Entropy and complexity of a path in sub-riemannian geometry

DOI: 10.1051/cocv:2003024

ENTROPY AND COMPLEXITY OF A PATH IN SUB-RIEMANNIAN GEOMETRY

Fr´ ed´ eric Jean

Abstract. We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subsetAof a metric space is the minimum number of balls of a given radiusεneeded to coverA. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub- Riemannian manifold as the infimum of the lengths of all trajectories contained in anε-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a normk · kεon the tangent space that depends on a parameterε >0. Our main result states then that the entropy of a path is equivalent to the integral of thisε-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen.

Mathematics Subject Classification. 53C17.

Received July 18, 2002. Revised February 27, 2003.

1. Introduction

In a sub-Riemannian geometry, submanifolds may have a Hausdorff dimension greater than their topological dimension. What values can the Hausdorff dimension have? More generally, what are the metric properties of sub-Riemannian submanifolds? In this paper we will answer these questions for one-dimensional submanifolds, or paths, using two geometric invariants, the entropy and the complexity.

The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to coverA. This value indicates how well one can approximateA by a finite set. The use of entropy in sub- Riemannian geometry was suggested by Gromov [7] (p. 277). It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general.

We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in anε-neighborhood of the path, having the same extremities as the path. Trajectories here refer to the control system associated with the sub-Riemannian manifold. The concept of complexity for paths was first developed for the nonholonomic motion planning problem in robotics (see Bella¨ıcheet al.[2]), to

Keywords and phrases. Complexity, Hausdorff dimension, metric entropy, non-linear control, nonholonomic systems, sub- Riemannian geometry.

1 ENSTA, Unit´e de Math´ematiques Appliqu´ees, 32 boulevard Victor, 75739 Paris, France; e-mail:[email protected]

c EDP Sciences, SMAI 2003

model the algorithmic complexity of the problem. This application is described in [10], which uses the results of this paper. Gromov also introduced a concept equivalent to the complexity in [7] (p. 278). As with the entropy, this concept is based on an approximation by finite sets, there calledε-chains.

The Hausdorff dimension of paths, and more generally of submanifolds, is known for the case where the sub- Riemannian manifold is equiregular (Gromov [7]). However neither the entropy nore the Hausdorff dimension have been estimated for the general case.

In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for the general case.

On a sub-Riemannian manifold, we construct a normk · k_ε on the tangent space that depends on a parameter ε >0 (see Sect. 3.1). Our first main result, Theorem 3.10, is an estimate of the entropy in terms of thisε-norm.

Theorem. The entropye q(·), ε

of a pathq(t),t∈[0, T], satisfies e q(·), ε

Z _T

0 kq(t)k˙ εdt,

wheredenotes the equivalence up to multiplicative constants (uniform with respect toε, for εsmall enough).

As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result links complexity to entropy.

Theorem. For a generic pathq(t),t∈[0, T], the complexity σ q(·), ε

satisfies

σ q(·), ε

e q(·), ε

Z _T

0 kq(t)˙ kεdt.

We give also a computable sufficient condition on the path for this equivalence to happen. This condition precises the meaning of generic here.

The organization of this paper is the following. In Section 2, we give some recalls of sub-Riemannian geometry and we define the entropy and the complexity of a path. We show there how the complexity of paths allows to compute the algorithmic complexity of nonholonomic motion planning. Section 3 contains the definition of the ε-norm and our main theorems which are illustrated by several examples. The proofs are postponed in Section 4 for a particular case and in Section 5 for the general case.

2. Paths in sub-Riemannian geometry 2.1. Sub-Riemannian manifolds

We recall here some definitions and basic results of sub-Riemannian geometry. More general presentations can be found in Bella¨ıche [1] (the main reference for this section) or in Kupka [13].

Let M be a real analytic manifold and X1, . . . , X_m analytic vector fields on M. We denote by (Σ) the associated control system

˙ q=

Xm i=1

u_iX_i(q), q∈M, (Σ)

where the control functionu(t) = (u1(t), . . . , u_m(t)) takes values inR^m. The choice of a measurable function u(t),t∈[0, τ], and of an initial pointp∈M defines a trajectory γof (Σ). Thelength ofγis

length(γ) = Z _τ

0 kγ(t)k˙ _γ(t)dt,

where forq∈M andv∈T_qM, we set kvk_q = inf

(q

u²₁+· · ·+u²_m| X^m

i=1

u_iX_i(q) =v )

· (1)

The sub-Riemannian distance is defined by d(p, q) = inf length(γ), where the infimum is taken on all the trajectories of (Σ) joining ptoq. The manifoldM endowed with the distanced, denoted (M, d), is called the sub-Riemannian manifold attached toX1, . . . , X_m.

LetL¹(X1, . . . , X_m) be the set of linear combinations, with real coefficients, of the vector fieldsX1, . . . , X_m. We define recursivelyL^s=L^s(X1, . . . , X_m) by setting L^s=L^s−1+ [L^s−1,L¹], fors= 2,3, . . . Due to Jacobi identityL^sis the set of linear combinations of all commutators ofX1, . . . , X_mwith a length≤s. The unionL of all L^s is a Lie sub-algebra of the Lie algebra of vector fields on M. It is generated by the commutators [[X_i1, X_i2], . . . , X_ik]. Such a commutator is denoted [X_I], where I is the multi-indexI = (i1, . . . , i_k) and its length is|I|=k.

Forp∈M, letL^s(p) be the subspace ofT_pM which consists of the valuesX(p) taken, at the pointp, by the vector fieldsX belonging toL^s. The following condition onX1, . . . , X_m:

[

s≥1

L^s(p) =T_pM at every pointp∈M,

is called the rank condition. When this condition is satisfied, at each point p∈M there is a smallest integer r = r(p) such that L^r(p)(p) = T_pM. This integer is called the degree of nonholonomy at p. For each point p∈M, there is an increasing sequence of dimensions, called thegrowth vector,

1≤dimL¹(p)≤ · · · ≤dimL^s(p)≤ · · · ≤dimL^r(p)(p) =n.

We say that pis a regular point for (M, d) if the growth vector remains constant in some neighborhood ofp.

Otherwise we say thatpis asingular point for (M, d).

Finally, one says that the system (Σ) iscontrollable if the sub-Riemannian distance between any two points ofM is finite.

Chow’s theorem (Chow [5]). If M is connected and if X1, . . . , X_m satisfy the rank condition, then the sys- tem (Σ) is controllable.

Notice that the conditions are also necessary here, sinceX1, . . . , X_mare analytic vector fields (Nagano [17], Sussmann [19]).

In the sequel we always assume that the rank condition is satisfied. The sub-Riemannian distancedis then continuous and the topology defined by dis the original topology ofM.

2.2. Entropy and Hausdorff dimension

Let (X, d) be a metric space andA⊂X a bounded subset.

Definition 2.1. Themetric entropy of A is the functionε7→e(A, ε), defined forε >0, where e(A, ε) is the minimal number of closed balls of radiusεinX needed to coverA.

Notice that originally, as conceived by Kolmogorov [12], theε-entropy is defined as the logarithm log₂e(A, ε).

It represents the amount of information we need to describe a point in A with the accuracyε or to digitally memorizeA with this accuracy.

Definition 2.2. The entropy dimension of A, denoted dim_eA, is the greatest lower bound of β for which e(A, ε)≤(1/ε)^β forεsmall enough, that is

dim_eA= lim

ε→0

loge(A, ε) log(¹_ε) ·

The entropy dimension characterizes the asymptotic behavior ofe(A, ε) asεtends to 0 and reflects the geometry ofAin X.

A maybe more usual characterization of the geometry of a space uses the Hausdorff dimension (and measure), introduced by Hurewicz and Wallman [8].

Forβ≥0, theβ-dimensional Hausdorff measure, denotedm_β(A), is defined as lim_ε→0m_β(A, ε), where

m_β(A, ε) = inf ( _∞

X

i=1

r_i^β, r_i≤εradius of ballsB_i, andA⊂ [∞ i=1

B_i )

·

For a givenA,m_β(A) is a decreasing function ofβ, infinite whenβ is less than a certain value, and zero when β is greater than this value.

Definition 2.3. TheHausdorff dimension ofAis the real number:

dimHA= sup{β |m_β(A) =∞}= inf{β |m_β(A) = 0}·

Entropy and Hausdorff dimension are related by the following properties (Comte and Yomdin [6]).

Proposition 2.4. (i) For all β≥0,m_β(A)≤lim_ε→0ε^βe(A, ε), which implies dimHA≤dim_eA.

(ii) If e(A, ε)∼ε^−β0, thendimHA= dim_eA=β0.

In the case where the metric space is a sub-Riemannian manifold (M, d), some results already exist for the Hausdorff dimension and the entropy of a submanifold ofM.

When (M, d) is equiregular (every point inM is regular), the Hausdorff dimension of the manifold is given by dimHM =X

s≥1

s dimL^s(p)−dimL^s−1(p)

for any point p∈ M (see Mitchell [16] and Vershik and Gershkovich [20]). Gromov [7] (p. 104) extends this result to Hausdorff dimension of a submanifoldN, again in an equiregular manifold:

dimHN = max

q∈N



 X

s≥1

s dimL^s_N(q)−dimL^s−_N ¹(q)



, (2)

whereL^s_N(q) is the linear subspaceL^s(q)∩T_qN.

On the other hand, Bella¨ıche [1] computes the entropy and the Hausdorff dimension of a non equiregular sub-Riemannian manifold, the Gruˇsin plane. This last example is interesting because it does not fit in the case (ii) of Proposition 2.4. It follows that the asymptotic behavior of entropy cannot be deduced only from the entropy (or Hausdorff) dimension.

2.3. Complexity

Let (M, d) be the sub-Riemannian manifold attached to a system (Σ). We call path a one-dimensional analytic submanifold ofM which is diffeomorphic to a closed interval inR. Aparameterization of a pathC is an analytic diffeomorphismq: [0, T]→ C,t7→q(t).

Submanifold means here embedded submanifold. This implies in particular the following property:

Proposition 2.5. Let C be a path and q(t)a parameterization of C. There exist constants ρand ν >0 such that, if d(q(t), q(t⁰))< ρ, then

d(q(t), q(t⁰))≥ν|t−t⁰|.

Proof. The assertion is clear when we replacedby a Riemannian distance onM. And there exists a Riemannian distance onM which is everywhere smaller than the sub-Riemannian distance (Gromov [7], p. 98).

Given a pathCin M andε >0, we introduce the tubular neighborhood ofC Tube(C, ε) = [

q∈C

B(q, ε),

where B(q, ε) is the sub-Riemannian ball centered at qof radius ε. We denote also byaand bthe extremities ofC.

Definition 2.6. We callcomplexity ofC the functionε7→σ(C, ε), defined forε >0, with σ(C, ε) = inf





length(γ) ε

γ trajectory of (Σ) γ joinsatob γ⊂Tube(C, ε)



· (3) We are interested here in the asymptotic behavior of the complexity. In particular we want to compare this behavior to the one of the entropy. With this aim in view, we introduce some notations. We writef(ε)g(ε) if there existsκandν >0 such that, forεsmall enough,f(ε)≤κg(νε). We say thatf(ε) andg(ε) areequivalent, and we writef(ε)g(ε), whenf(ε)g(ε) andg(ε)f(ε).

Remark 2.7.

• In [9], following ideas of Gromov [7] (p. 278), we propose a different definition of complexity: it is the minimum number of points in anε-chain inC, where anε-chain inC is a sequence of pointsv0=a,v1, . . . ,v_k =b in C with d(v_i, v_i+1)≤εfor i= 0, . . . , k−1. Both definitions are however equivalent: the minimum number of points in anε-chain is greater thanσ(C, ε) but smaller thanσ(C,3ε).

• As suggested by Gromov [7] (p. 278), the entropy and the complexity allow to defineε-lengths of a path, generalizing the notion of length for the trajectories:

length^e_ε(C) =ε×e(C, ε) and length^σ_ε(C) =ε×σ(C, ε).

A natural question is to ask if theseε-lengths are equivalent. The answer will follow directly from the comparison between entropy and complexity.

Proposition 2.8. Let C be a path in the sub-Riemannian manifold (M, d). Then σ(C, ε)e(C, ε).

Proof. LetC ⊂M be a path with extremitiesaandb, andB1, . . . , B_N be balls of radiusεcoveringC.

Any two points in the same ball can be linked by a trajectory of length≤2εincluded in the ball. Consider now two balls B_i, B_j with a non empty intersection. Any couple of points in B_i ∪B_j can be linked by a trajectory of length≤4εincluded inB_i∪B_j.

By induction we show then that a and b can be linked by a trajectory of length ≤2N εincluded in S B_i.

The result follows from the definitions ofσandM.

Unfortunately the reverse inequalitye(C, ε)σ(C, ε) does not always hold. The obstruction to this inequality is the presence in the path of points with particular metric properties.

Definition 2.9. A point q in a path C is called a metric cusp if, for every positive constants k and ρ, there exist two points q1 andq2in C different fromqand such that:

• qlies betweenq1 andq2;

• d(q1, q2)≤ρ;

• d(q1, q)≥kd(q1, q2).

The sentence “q∈ Cbetweenq1andq2” is well defined sinceC is diffeomorphic to an interval ofR. Notice also that the extremities of the path are never metric cusps.

In Euclidean geometry (a special case of sub-Riemannian geometry), metric cusps coincide with the usual algebraic cusps (takey=p

|x|inR² for instance). Of course a curve containing an algebraic cusp can not be a path (it is not a submanifold). In sub-Riemannian geometry however, paths may have metric cusps, as we will see in Section 3.6.

2.4. Application to nonholonomic motion planning

The complexity of paths is related to the algorithmic complexity of a Control Theory problem: nonholonomic motion planning in the presence of obstacles.

We consider a nonholonomic control system, that is a system (Σ). We assume it is controllable. Obstacles are closed subsetsF of the configuration spaceM. The open set M −F is called thefree space. The motion planning problem is: givenaandb inM−F, find a trajectory of (Σ) contained in the free space, and joininga tob.

This problem has a solution if and only ifaandb are in the same connected component ofM−F (Chow’s theorem). SinceM −F is an open set, connectivity is equivalent to arc connectivity. Thenaand b belong to the same connected component if there is an arc inM −F linkingatob.

This argument suggests a general method to solve the complete problem. This method, called “Approximation of a collision-free holonomic path” (see Laumondet al.[15]), has two steps:

• find a curveC in the free space linkingatob(C is the “collision-free holonomic path”);

• approximateC by a trajectory of (Σ), close enough to be contained in the free space.

What is the algorithmic complexity of this method? The complexity of the first step (motion planning problem for holonomic systems) is well modeled and understood. It depends on the geometric complexity of the obstacles and the robot in the real world (see Canny [4] or Schwartz and Sharir [18]).

We are interested here in the second step which is due to the nonholonomy. Let us precise first what is a trajectory approximating a given curve. LetCbe a curve and Tube(C, ε) the tube of radiusεcentered atC. We denote by ρthe biggest radius εfor which Tube(C, ε) is contained in the free space. We say that a trajectory of (Σ) approximatesC if it is contained in Tube(C, ρ) and if it has the same extremities asC.

Now, recall that the complexity of an algorithm is the minimal number of elementary steps needed to get the result. Since the parameter ρrepresents the size of the free space aroundC, we consider that the elementary step of our method is “to build a trajectory of lengthρ”. The number of elementary steps in a trajectory is

σ(γ) = length(γ)

ρ ·

We define the algorithmic complexity σ_m(C) of the second step as the greatest lower bound of σ(γ) for all trajectoriesγapproximatingC(other kind of complexities can be defined, see Bella¨ıcheet al.[3] and Jean [10]).

WhenC is a path, this algorithmic complexity can be deduced from the values of the complexity of the pathC. Indeed,σ_m(C) is equal toσ(C, ε) taken atε=ρ.

3. Entropy and complexity of a path

Let (M, d) be the sub-Riemannian manifold attached to a system X1, . . . , X_m. We assume that M is an orientable manifold and we denote by det the determinantn-form.

3.1. Shape of sub-Riemannian balls

We fix a pathC ⊂M and we denote byrthe maximum of the degree of nonholonomy onC.

We consider the families of vector fields ([X_I1], . . . ,[X_In]) such that each bracket [X_Ij] is of length|Ij| ≤r and we denote byIr the (finite) set of these families.

Definition 3.1. Let p∈ C and ε >0. We say that the family I = ([X_I1], . . . ,[X_In])∈ Ir is associated with (p, ε)on Cif it achieves the maximum onIr of the function

det

[X_I1]ε^|I1|, . . . ,[X_In]ε^|In|

(p).

In particular the value atpof a family associated with (p, ε) forms a basis ofT_pM.

Example 3.2 (A nilpotent system inR³). Fix an integer r≥2 and consider the system defined inR³ by the vector fields

X1=∂_x, X2=∂_y+ x^r−1 (r−1)!∂_z.

The Lie algebraL(X1, X2) is nilpotent of orderr(forr= 2, it is the Lie algebra associated with the Heisenberg group and forr= 3,X1, X2 generate the Martinet distribution). The only non zero commutators are

[X_Ii] = (−1)ⁱ⁻² x^r−i

(r−i)!∂_z, i= 2, . . . , r, whereI_i=



1,2,1, . . . ,1

| {z }

(i−2)



.

The set of singular points is the plane {x= 0}. The degree of nonholonomy is 2 at regular points and r at singular ones. The families of vector fields with non identically zero determinant areI_i= (X1, X2,[X_Ii]) and

det εX1, εX2, εⁱ[X_Ii]

(x, y, z)= |x|^r−i (r−i)!ε²⁺ⁱ.

Choose any pathC inR³ crossing the singular locus. The families associated with (x, y, z), ε

onC are:

I₂ if |x| ≥(r−2)ε, I_i if |x| ∈

(r−i)ε,(r−i+ 1)ε

, i= 3, . . . , r.

Notice that, when |x|= (r−i)ε, both familiesI_i andI_i+1 are associated onC.

The notion of associated basis allows to describe the shape of the sub-Riemannian balls.

Theorem 3.3. There exist a constantδ0>0and functionsκ(δ), K(δ),0< κ(δ)< K(δ), withlim_δ→0K(δ) = 0, such that:

for every p∈ C,ε <1,δ < δ0 and every family I associated with (p, ε)onC, B_I p, κ(δ)ε

⊂B(p, δε)⊂B_I p, K(δ)ε

, (4)

whereB_I(p, ε) =

pexp(u_n[X_In])· · ·exp(u1[X_I1]), |u_i|< ε^|Ii|, 1≤i≤n . This theorem is proved in [11].

Remark 3.4.

• We note on the right the action of diffeomorphisms:pexp(tX) results of the action of exp(tX) on point p. This notation is consistent with the notation for Lie group: diffeomorphisms which come from flows of left-invariant vector fields are defined by right multiplication.

• Theorem 3.3 gives a uniform estimate of sub-Riemannian balls which extends the classical Ball–Box theorem (Bella¨ıche [1], Gromov [7]). Indeed,pbeing fixed (and δ=δ0), forεsmaller than some ε(p), the estimate (4) is equivalent to the one of Ball–Box theorem. Howeverε(p) can be infinitely small for pclose to a singular point, though (4) holds forε <1.

Letp∈M and v∈T_pM. For a family I which values atpform a basis of the tangent spaceT_pM, we denote byv₁^I, . . . , v_n^I the coordinates ofv in this basis, that is

v= Xn

i=1

v^I_i [X_Ii](p).

Definition 3.5. Theε-norm ofv∈T_pM is defined as

kvk_p,ε= maxnv_i^Iε^−|Ii|, 1≤i≤n, I associated with (p, ε) onCo

·

Whenvis the tangent ˙q(t) to the path, the dependence with respect toq(t) is implicit and we write theε-norm as kq(t)k˙ ε. It becomes then a function oft, which is piecewise continuous and so integrable on [0, T].

Example 3.6. Let us carry on the previous example. We choose for C the path{x=z^p, y = 0, 0≤z≤1}, p≥1 being an integer. The tangent is ˙q(z) =pz^p−1∂_x+∂_z, that is, fori= 2, . . . , r,

˙

q(z) =pz^p−1X1+(r−i)!

z^p(r−i)[X_Ii].

When the family associated with q(z), ε

is I_i, i.e. whenz∈

(r−i)ε1/p

, (r−i+ 1)ε1/p

fori= 3, . . . , r andz≥ (r−2)ε1/p

fori= 2, theε-norm is

kq(z)k˙ _ε= max pz^p−1

ε , (r−i)!

z^p(r−i)εⁱ

!

= (r−i)!

z^p(r−i)εⁱ forεsmall enough.

In the same way, if we consider now the path{y= 0, z=x^p,0≤x≤1}, theε-norm is kq(x)k˙ _ε= p(r−i)!

x^r−i−p+1εⁱ if x∈

(r−i)ε,(r−i+ 1)ε , i≥3, kq(x)k˙ ε= p(r−2)!

x^r−1−pε² if x≥(r−2)ε.

3.2. (H)-generic and regular points for a path

Definition 3.7. A pointq∈ Cis said to beregular for Cif the growth vector is constant onCnearq. Otherwise qis said to be singular for C.

Regular points forC form an open dense set inCand singular points are isolated inC. Notice that a regular point forC can be singular forM.

Let us consider the following property for a pointq in the interior ofC:

(H) if T_qC belongs to L^s(q), then there exist commutators [X_I1], . . . ,[X_Il] in L^s whose values taken at q form a basis ofL^s(q) and such that, for all q⁰∈ C nearq,T_q⁰Cbelongs to span{[XI1](q⁰), . . . ,[X_Il](q⁰)}.

Forqa regular point forC, this property simply reads as: ifT_qC belongs toL^s(q), thenT_q0Cbelongs toL^s(q⁰) for allq⁰∈ C nearq.

Definition 3.8. We say that a point is(H)-generic either if it satisfies (H) or if it is an extremity of the path.

The (H)-generic points form an open dense set inC. A consequence of this property is the following lemma, proved in Section 5.

Lemma 3.9. The metric cusps are not (H)-generic and so are isolated points inC.

Finally we have two finite sets of points inC: singular and non (H)-generic points. We say thatCisgeneric at singular pointswhen these two sets have an empty intersection, that is if each singular point forCis (H)-generic.

It is a generic property on the paths space.

3.3. Estimate of entropy and complexity

Recall from Section 2.3 that we use the notations , to denote the corresponding inequalities up to multiplicative constants (uniform with respect toε, forεsmall enough). Our first main result gives an equivalent for the entropy.

Theorem 3.10. For a pathC with a parameterization q(t),t∈[0, T], e(C, ε)

Z _T

0 kq(t)k˙ εdt.

The proof is given in Section 5.1.

Remark 3.11. Whenεtends to zero theε-normkvkp,ε is equivalent tokvkp/ε(kvkp is defined in (1)). IfC is a trajectory of the system, the integral of theε-norm – and so the entropy – is then equivalent to length(C)/ε.

Thus theε-length length^e_ε(C) defined in Section 2.3 seems to be a good generalization of the notion of length.

This estimate allows to compute the entropy dimension. It gives also upper and lower bounds for the Hausdorff dimension.

Corollary 3.12. The Hausdorff dimension of a pathC belongs to the interval [βreg, r], where

• βreg is the smallest integerβ such thatT_qC ∈L^β(q)for all pointq regular forC;

• ris the maximum of the degree of nonholonomy onC.

Moreover, for a path containing no singular points, the Hausdorff dimension is the smallest integerβ such that T_qC ∈L^β(q)for allq inC.

The proof is given in Section 5.1.

Notice that the Hausdorff dimension can take not only the integer values belonging to the interval [βreg, r], but also rational ones.

Remark 3.13. In an equiregular manifold, the Hausdorff dimension of a path results from formula (2) of Gromov: dimHCequals the smallest integerβ such thatT_qC ∈L^β(q) for allqinC. Corollary 3.12 appears then as a generalization of Gromov’s formula. However, when the path contains singular points, we may have both T_qC ∈L^β(q) for allqinCand dimHC> β(see example (d) in Sect. 3.5 below). Thus, in general, the Hausdorff dimension ofC is not characterized only by the dimension of the spacesL^s_C(q) =L^s(q)∩T_qC.

Our second main result links the complexity to the integral of theε-norm.

Theorem 3.14. Let C be a path, with a parameterization q(t), t∈[0, T], andt1<· · ·< t_s be the parameters of the points which are both metric cusps and singular for C (0< t1 andt_s< T). The complexity ofC satisfies

Z _T

0 kq(t)˙ kεdt− Xs

i=1

Z _ti+ε

ti−ε kq(t)˙ kεdtσ(C, ε) Z _T

0 kq(t)˙ kεdt.

The proof is given in Section 5.2.

This result does not always allow to estimate the complexity and to compare it with the entropy. Anyway complexity and entropy may be non equivalent, as shown by the example in Section 3.6. Moreover it is in general difficult to decide if a point is a metric cusp or not.

However Theorem 3.14 provides sufficient conditions for complexity and entropy to be equivalent. These conditions, expressed in the corollary below, are interesting for applications since they are generically satisfied and computable.

Corollary 3.15. Let C be a path, with a parameterization q(t), t ∈[0, T]. If one of the following conditions holds:

• C is generic at singular points;

• denotingt⁰₁<· · ·< t⁰_N the parameters of the points which are both non (H)-generic and singular for C, t⁰₀= 0 andt⁰_N+1=T, we have, for i= 1, . . . , N+ 1,

Z _t⁰_i

t⁰_i−1kq(t)˙ kεdt Z _t⁰_i−ε

t⁰_i−1+εkq(t)˙ kεdt;

thenσ(C, ε)e(C, ε).

3.4. Example 1: The car control system

A classical example of nonholonomic control system is the car (see Laumond [14]). It is represented as two driving wheels connected by an axle. A state of the system is parameterized by the coordinates (x, y) of the center of the axle and by the orientation angleθof the car. In the manifoldM =R²× S¹, the control system is

˙

q=u1X1+u2X2, with X1= cosθ∂_x+ sinθ∂_y, X2=∂_θ.

The associated sub-Riemannian manifold has no singular point: the values of the familyI={X1, X2,[X1, X2]} form a basis everywhere. It is the only possible associated family.

Let C be a path inM and q(t) a parameterization ofC. We define ϕ(t) as the angle between ˙q(t) and the plane generated by X1(q(t)) andX2(q(t)). The coordinate of ˙q(t) on [X1, X2](q(t)) is sinϕ(t). According to Theorem 3.14 we have

σ(C, ε) 1 ε²

Z _T

0 max(ε,|sinϕ(t)|)dt.

If the path is a trajectory, thenϕ≡0, and the complexity equals length(C)/ε. On the other hand if the path is always perpendicular to the direction of the car, the complexity is equivalent to 1/ε². So we show that to reverse into a parking place needs more maneuvers than going in a straight line!

This application of the complexity is developed in a more complete way in [10], including the case of the car pulling trailers.

3.5. Example 2: A nilpotent system in R

As in the examples of Section 3.1, we consider the system of vector fields inR³ X1=∂_x, X2=∂_y+ x^r−1

(r−1)!∂_z.

We are interested in paths with singular points. We consider for instance the paths{x=z^p, y= 0,0≤x, z≤1} where eitherpor 1/pis an integer and

− ifp≥1, thenpis the multiplicity of the intersection between C and the singular locus{x= 0};p=∞ means thatCis included in the singular locus;

− if 1/p≥1, then 1/pis the multiplicity of the intersection betweenCand the plane{z= 0}(that is the plane{X1(0), X2(0)}); 1/p=∞(orp= 0) means that Cis a trajectory of the system.

For these paths, the only singular point is an extremity, the origin. They are then generic at singular points and so the complexity is always equivalent to the entropy. Theε-norm has already been computed in the examples of Section 3.1. Integrating it along the path yields the following entropy estimates.

(a) Ifp= 0 (C is a trajectory of the system):

e(C, ε) 1

ε and dimHC= 1.

It is the only case where the Hausdorff dimension equals the topological one.

(b) If 0< p <1/(r−2) (Cis tangent to the distribution at 0):

e(C, ε) 1

ε² and dimHC= 2.

The path is tangent to the distribution {X1, X2} at the singular point 0, but not at regular points.

The leading term ine(C, ε) is the entropy of a path contained in the regular locus (but which is not a trajectory). Notice that when there is no singular points, that is whenr = 2, only this case or the previous one can occur.

(c) Ifp= 1/(r−2) (C is transverse to the singular locus at 0 and, ifr≥4, tangent to the distribution at 0):

e(C, ε) 1 ε²log(1

ε) and dimHC= 2.

In this case, the entropy dimension is dim_eC= 2 althoughe(C, ε)ε²tends to infinity. For the Hausdorff dimension we have dimHC ≤dim_eC= 2 and the equality dimHC= 2 results from Corollary 3.12.

(d) If 1/(r−2)< p <∞(C may be either transverse or tangent to the singular locus):

e(C, ε) 1

ε^r−1/p and dimHC=r−1/p.

The Hausdorff dimension is greater than 2, the Hausdorff dimension of a path included in the regular set, but less thanr, the one of a path included in the singular locus (see next case). It may not be an integer whenpis greater than one (in this caser−1≤dimHC< r).

Moreover we may have bothT_qC ∈L^β(q) for allqin Cand dimHC> β. Take for instancep= 1/2 and r≥5: the path is tangent everywhere toL²(q) and dimHC=r−2≥3.

(e) Ifp=∞(the path is included in the singular locus):

e(C, ε) 1

ε^r and dimHC=r.

In this case every point in the path is regular forC.

3.6. Complexity and entropy may be non equivalent

X1=∂_x, X2=∂_y+ (x⁹−xz²)∂_z.

The singular locus is the set {9x⁸ =z²}. The degree of nonholonomy equals 10 on the subset {x=z= 0}, 3 on the remainder of the singular locus, and 2 elsewhere.

We choose a path tangent to {x = z = 0} at 0 but which does not go through the singular locus, say C={(0, y, y²), y∈[−1,1]}. The origin is a singular point forC and is not (H)-generic. ThusC is not generic at singular points.

A short calculation gives theε-norm

kq(y)k˙ ε=



 max

ε⁻¹, 2

9!yε⁻¹⁰

if |y| ≤(9!)^1/4ε², 2y⁻³ε⁻² if (9!)^1/4ε²≤ |y| ≤1.

On the other hand we have d(q(ε), q(−ε)) = 2ε. It is then easy to show that σ(C, ε) σ(q([−1,−ε]), ε) + σ(q([ε,1]), ε). By applying Theorems 3.10 and 3.14, we obtain

σ(C, ε) Z 1

ε kq(y)k˙ εdy and e(C, ε) Z 1

0 kq(y)k˙ εdy.

Using the expression of theε-norm, this yields to

e(C, ε)ε⁻⁶ and σ(C, ε)ε⁻⁴.

Thus entropy and complexity are not equivalent. It results also from Theorem 3.14 that 0 is a metric cusp.

4. Paths with only (H)-generic points

In this section we study paths containing only (H)-generic points (see Sect. 3.2 for the definition). In this case Theorems 3.10 and 3.14 take the following form, proved in Section 4.2 below.

Lemma 4.1. Let C be a path containing only (H)-generic points andq(t)a parameterization of C. Then σ(C, ε)e(C, ε)

Z _T

0 kq(t)k˙ _εdt.

4.1. The ε-norm along a path

Lemma 4.2. Let C be a path containing only (H)-generic points andq(t)a parameterization of C. Then (a) the following metric property (MP) is verified:

there existkandρ0>0 such that, forq(t0)andq(t1)∈ C,

d(q(t0), q(t1))< ρ0 ⇒ d(q(t0), q(t))≤kd(q(t0), q(t1)) ∀t∈[t0, t1];

(b) there exist constantsν,k1andk2>0such that, ift0∈[0, T]andε≤ρ0, thend(q(t0), q(t1)) =εimplies k1≤

Z _t1

t0

kq(t)k˙ _νεdt≤k2. (5)

Remark 4.3. The property(MP)implies obviously that no points ofCis a metric cusp.

Lemma 4.2 results from Propositions 4.6, 4.7 and 4.8 below.

4.1.1. Notations

For a familyI ofnbrackets ([X_I1], . . . ,[X_In]), we denote byD(I) =|I1|+· · ·+|In|the total length and by det_I(q) = det([X_I1], . . . ,[X_In])(q) the determinant.

Givenp∈M, the applicationφ_I from an open neighborhood of 0∈Rⁿ toM is defined by φ_I(u) =pexp(u_n[X_In])· · ·exp(u1[X_I1]).

Forε >0, Box(ε) denotes the set{|u_i|< ε^|Ii|, 1≤i≤n}in Rⁿ and its image isB_I(p, ε) =φ_I Box(ε) . Consider a pathC with a parameterizationq(t),t∈[0, T]. Given a familyI of vector fields andi= 1, . . . , n, we set

det^I_i(t) = det

[X_I1] q(t)

, . . . ,q(t)˙

|{z}

i−th

, . . . ,[X_In] q(t) ,

˙

q^I_i(t) = det^I_i(t)

det_I q(t) if det_I q(t) 6= 0.

This is consistent with notations of Section 3.1: when defined, the ˙q^I_i(t)’s are the coordinates of ˙q(t) in the basis [X_I1](q(t)), . . . ,[X_In](q(t))

, that is

˙ q(t) =

Xn i=1

˙

q^I_i(t) [X_Ii] q(t)

. (6)

LetF be the set of analytic functions det_I q(t)

and det^I_i(t), fori∈ {1, . . . , n} andI such that each |I_k| ≤r, where ris the maximum of the degree of nonholonomy onC. Zeroes of functions in this set define on [0, T] a finite number of real numbers 0 =T0 < T1<· · ·< T_N =T such that, on ]T_j, T_j+1[, a function in F is either identically zero or everywhere non zero. In particular a function non identically zero on ]T_j, T_j+1[ has a constant sign on this interval.

There is also a constant ∆t0, 0<∆t0<¹₂max_j(T_j+1−T_j), such that:

for allj∈ {0, . . . , N},t∈[T_j−∆t0, T_j+ ∆t0], andf andg∈ F,

if f(T_j)6= 0 andg(T_j) = 0, then|f(t)|>|g(t)|. (7) Set

dmax = maxndet^I_i(t) s.t. t∈[0, T], i= 1, . . . , n, each|I_k| ≤r o

dmin = minndet^I_i(T_j)6= 0 s.t. j = 1, . . . , N, i= 1, . . . , n, each|I_k| ≤r o·

Both dmax anddmin are positive and finite. It follows from (7) that, if det^I_i(T_j)6= 0 for some i, j and I, then

|det^I_i(t)|> dmin/2 for everyt∈[T_j−∆t0, T_j+ ∆t0].

Now, setη0=dmin/(2dmax)>0. For anyi,l= 1, . . . , n, anyj∈ {0, . . . , N}and any familiesI,J of brackets of length≤r, if det^I_i(T_j)6= 0, then for everyt∈[T_j−∆t0, T_j+ ∆t0]

|det^I_i(t)|

|det^J_l(t)| > η0. (8)

4.1.2. Preliminary results

Fix a path C containing only (H)-generic points and a parameterization q(t) of C. Let δ0 and K(·) be respectively the constant and the function given by Theorem 3.3. We will use the following lemma, which results from [11] (Lem. 7).

Lemma 4.4. There exist a constantC >0 and a functionδ: ]0,1[→]0, δ0]such that, ifp∈ C,ε <1and ifI is a family associated with (p, ε)on C, then the following properties are satisfied.

(i) For every q∈B_I p, K(δ0)ε ,

1

2det_I(p)≤det_I(q)≤2det_I(p) det_I(q)ε^D(I)≥ 1

Cmaxdet_K(q)ε^D(K), K s.t.each |Ki| ≤r · (ii) φ_I is a local diffeomorphism in a neighborhood of every point ofBox K(δ0)ε

.

(iii) Denoting by(ψ1, . . . , ψ_n)the local inverse application ofφ_I, we have, for everyq∈B_I p, K(δ0)ε ,

1

2 ≤[X_Ii]·ψ_i(q)≤2.

Moreover, givenτ >0, if q∈B_I p, K δ(τ) ε

, then,

[X_Ij]·ψ_i(q)≤τ ε^|Ii|−|Ij| forj6=i.

Let [t0, t1]⊂[0, T]. We assume that q([t0, t1]) is included in B_I q(t0), K(δ0)η

for someη < 1 and a familyI associated with (q(t0), η) onC.

The application φ_I is a local diffeomorphism on Box K(δ0)η

. So there is a unique absolutely continuous applicationθ: [t0, t1]→Rⁿ such thatθ(t)∈Box K(δ0)η

andφ_I(θ(t)) =q(t), for allt∈[t0, t1].

Now,q(t) andθ are one-to-one on [t0, t1] and φ_I is a local diffeomorphism on Box K(δ0)η

. Thereforeφ_I is a global diffeomorphism on a neighborhood ofθ([t0, t1]) and its inverse applicationφ⁻_I¹ = (ψ1, . . . , ψ_n) is well defined on this neighborhood.

To shorten we denote by ψ1(t), . . . , ψ_n(t)

the applicationφ⁻_I¹ q(t)

=θ(t) fort∈[t0, t1].

Proposition 4.5. Considert0∈[0, T[,t1∈[t0, t0+ ∆t0],η < η0 and a familyI associated with(q(t0), η)onC. Let pbe an integer such that

Z _t1

t0

q˙_p^I(s)η^−|Ip|ds= max

1≤i≤n

Z _t1

t0

q˙^I_i(s)η^−|Ii|ds.

If q([t0, t1])is included in B_I q(t0), K(δ0)η , then

|ψ_p(t1)| ≥ 1 4

Z _t1

t0

q˙_p^I(s)ds, (9)

|ψ_i(t)| ≤3η^|Ii| Z _t1

t0

q˙_p^I(s)η^−|Ip|ds, ∀t∈[t0, t1], i= 1, . . . , n. (10) As a consequence we have

|ψ_i(t)| ≤12η^|Ii|−|Ip||ψ_p(t1)|.

Proof. In the proof of this proposition – and below in those of Propositions 4.6 and 4.7 – the familyI is fixed.

We then write ˙q_i and det_i instead of ˙q^I_i and det^I_i.

Lett∈[t0, t1] andi∈ {1, . . . , n}. The applicationφ⁻_I¹is defined at every pointq(s) withs∈[t0, t]. We can write

ψ_i(t) = Z _t

t0

dψ_i dt (s)ds.

Due to (6) the derivative ofψ_i(s) is

dψ_i dt (s) =

Xn l=1

˙

q_l(s)[X_Il]·ψ_i(s).

We integrate fromt0 tot and obtain

ψ_i(t) = Xn l=1

Z _t

t0

˙

q_l(s)[X_Il]·ψ_i(s)ds.

According to Lemma 4.4(iii), reducing eventuallyδ0(in such a way that δ0≤δ(1/4n)), we have,

Z _t

t0

˙

q_i(s)[X_Ii]·ψ_i(s)ds≤2 Z _t

t0

q˙_i(s)ds,

Z _t

t0

˙

q_l(s)[X_Il]·ψ_i(s)ds≤ 1

4nη^|Ii|−|Il| Z _t

t0

q˙_l(s)ds, forl6=i. (11) This implies that

ψ_i(t)≤2 Z _t

t0

q˙_i(s)ds+X

l6=i

1

4nη^|Ii|−|Il| Z _t

t0

q˙_l(s)ds

≤2η^|Ii| Z _t1

t0

q˙_p(s)η^−|Ip|ds+ 1 4n

X

l6=i

η^|Ii| Z _t1

t0

q˙_p(s)η^−|Ip|ds,

by definition ofp. It proves (10).

On the other hand we have

|ψ_p(t1)| ≥ Z _t1

t0

˙

q_p(s)[X_Ip]·ψ_p(s)ds −X

l6=p

Z _t1

t0

˙

q_l(s)[X_Il]·ψ_p(s)ds . Claim. The functionq˙_p(s)has a constant sign on[t0, t1].

The function [X_Ip]·ψ_p(s) has also a constant sign since|[XIp]·ψ_p(s)| ≥1/2 (Lem. 4.4(iii)). This observation and (11) applied toi=pyield to

ψ_p(t1)≥1 2

Z _t1

t0

q˙_p(s)ds−X

l6=p

1 4nη^|Ip|

Z _t1

t0

q˙_p(s)η^−|Ip|ds,

which implies (9).

It remains to prove the claim. Observe first that the parametert0 belongs to some interval [T_i, T_i+1[, say [T0, T1[. As ∆t0 < T2−T1, we have [t0, t1]⊂[T0, T2]. We distinguish two cases according to the position of t1

with respect to T1.

− Ift1≤T1: each ˙q_i – and then ˙q_p – has a constant sign on [t0, t1].

− Ift1> T1, thenq(T1) belongs toB_I q(t0), K(δ0)η

. It follows from Lemma 4.4(i) that det_I q(T1)

6

= 0, and so that the sign of det_I(q(t)) is constant on [t0, t1]. Since ˙q_p(t) = det_p(t)/det_I(q(t)), its sign is constant if the one of det_p(t) does.