearson's gresham lectures: Wfr weldon, speciation and the origins of pearsonian statistics[END_REF]. As for sensitivity analysis, we are interested in studying the effect of each parameter on the outputs. We are primarily focused on finding the total effects of inputs [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF], i.e. examining the effect of variability of just a parameter or combination of them on H. In mathematical terms, we are interested in calculating,

S T i = E[V[H|ρ \i ]] V[H] , S T ij = E[V[H|ρ \(i,j) ]] V[H] , . . . (5)
where,

V[•] = E • 2 -E [•] 2
where i, j ∈ {1, 2, . . . , P} represent the parameter's indexes, \i = (1, . . . , i-1, i+1, . . . , P) and \(i, j) = (1, . . . , i-1, i+ 1, . . . ; 1, . . . , j -1, j + 1, . . . , P). S T i and S T ij represent total effects of ρ i alone, and combination of ρ i and ρ j , respectively. For example, S T i is calculated varying only ρ i over its domain and calculating its effect.

Here, we are interested to know the value of total effects of each input parameter on the variability of the adjustment factors.

B. Methodology for Verification of Calibration

The inbuilt calibration in TRANUS stops when the following criteria are met,

max n,m P τ nm -P τ -1 nm P τ -1 nm ≤ ǫ 1 max n,m X τ nm -X τ -1 nm X τ -1 nm ≤ ǫ 2 , ( 6
)
where τ is the iteration index (in the loop represented in fig. 1). However, as previously mentioned, if the model is sufficiently well calibrated, the production values calculated by the model, and the observed production should match.

Hence, the calibration step must be verified for consistency in production values.

In this work, it has been tested that given eqn. ( 6) holds then the L 2 norm of the difference between the productions is less than a pre-specified value ǫ, i.e. :

X obs -X 2 < ǫ. (7)
A statistical method has been employed here, where inference is drawn based on whether the sample passes a hypothesis test. The verification methodology, in the context of model calibration of TRANUS is presented next.

Given the value of ǫ we define the following random variable,

I X = 1, ||X obs -X|| 2 < ǫ, 0, otherwise. ( 8
)
Let us further define another random variable Y , which is a function of I X . For a sample of size K of X, received from sampling the parameter space and subsequently applying the calibration methodology,

Y = 1 K K k=1 I X (k) . (9)
Clearly, the random variable Y follows a binomial distribution: it can be proved as the random variable I X is a Bernoulli random variable, hence a finite sum of them will result in a binomial random variable. Hence, Y ∼ B(p, K), where p is the probability that eqn. ( 7) holds.

For the discrete random variable Y , a z-test for proportion is then carried out [START_REF] Ang | Probability concepts in engineering[END_REF], to verify TRANUS calibration. The null (H 0 ) and the alternate (H 1 ) hypothesis are set as follows:

• H 0 : Y > Y 0 , • H 1 : Y ≤ Y 0 ,
where Y 0 is a user specified proportion, quantifying the fraction of sample, that the user believes should pass the z-test.

In other words, Y 0 is the user's perception on what fraction of an ensemble of input parameter vector ρ, will satisfy eqn. [START_REF] Campolongo | From screening to quantitative sensitivity analysis. a unified approach[END_REF] given the ensemble satisfies eqn. ( 6), after outputs X and H have been obtained applying eqn. (1) and eqn. [START_REF] Echenique | An integrated land use and transport model[END_REF].

Typically, for a model that has been well designed and calibrated, Y 0 is set at a high value. The null hypothesis is accepted at a significance level α for a sample. If the null hypothesis is accepted we infer that the calibration task has been verified.

IV. TRANUS GRENOBLE MODEL

In this work, TRANUS has been applied to model the urban area of Grenoble, France [START_REF] Saujot | Simuler les dynamiques urbaines avec tranus: une méthodologie d'application comme passerelle entre théorie et territoire[END_REF]. The region is divided into 225 zones, and the transport network is composed of 2413 nodes. The meshes are of variable size, so that the transport network and the geographic distribution are coherent. Public transport modes (buses, tramways, trains) as well as private modes (walk, bike, and 4 categories of cars) have been implemented. The data of the transport network were obtained from the SMTC 1 . The transport data on peak hours were mainly gathered from SMTC, AURG 2 , INRETS 3 , and the survey EMD 4 made on Grenoble in 2010.

The Grenoble model considers 22 economical sectors. The population has been divided into 7 sectors: the households are classified into 4 categories depending on their income, as well as the retired households that are classified into 2 categories. The students are considered as a separate sector. The retired households and the students are made exogenous, as their location choice is more explained by attractivity criterion than by accessibility criterion. As for the 8 employment sectors, the industries and immaterial industries are made basic. Service employments are classified into 3 sectors, all induced by the model: community-based services for daily use, communitybased services for occasional use and supermarkets. Schools, university and other public employments corresponds to 3 induced sectors. Finally, the real estate is divided into 7 categories: individual / collective / social housing, and 4 categories of net gross floor areas depending in their location, and use. The data for calibrating the land use model were obtained from INSEE 5 , AURG, UNEDIC 6 and EMD. These data have been processed to obtain population and employment by type and by zone, as well as the prices of the real estate.

The economic sectors can also be classified as transportable and non-transportable sectors. Non-transportable sectors are the ones that must be consumed where they are produced. Typically they correspond to the real estate, i.e. housings and net gross floor areas. All sectors for employment and population are transportable. In the case of transportable sectors, because the model is distributing a total production amount to zones, the magnitude of the adjustment factors are of little significance, as long as their standard deviation over all zones is small. Otherwise it means that the adjustment factors are dominating the zonal distribution of production. In the case of non-transportable sectors, there is no zonal distribution, so the values of the adjustment factors for all zones, must be as small as possible. Detailed explanation can be found in [START_REF] De La Barra | Integrated land use and transport modelling. Decision chains and hierarchies[END_REF]. For the Grenoble model, 7 of the 22 sectors are nontransportable and rest are transportable [START_REF] Saujot | Simuler les dynamiques urbaines avec tranus: une méthodologie d'application comme passerelle entre théorie et territoire[END_REF].

A. Uncertain Parameters in the Model

As said previously, TRANUS contains a large number of economic parameters (e.g. demand elasticity parameters, location dispersion parameters, etc.). Among these parameters, some are more difficult to estimate, but has a major impact on the calibration result. This is the case of the parameters involved in the calculation of the induced demand, that are taken here as uncertain parameters of interest [START_REF] De La Barra | Integrated land use and transport modelling. Decision chains and hierarchies[END_REF]. The demand function is given by an exponential model,

a i nn ′ = min nn ′ + (max nn ′ -min nn ′ ) e -δ nn ′ U i n ′ (10)
where a i nn ′ is the amount of production of sector n ′ demanded by a unit of sector n in zone i, max nn ′ and min nn ′ are 2 Agence d'Urbanisme de la Région Grenobloise 3 Institut National de REcherche sur les Transports et leur Sécurité 4 Enquète Ménage Déplacement 5 Institut National de la Statistique et des Etudes Economiques 6 Union Nationale Interprofessionnelle pour l'Emploi dans l'Industrie et le Commerce minimum and maximum amount of n ′ required by a unit production of n respectively, δ nn ′ is the elasticity parameter of n with respect to the cost of input n ′ , and U i n ′ is the consumption disutility of n ′ in i. Details on how consumption disutility U i n ′ is calculated can be found in [START_REF] De La Barra | The mathematical and algorithmic structure of tranus[END_REF]. In this work, max nn ′ , min nn ′ and δ nn ′ are assumed to be uncertain. Amongst 231 possible inter-sector combination for n and n ′ , 16 were selected. We focused on the parameters describing the demand of the housing sectors required by the various population sectors. Figure 2 describes these sector's interaction. Hence there were 48 input parameters which were uncertain. The resulting uncertainty in adjustment factors were studied, consisting of 225 × 22 = 4950 outputs for all zones and sectors combined. However, a comprehensive analysis, assuming all the parameters as uncertain, leading to calculation of production has not been done here, and is a topic of our future research. V. SIMULATION RESULTS Monte Carlo simulations were performed with 9 samples, each of sample size 200. Hence total number of input parameter sets were 1800. As stated earlier, 16 inter-sector interactions were considered and a multivariate Gaussian distribution was assumed about a pre-specified mean with 5% standard deviation about the mean value. The parameters considered uncertain, and their mean values are given in table I, where the symbols have the same meaning as in eqn. [START_REF] Zhao | The propagation of uncertainty through travel demand models: An exploratory analysis[END_REF]. The calibration mean values correspond to approximate values obtained from opinions of practitioners and experts after analyzing the parameters for each sector-sector combination individually. Also, in the present case we assume that the random variables considered are independent of each other.

As stated in section III, the calibration methodology of TRANUS is computationally intensive. The computational time taken to run TRANUS for one set of input parameters, on an average is 350.8s. Hence, in this work large number of samples were not considered. However, in future, we plan to work with larger number of samples. First we apply the proposed verification methodology on each sample. Then, we will present the results of uncertainty propagation and sensitivity analysis.

A. Verification Results

At first we check whether for each input set in a sample, if TRANUS converges, or if eqn. ( 6) holds. Then we perform the hypothesis test to verify if indeed convergence has been achieved in the sense of eqn. [START_REF] Campolongo | From screening to quantitative sensitivity analysis. a unified approach[END_REF]. Amongst the 1800 input sets 1509 of them satisfied eqn. [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]. For the hypothesis test, the significance level α was fixed to 0.95 and ǫ and Y 0 were accordingly varied. The results received for some ǫ and Y 0 combination are given in table II. From the data, it is observed that, the L 2 norm of X obs for Grenoble region is at least ||X obs || 2 > 1.2 × 10 8 . Hence . Thus, clearly ǫ = 90 represents a very small fraction of the total production of the Grenoble region. It can be said that for given set of input parameters, if the TRANUS calibration process converges in the sense of eqn. ( 6), then for an acceptable value of ǫ, eqn. ( 7) is true with a significance level of α = 0.95.

B. Uncertainty Propagation

This section assesses, how the PDF of the adjustment factors are affected when the input is uncertain. To do this, we employ the Monte Carlo approach explained in section III-A. From the verification methodology only those input sets which satisfy eqn. ( 6) are chosen. The price adjustment factors are normalized and presented as increase over the observed price P obs . In other words, Hs in eqn. (1) and eqn.

(2) are normalized to get H/P obs . The adjustment factors are expected to lie between [-1, 1] ideally [START_REF] De La Barra | Integrated land use and transport modeling: the tranus experience[END_REF]. The results are presented in an aggregate level, for the sake of ease in representation.

Figure 4 shows the frequency distribution for mean and variances of the adjustment factors, calculated for every sectorzone combination. It can be observed that for most cases the variances are low with a maximum frequency for 0.05 -0.06. However for some cases the adjustment factors have high variance. Amongst 4950 combination 5 of them were observed to have variance over 0.2. The maximum variance in adjustment factors was observed for zone 45 and sector 2 which was 0.2417, fig. 5 shows the histogram plot.

Next the correlation structure between the adjustment factors was studied. All the sector-zone combination were considered here giving rise to 3184026 possible pairings. Amongst them 3621 sector-zone combination were highly correlated having correlation coefficient greater than 0.9. However, 11 pairings had correlation coefficient less than -0.4. The frequency distribution of the correlation coefficient has been plotted in fig. 6. It can be observed that most of the adjustment factors do not exhibit a strong dependence with other factors, as the correlations are crowded around zero.

The input-output correlation structure has been plotted in fig. 7. It can be seen that they are clustered around zero. This indicates the absence of a linear relationship between outputs and inputs. However, though strong dependence is not exhibited, nothing can be intuitively concluded regarding actual relation between inputs and outputs.

C. Adjustment Factors for Transportable and Non-Transportable Sectors

In this section we investigate the variation of the standard deviation (σ n ) and maximum value H max n of the adjustment factors, for a given sector over all the zones. If the adjustment factors are defined as H nm , for sector n and zone m (refer section III), then the two random variables can be defined as,

σ n = 1 M -1 M m=1 [H nm -E [H n ]] [H nm -E [H n ]] ⊤ (11)
H max n = max m∈[1,M ] H nm (12)
where,

E [H n ] = 1 M m H nm
As stated in section IV, the transportable sectors should have a low σ n , which is the standard deviation of H nm for the sector n over all zones. Hence, we are interested in σ n for transportable sectors. However, for non-transportable sectors, the adjustment factors itself should be low. Thus, here we investigate the maximum adjustment factor obtained in a given sector for all zones, i.e. H max n . The non-transportable sectors in the Grenoble model are, sector number 11, 12, 13, 14, 15, 21 and 22. Rest of the sectors are transportable. We consider the Monte Carlo input sets obtained from uncertainty propagation in the previous section for the present analysis.

Figure 8 shows the mean and the 99.7% confidence intervals of σ n for all sectors, and fig. 9 shows their maximum values obtained from the samples. It can be observed that amongst transportable sectors, sector number 3, 8, 19 and 20 have maximum values for σ n greater than 0.4. Moreover the variability of σ n for these sectors are higher than any other transportable sector. For instance, the standard deviation of the adjustment factor for sector number 8 can go as high as 0.47 with a non zero probability.

Figure 10 shows, plots for mean and 99.7% confidence intervals for H max n for all sectors and fig. 11 value of the adjustment factors for these sectors are low, but large variations may occur.

D. Sensitivity Analysis

The total effect of the uncertain input variables on the output is investigated here, using methods explained in section III-A. Here, effect of each input parameter, individually has been considered, hence, for subsequent input parameter sets just one parameter value is varied where as others are kept fixed. We work with the samples that were drawn initially for uncertainty propagation. The number of inputs were 48, corresponding to 16 elasticity, minimum and maximum consumption value parameters. For the sake of convenience in representation, the uncertain parameters have been replaced by numbers corresponding to column 1 of table I. The number of total effects observed were 4950 × 48 = 237600. As it is not possible to present all the results individually, the results are presented in an aggregate level, covering all zones and sectors, with notes on important observations.

Figure 12 shows the plot of means and 99.7% CIs for total effect of each elasticity parameter δ nn ′ , over all sectors and zones. It can be seen that elasticity parameter 8 i.e. δ 9-12 has the +99.7% CI spanning above 0.025, which is maximum amongst any other parameters. Amongst elasticity parameters, the total effect of δ 9-12 most significant. The effect of minimum consumption amount parameters in eqn. [START_REF] Zhao | The propagation of uncertainty through travel demand models: An exploratory analysis[END_REF] are observed to follow a similar trend. It is seen that the parameter for sector-sector combination of 9 and 12 have the maximum total effect. From fig. 13 it can be observed that min 9-12 can potentially be as high as 0.85, which is 8.5 times more than the next largest value any other parameter can take. However, the total effects of maximum consumption amount parameters, behave differently. Figure 14 shows the 99.7% CI and mean for total effect of max nn ′ over all sectors and zones. It is observed that, unlike the previous two cases, the positive end of the confidence intervals do not show large variation. The spread of the total effect of max 6-12 , is larger than any other maximum consumption amount parameter, with 0.026 being the +99.7% CI, but the positive end for max 9-12 is just 0.003 less.

VI. CONCLUSION

In the current work, we have performed, propagation of uncertainty and sensitivity analysis during calibration process of TRANUS. Moreover a probabilistic verification methodology, has been proposed. The calibration process was presented as an input output model with uncertain inputs as the parameters of the induced demand function and outputs being the adjustment factors computed during calibration of TRANUS. The calibration verification methodology, sought to equate the observed production data with the calculated production during calibration. The land use and transportation model of Grenoble, France was used as an application here.

A z-test for hypothesis was employed for verification. It was found that with a significance level of α = 0.95 the calibration process is verified if the TRANUS calibration algorithm converges.

The weak correlation was observed between the adjustment factors and between adjustment factors and inputs, with correlation coefficients clustered around zero. This serves as an indicator that no strong linear relationship exists between the adjustment factors and the inputs.

The variation of the standard deviation and maximum value of the adjustment factors, for a given sector over all the zones was investigated next. Amongst transportable sectors sector number 8 was observed to have the highest +99.7% confidence interval for standard deviation. As for the non-transportable sectors, sector number 22 had the largest +99.7% confidence interval for maximum value of adjustment factors.

The total effect of the input parameters on outputs was investigated next. It was found that the min 9-12 parameter has the maximum effect amongst all other parameters. Amongst elasticity parameters, elasticity for sector combination 9 -12 has the maximum total effect, and maximum consumption amount for sector combination 6 -12 had higher total effect than any other maximum consumption parameter.

In future, we plan to use the results obtained from this analysis, to estimate parameters, and to employ techniques for Bayesian inference to get the PDF of input parameters. It is our future aim to calibrate the Grenoble model such that the land use and transportation scenarios, can be predicted and validated.

Fig. 1 .

 1 Fig. 1. Calibration process of TRANUS

Fig. 2 .

 2 Fig. 2. Sector interactions for which we consider uncertain the parameters of the demand function. The numbers correspond to the sector id in the model. The direction of arrows correspond to directions of consumption, e.g. students consume collective or social housing.

  It can be observed that given Y 0 = 0.85 for a low ǫ value, 3 amongst 9 fail the test and for ǫ = 70 and Y 0 = 0.95 two of them fail. It is seen, most of the samples pass the test if we increase the value of ǫ. Figure3, shows number of samples passing the z-test when ǫ and Y 0 are varied. The colored lines have the same Y 0 value. It is observed that, all the samples pass the test if ǫ > 90 with Y 0 = 0.95.

Fig. 3 .

 3 Fig. 3. Plot showing variation number of samples passing hypothesis test with ǫ for different values of Y 0

Fig. 4 .Fig. 5 .

 45 Fig. 4. Histogram of a) variance and b) mean of the adjustment factors.

Fig. 6 .

 6 Fig. 6. Frequency distribution plot for correlation coefficient between outputs.

Fig. 7 .

 7 Fig. 7. Frequency distribution plot for correlation coefficient between inputs and outputs.

Fig. 8 .

 8 Fig.8. Plot for mean and 99.7% confidence intervals (CI) for standard deviation of adjustment factors over all zones. Transportable sectors-blue dots and red lines. Non-transportable sectors-red dots and blue line.

  std. dev.

Fig. 9 .Fig. 10 .

 910 Fig.9.Maximum of the standard deviation of adjustment factors for each sector over all zones. Transportable sectors-red dots. Non-transportable sectors-blue dots.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Maximum H max n for each sector over all zones. Transportable sectors-magenta dots. Non-transportable sectors-green dots.

Fig. 13 .Fig. 14 .

 1314 Fig.[START_REF] Waddell | Modeling urban development for land use, transportation, and environmental planning[END_REF]. Confidence intervals of the total effect of minimum consumption amount parameters (min nn ′ ), over all sectors and zones. Mean-black star. CI-red lines.

TABLE II RESULTS

 II FOR THE z-TEST BY VARYING ǫ AND Y 0 .

	Sample ǫ = 50,Y 0 = 0.85	ǫ = 70,Y 0 = 0.85	ǫ = 70,Y 0 = 0.95
		p value	Result	p value	Result	p value	Result
	1	0.957	Pass	1	Pass	0.97045	Pass
	2	0.1943	Fail	1	Pass	0.82237	Fail
	3	0.9996	Pass	1	Pass	0.97495	Pass
	4	0.7543	Fail	1	Pass	0.95027	Pass
	5	0.5706	Fail	1	Pass	0.99143	Pass
	6	0.999	Pass	1	Pass	0.98859	Pass
	7	0.9993	Pass	1	Pass	0.98296	Pass
	8	0.9966	Pass	1	Pass	0.9303	Fail
	9	0.9998	Pass	1	Pass	0.99411	Pass

  shows their maximum values. It can be observed that, amongst nontransportable sectors, sector number 12, 15 and 22 have value of H max n , which is greater than 3. Moreover, the variability of H max n for sector 22 is higher than any other sector. It can be inferred from fig. 10 that +99.7% confidence interval of H max n , for sectors 11, 12, 14 and 15 extend significantly beyond their mean values. So, on an average the maximum

ACKNOWLEDGMENT

The authors would like to thanks Tomás de la Barra, the designer of TRANUS, for his immense help in developing the Grenoble model and expert opinion on calibration. The also authors would like to thank Mathieu Vadon and Hugo-Luis Manterola for their help with the codes of TRANUS optimization, and Alejandro Deymonnaz for help in scientific computing.