Entropy and Complexity of a Path in. Sub-Riemannian Geometry. Unit de Math matiques Appliqu es, ENSTA. 32, bd Victor, Paris Cedex 15

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1 Entropy and Complexity of a Path in Sub-Riemannian Geometry Fr d ric JEAN Unit de Math matiques Appliqu es, ENSTA 32, bd Victor, Paris Cedex 5 fjean@ensta.fr Abstract We characterize the geometry of a one-dimensional submanifold in a sub-riemannian manifold by two metric invariants, the entropy and the complexity. The entropy is related to the Hausdor dimension though the complexity allows to compute the complexity of nonholonomic motion planning. We give estimates for these invariants in function of coordinates of the tangent to the submanifold in a basis of the control Lie algebra and of the growth vector. The proof is based on an estimation of sub-riemannian balls depending uniformly of their radius. Contents Introduction 2 2 Denitions 3 2. Sub-Riemannian manifolds Entropy and Hausdor dimension Complexity of a path Links between complexity and entropy Application to nonholonomic motion planning Principal resul 3. Basis of L(p) and "-norm Regular and singular points for C Statement of the results Example : the car control system Example 2: nilpotent systems in IR Complexity and entropy can be non equivalent Proof of Theorems 4 and Shape of sub-riemannian balls Estimate of k _q(t)k " along a path Entropy and complexity of a path A Lifting of the manifold (Lemma 8) 28

2 2 INTRODUCTION B Canonical coordinates of the second kind 30 B. Canonical coordinates are privileged B.2 Jacobian of B.3 Taylor expansion of f((u)) C Choice of a basis (Lemma 0) 35 C. Notations C.2 Determinant estimates C.3 Jacobian and inverse application C.4 Comparison of sets Introduction In sub-riemannian geometry, submanifolds may have a Hausdor dimension greater than their topological dimension. Which values can take this Hausdor dimension? More generally, what are the metric properties of these submanifolds? We restrict ourselves to some one-dimensional submanifolds, the paths, and we study two geometric invariants, the entropy and the complexity. The entropy has been introduced by Kolmogorov [4] in 956. It shows how well one can approximate a given subset of a metric space by a nite set. It has been mostly used to describe non-linear approximation properties in function spaces and has been computed for many such sets (see Kolmogorov and Tihomirov [5], Vitushkin [30], Lorentz [20, 2]). The use of entropy in sub-riemannian geometry was suggested by Gromov [0, p. 277]. He prefers it to the Hausdor measure because it is easier to evaluate. However we will see that entropy generally allows to compute the Hausdor dimension. The second metric invariant of a path, the complexity, arises from questions of robotics like: how many maneuvers are needed to park a car? or, in a more general way, what is the complexity of a nonholonomic motion planning algorithm? Dierent approaches are used to plan nonholonomic motions (see Laumond, Sekhavat and Lamiraux [9] for guidelines). However, most of them are based on the same scheme; namely rst nd an holonomic path and then approximate it by a nonholonomic motion (the meaning of approximation diers according to algorithms). What interested us is the complexity of the second step, the one due to nonholonomy. Laumond and others [3, 8] have given a rst topological denition of this complexity, but no estimates. We have also suggested a metric denition in a previous work (Bella che, Jean and Risler [2]). We resume here this denition. We consider the problem as being part of the sub-riemannian geometry attached to the nonholonomic system. The complexity due to nonholonomy appears then as a function of some path. We call it the complexity of the path. For an equiregular sub-riemannian manifold, the Hausdor dimension of submanifolds is known (Gromov [0]). However there is neither estimates of the entropy nor results in the general case. We give in this paper an equivalent (to within a multiplicative constant) of the entropy of a one-dimensional submanifold in any sub-riemannian manifold (Theorem 4). This equivalent depends on the coordinates of the tangent to the submanifold in basis of the control Lie algebra. We show also that generically the complexity of a path is equivalent to its entropy (Theorem 6).

3 3 To obtain the entropy estimate, we need a description of the shape of sub-riemannian balls depending uniformly of their radius. This result (Theorem 7) is a generalization of the Ball-Box Theorem. It can be read on its own. The outline of this paper is as follows. After some recalls of sub-riemaniann geometry, we dene, in 2, the entropy and the complexity of a path and we compare these two quantities. We explain how the complexity of paths allow to compute the complexity of nonholonomic motion planning. In 3, we state our results on entropy and complexity, Theorems 4 and 6. We describe rst the basis of the tangent space and present a "-norm for the tangent to a path. We apply then our theorems to the complexity of car parking and to the Hausdor dimension of paths going through the singular locus. Finally we show with an example that complexity and entropy are not always equivalent. We show Theorems 4 and 6 in 4. The proof is based on estimates of the "-norm along the path. The key part is the description of sub-riemaniann balls given in Theorem 7. We just sketch out the proof of this theorem and postpone intermediate demonstrations to the appendix. In Appendix A, we construct an extended sub-riemannian manifold. This manifold is equiregular and its balls are projected onto balls of the original manifold. In Appendix B, we present a particular class of privileged coordinates, the canonical coordinates of the second kind. Finally Appendix C is dedicated to the choice of a coordinates system and to their estimates. 2 Denitions 2. Sub-Riemannian manifolds We recall here some denitions and basic results of sub-riemannian geometry. One of our goal is an application to motion planning ( 2.5). So we look at things from Control Theory point of view. More general presentations can be found in Bella che [] (the main reference for this section) or in Kupka [6]. 2.. Sub-Riemannian distance Let M be a real analytic manifold and ; : : :; m analytic vector elds on M. They dene a control system dened by the following equation _q = m i= u i i (q); q 2 M; () where the control function u(t) = (u (t); : : : ; u m (t)) takes values in IR m. For a choice of a measurable function u(t) dened on some interval [0; ], equation () becomes a dierential equation _q = m i= u i (t) i (q); t 2 [0; ]: () Given any point p 2 M, we can integrate (), taking as an initial condition q(0) = p:

4 4 2 DEFINITIONS Call the solution obtained in this way. One says that is the trajectory of () with initial point p and controlled by u. It joins p to ( ), so ( ) is said to be accessible from p. For q 2 M and v 2 T q M, we set kvk q = inf q u u2 m j We dene the length of a trajectory as Z length() = 0 m i= k _(t)k (t) dt: u i i (q) = v : Finally the sub-riemannian distance attached to the system ; : : :; m is dened by d(p; q) = inf length(); where the inmum is taken on all the trajectories of () joining p to q. The manifold M provided with the distance d is called the sub-riemannian manifold attached to the system ; : : :; m. It is denoted (M; d). Observe now that d(p; q) < if and only if p and q are accessible from one another. This property of accessibility is characterized by Chow's Theorem. Chow's Theorem (Chow [7], Rashevsky [27]). If the vector elds ; : : :; m and their iterated brackets [ i ; j ], [[ i ; j ]; k ], etc. span the space T p M at every point p of M, then every two points are accessible from one another if and only if they belong to the same connected component of M. When the condition of Chow's Theorem is satised, one says that the system () is controllable. An important consequence of this theorem is that, when () is controllable, the topology dened by the sub-riemannian distance d is the original topology of M [] Regular and singular points From now on we assume that () is controllable. Let L ( ; : : :; m ) be the set of linear combinations, with real coecients, of the vector elds ; : : :; m. We dene recursively L s = L s ( ; : : :; m ) by setting, for s = 2; 3; : : :, L s = L s? + i+j=s [L i ; L j ]: Hence L s is the set of linear combinations of commutators of ; : : :; m with a length s. The union L of all L s is a Lie sub-algebra of the Lie algebra of vector elds on M and is called the control Lie algebra associated to (). Remark. Due to Jacobi identity, we could also have dened L s by setting L s = L s? + [L s? ; L ]: L s is then generated by the commutators [[: : :[ i ; i2 ]; : : :]; ik ], each i j belonging to f; : : :; mg. Such a commutator is denoted [ I ], where I is the multi-index I = (i ; : : :; i k ). The length of this commutator is denoted jij = k.

5 2. Sub-Riemannian manifolds 5 For p 2 M, let L s (p) be the subspace of T p M which consists of the values (p) taken, at the point p, by the vector elds belonging to L s. Since () is controllable, for each point p 2 M there is a smallest integer r = r(p) such that L r(p) (p) = T p M. This integer is called the degree of nonholonomy at p. For each point p 2 M, there is an increasing sequence of dimensions, called the growth vector, diml (p) diml s (p) diml r(p) (p) = n: We say that p is a regular point for (M; d) if the growth vector remains constant in some neighborhood of p. Otherwise we say that p is a singular point. Remark. Since M and the vector elds ; : : :; m are analytic, regular points form an open dense set in M Privileged coordinates We x p 2 M and we set n s = diml s (p) for s = ; : : :; r. The growth vector at p is (n ; : : :; n r ). We dene also the sequence w w n by setting w j = s if n s? < j n s. Call f; : : :; m f the nonholonomic partial derivatives of order of f (compare x f; : : xn f). Call further i j f, i j k f, : : : the nonholonomic derivatives of order 2, 3, : : : of f. If the nonholonomic derivatives of order s? of f vanish at p, we say that f is of order s at p. A function f is of order s at p if it is of order s but not of order s +. We say that local coordinates (q ; : : :; q n ) centered at p are privileged coordinates at p if the order of q j at p is equal to w j, for j = ; : : :; n (this denition has been introduced by Bella che [4]). The numbers (w ; : : :; w n ) are the weights of the coordinates (q ; : : :; q n ). We say then that a polynomial is homogeneous of weighted degree s if it is a linear combination of monomials q l q ln n, with w l + + w n l n = s. We can compute the order at p of a smooth function f: it is the least weighted degree arising in the expansion of f as a sum of homogeneous polynomials. Using privileged coordinates, the control system () may be rewritten near p as _q j = m? u i fij (q ; : : :; q j? ) + O kqk wj i= (j = ; : : :; n); where? the functions f ij are homogeneous polynomials of weighted degree w j?. By dropping the O kqk wj, we get a control system (b ) m _q = i= u i b i (q); with i b P n = f j= ij(q ; : : :; q n )@=@q j. The Lie algebra L( b ; : : :; m b ) is nilpotent. approximation of () at p. This system (b ) is called the nilpotent The main interest of privileged coordinates is that they allow to estimate the sub- Riemannian distance. Ball-Box Theorem (Bella che [], Gromov [0]). The following estimate holds if and only if (q ; : : :; q n ) form a system of privileged coordinates at p:

6 6 2 DEFINITIONS there exist constants c; C > 0 and a neighborhood U of p such that, for q 2 U, c? jq j =w + + jq n j =wn d (0; (q ; : : :; q n )) C? jq j =w + + jq n j =wn : This result allows us to describe locally the shape of sub-riemannian balls. However this description is not uniform on M. Indeed the size of U and the constants c; C could depend on p. In particular, if we consider a sequence of regular points p N approaching a singular point p, the size of U pn tends to 0 (but the size of U p is non-zero). Thus we can not use Ball-Box Theorem to answer questions like: how many balls of a given radius are needed to cover a curve containing one singular point? Actually, to estimate balls B(p; "), we have to choose coordinates depending on both p and ". This is the purpose of section 4.. Remark. On the contrary there is a continuity property near regular points. Assume we have a system of privileged coordinates at p varying continuously with p near a regular point. Then the constants c; C and the size of U vary continuously with p (it results from the Ball-Box Theorem's proof in Bella che []). We will see such a system of coordinates in B Entropy and Hausdor dimension The rst aim of this paper is to characterize the geometry of one dimensional submanifolds of a sub-riemannian manifold. We rst present metric invariants and measures for subsets in metric space. Let (; d) be a metric space and A a bounded subset. For all " > 0, we dene the metric entropy M (A; ") as the minimal number of closed balls of radius " in needed to cover A. Notice that originally, as conceived by Kolmogorov [4], the "-entropy is dened as H " (A) = log 2 M (A; "). H " (A) represents the amount of information we need to describe a point in A with the accuracy " or to digitally memorize A with this accuracy. The asymptotic behavior of M (A; ") as " tends to 0 reects the geometry of A in. This behavior is characterized essentially by the entropy dimension dim e A = lim "!0 log M (A; ") log( " ) : In other words dim e A is the inmum of for which M (A; ") (=") for " small enough. A maybe more usual characterization of the geometry of a space use the Hausdor dimension (and measure), introduced by Hurewicz and Wallman [2]. For 0, we dene the -dimensional Hausdor measure as m (A) = lim "!0 m (A; "), where m (A; ") = inf n i= r i ; r i " radius of balls B i ; and A [ i= B i o: Notice that m 0 (A; ") = M (A; ") and m 0 (A) is the number of points in A. For a given A, m (A) is a decreasing function of, innite when is less than some 0, and zero when is greater than 0. We call 0 = dim H A the Hausdor dimension of A: dim H A = supf j m (A) = g = inff j m (A) = 0g:

7 2.3 Complexity of a path 7 Of course the two dimensions we introduced are linked and they coincide in a lot of case. We have in particular the following properties (see Gromov [0, p. 277], Yomdin [3] and Lorentz [22]). (i) For A a compact l-dimensional submanifold in the Euclidean space IR n, dim e A = dim H A = l: (ii) For all 0, m (A) lim "!0 " M (A; "), which implies dim H A dim e A: (iii) If M (A; ") "?0, then dim H A = dim e A = 0. Thus, given M (A; "), we can majorate the Hausdor dimension and even compute it in case (iii) (for this reason in the literature the Hausdor dimension is often dened as the entropy dimension). We take now as metric space a sub-riemannian manifold (M; d). What are the Hausdor dimension and the entropy of a submanifold of M? When (M; d) is equiregular (every point in M is regular), Mitchell [24] determines the Hausdor dimension of the manifold. Gromov [0] extends this result to Hausdor dimension of submanifolds (see Formula (4)). Notice also that Hausdor measure are involved in isoperimetric inequalities (see Pansu [26]). Finally Bella che [] gives an example of computation of entropy and Hausdor dimension for a non equiregular sub-riemannian manifold, the Gru in plane. This last example is interesting because it does not t in the case (iii) above. It follows that the asymptotic behavior of entropy can not be deduced only from the entropy (or Hausdor) dimension. It remains to study submanifolds of non-equiregular sub-riemannian manifolds. We focus our study on the entropy rather than the Hausdor measure because the latter is much more dicult to evaluate. We show also in particular cases how to obtain Hausdor dimensions (Corollary 5 and examples in 3.5). 2.3 Complexity of a path Let (M; d) be the sub-riemannian manifold attached to a system (). We consider onedimensional analytic submanifolds of M with the following properties: they are connected and they have two extremities (these will be specications of an holonomic path in 2.5). These properties imply that the submanifold is dieomorphic to a closed interval in IR (see for instance Milnor [23]). We call such a submanifold a path. A path C can be seen as a parameterized arc, whose parameterization q : [0; T ]! C, t 7! q(t), is an analytic dieomorphism. The extremities q(0) and q(t ) are often denoted by a and b. Notice that submanifold means here embedded submanifold. This implies in particular the following property. Proposition. Let C be a path and q(t) a parameterization of C. There exists a function f = O(") such that: if d(q(t); q( )) "; then jt? j f("): (The usual notation f = O(") means f(")! 0 as "! 0.)

8 8 2 DEFINITIONS We are now in a position to dene the complexity of a path. Consider C a path in M, " > 0, and the neighborhood of C Tube(C; ") = [ q2c B(q; "): Notice that B(q; ") are sub-riemannian balls. The complexity of C is dened as (C; ") = inf 8 < : length() " trajectory of () joins a to b Tube(C; ") 2.4 Links between complexity and entropy 9 = ; : (2) In 3 we will show that the entropy of a path is closely related to the complexity of this path. We can already establish two relations. Proposition 2. Let C be a path in the sub-riemannian manifold (M; d). Then, (C; ") 2M (C; "): Proof. Let C M be a path with extremities a and b, and B ; : : :; B N be balls of radius " covering C. Any two points in the same ball can be linked by a trajectory of length 2" included 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 4" included in B i [ B j. By induction we show then that a and b can be linked by a trajectory of length 2N" included in S B i. The result follows from the denitions of and M. To obtain an inequality in the opposite direction, an additional property of the path is needed. Proposition 3. Let k be a constant. If the path C satisfy (P k ) there exists > 0 such that, for q and q 2 2 C, d(q ; q 2 ) < ) d(q ; q) kd(q ; q 2 ) for all q 2 C between q and q 2 ; then, for " small enough, we have M (C; 3k") (C; "): Notice that, since C is dieomorphic to a closed interval, C between q and q 2 is well dened. Proof. Choose a parameterization q(t) of C. Let be a trajectory of (), contained in Tube(C; ") and connecting the extremities of C. We consider a piece of of length " connecting some balls B(q(t ); ") and B(q(t 2 ); "). The distance between q(t ) and q(t 2 ) is smaller than 3". If 3" <, it follows from property (P k ) that q(t) 2 B(q(t ); 3k") for all t 2 [t ; t 2 ]. Iterating this argument from t = 0 until t 2 = T, we cover C with N balls of radius 3k", where N is not greater than length()=". This proves the proposition.

9 2.5 Application to nonholonomic motion planning 9 Remark. When d is an Euclidean distance, each one-dimensional analytic submanifold satisfy property (P k ) for some k. It is not true for sub-riemannian distances. This can be understand thanks to Ball-Box Theorem (page 5). Assume that, near p = q(0), a path C is given by the analytic parameterization (q (t); : : :; q n (t)), where the q i 's are privileged coordinates at p. The distance d(q(0); q(t)) is equivalent to jq (t)j =w + + jq n (t)j =wn. Thus d(q(0); q(t)) can be seen as the Euclidean distance d E (p; q 0 ) where q 0 belongs to the parameterized arc o nsgn(q (t))jq (t)j =w ; : : :; sgn(q n (t))jq n (t)j =wn ; t near 0 : This arc can be non analytic at t = 0, so property (P k ) with the Euclidean distance can be not satised for any k (think to a curve with a cusp, like y = p jxj in IR 2 ). 2.5 Application to nonholonomic motion planning This section is concerned with a problem of Control Theory: nonholonomic motion planning amidst obstacles. We are going to show how (C; ") can be used to determine the complexity of this problem. We consider a nonholonomic control system, that is a system (). We assume it is controllable. Obstacles are closed subsets F of the conguration space M. The open set M? F is called the free space. The motion planning problem is: given a and b in M? F, nd a trajectory of () contained in the free space, and joining a to b. This problem has a solution if and only if a and b are in the same connected component of M? F (Chow's Theorem). Since M? F is an open set, connectivity is equivalent to arc connectivity. Then a and b belong to the same connected component if there is an arc in M? F linking a to b. This implies that the decision part of the motion planning problem is the same for nonholonomic controllable systems as for holonomic ones. This argument suggests a general method to solve the complete problem. This method, called Approximation of a collision-free holonomic path (see Laumond et al. [9]), has two steps: nd a curve C in the free space linking a to b (C is the collision-free holonomic path); approximate C by a trajectory of (), close enough to be contained in the free space. The existence of a trajectory approximating a given curve is a consequence of Chow's Theorem. Indeed, choose a small open neighborhood U of C contained in M? F. We can assume that U is connected and then there is a trajectory contained in U and linking a to b. What is the complexity of this method? The complexity of the rst step (motion planning problem for holonomic systems) is well modeled and understood. It depends on the geometric complexity of the environment, that is on the complexity of the geometric primitives modeling the obstacles and the robot in the real world (see Canny [6] or Schwartz and Sharir [29]). We are interested here in the second step: its complexity represents the increase of complexity due to the nonholonomy. This complexity can be modeled as the one of an output trajectory (that is a trajectory solution of the problem). We have then to dene the complexity of a trajectory approximating a given curve.

10 0 3 PRINCIPAL RESULT Let C be a curve and Tube(C; ") the tube of radius " centered at C. We denote by the biggest radius " for which Tube(C; ") is contained in the free space. We say that a trajectory of () approximate C if it is contained in Tube(C; ) and if it has the same extremities as C. Remark that, if there is no obstacle, can be innite. In this case a trajectory approximate C if it has the same extremities. Let us assume that we have already dened a complexity () of a trajectory. Then we dene the complexity (C) of the second step as the inmum of () for all trajectories approximating C. It remains to dene the complexity of a trajectory. We present here some possibilities. Let us consider rst bang-bang trajectories, that is trajectories obtained with a nite number of successive trajectories of ; : : :; m. The number of switches in the controls denes a complexity for this kind of trajectories. We can extend this denition to any kind of trajectory. Following Bella che, Laumond and Jacob [3], a complexity can be derived from the topological complexity of a real-valued function (it is the number of changes in the sign of variation of the function). The complexity () appears then as the total number of sign changes for all the controls associated to the trajectory. Notice that, for a bang-bang trajectory, this denition coincides with the previous one. We call topological complexity the complexity t (C) obtained with this denition. Let us recall that the complexity of an algorithm is the number of elementary steps needed to get the result. For the topological complexity, we choose as elementary step the construction of a piece of trajectory without change of sign in the controls (that is without maneuvering, if we think to a car-like robot). Another way to dene the complexity () is to use the length of the trajectory. Roughly speaking, is the size of the free space around C. So we consider that the elementary step of our method is to build a trajectory of length. The number of elementary steps in a trajectory is then () = length() : We call metric complexity the complexity m (C) obtained in this way. Assume that C is analytic. It is a path (as dened in 2.3). The metric complexity m (C) is then the complexity (C; ") of the path C taken at " =. Thus (C; ") allows to compute the metric complexity of nonholonomic motion planning. Moreover both metric and topological complexities yield to similar estimates. Let us justify this argument with an example. Consider a path C such that, for any q 2 C and any i 2 f; : : :; mg, the angle between T q C and i (q) is greater than a given 6= 0. Then, for the trajectory of some i contained in Tube(C; ), the length cannot exceed = sin + O() 2= sin. Thus, if is a bang-bang trajectory contained in the tube, the number of switches in is greater than sin length()=(2). This links m (C) to the topological complexity. 3 Principal result 3. Basis of L(p) and "-norm Let (M; d) be the sub-riemannian manifold attached to a system ; : : :; m and M a compact set. We denote by r the maximum of the degree of nonholonomy on (this maximum exists because ; : : :; m are analytic vector elds).

11 3. Basis of L(p) and "-norm Let J be the set of multi-indices I with jij r. The family [ I ](q), I 2 J, generate T q M at every point q in. Let us introduce some notations. - I 2 J n denotes a family of n multi-indices (I ; : : :; I n ). - D(I ) = ji j + + ji n j is the length of the family I. - I (q) = det([ I ]; : : :; [ In ])(q). - Let q 2 and " > 0. We say that I is a family associated with (q; ") if j I (q)j" D(I) = max K2J n j K(q)j" D(K) : (3) Remark. For a family I associated with some (q; "), the determinant I (q) is non-zero. ([ I ](q); : : :; [ In ](q)) form then a basis of T q M. Example. A nilpotent system in IR 3 : the Martinet distribution Let us consider the system dened in IR 3 by the vector ; + : The only non zero commutators are [ ; 2 ] and [ ; 2 ; The set of singular points is the plane fx = 0g. The degree of nonholonomy is 2 at regular points and 3 at singular points. Two families of vector elds have a non identically zero determinant, ( ; 2 ; [ ; 2 ]) and ( ; 2 ; [ ; 2 ; ]), corresponding to I = (; 2; (; 2)) and J = (; 2; (; 2; )). We have j I (x; y; z)j" D(I) = jxj" 4 ; j J (x; y; z)j" D(J) = " 5 ; and then maxj K (q)j" D(K) = " 4 max(jxj; "). Finally, the families associated with (x; y; z; ") are: J if jxj < "; J and I if jxj = "; I if jxj > ": We consider now a path C and a parameterization q(t) of C. We x a parameter t and a family I such that I (q(t)) 6= 0. We denote by I (t); : : :; I n(t) the coordinates of _q(t) in the basis of the [ Ii ](q(t))'s, that is We dene the "-norm of _q(t) as _q(t) = n i= I i (t)[ I i ](q(t)): k _q(t)k " = maxn j I i (t)j"?jiij ; i n; I associated with (q(t); ") It is worth to notice that k _q(t)k " is a piecewise continuous function of t (maybe continuous). It is then integrable on [0; T ]. Example. As in the previous example, we consider the Martinet distribution. We choose for C the line segment fy = 0; z = x;? x g. The tangent to C is @. The "-norm is then k _q(x)k " = if jxj "; and k _q(x)k " = if jxj ": " 3 jxj" 2 o :

12 2 3 PRINCIPAL RESULT 3.2 Regular and singular points for C We show below (Theorems 4 and 6) that the integral of the "-norm along C gives estimates for the entropy and the complexity. To prove these results we have to distinguish dierent sort of points. A point q 2 C is said to be regular for C if the growth vector is constant on C near q. Otherwise q is said to be singular for C (compare with denitions of 2..2). Regular points for C form an open dense set in C and singular points are isolated in C. Notice that a regular point for C can be singular for M. Let us consider now the following property of a point q in the interior of C. (H) If T q C belongs to L s (q), then there exist commutators [ I ]; : : :; [ Il ] in L s whose values taken at q form a basis of L s (q) and such that, for all q 0 2 C near q, T q 0C 2 spanf[ I ](q 0 ); : : :; [ Il ](q 0 )g: For q a regular point for C, this property simply reads as: if T q C belongs to L s (q), then T q 0C belongs to L s (q 0 ) for all q 0 2 C near q. The points satisfying (H) form an open dense set in C. We have then two nite sets of points in C: singular points and points not satisfying (H). We say that C is generic at singular points when these two sets have an empty intersection, that is if (H) is satised at every singular point for C (except maybe at the extremities). We introduced (H) because it gives a sucient condition for property (P k ) (see Proposition 3). Indeed, we will see in Theorem that, if each point of a path C satises (H), then C veries (P k ) for some k. 3.3 Statement of the results Let f and g be functions of C and ". We say that f and g are equivalent, and we write it f(c; ") g(c; "), if there exist positive non-zero constants and such that, for all C and for " small enough, f(c; ") g(c; ") f(c; "): Let us stress the fact that the constants and are independent of C and ". Theorem 4. Let M be a compact set. Then, for all path C, with a parameterization q(t), t 2 [0; T ], we have M (C; ") Z T 0 k _q(t)k " dt: This estimate of the entropy gives directly the Hausdor dimension of paths without singular points: Corollary 5. Let C be a path containing no singular points for C. Then the Hausdor dimension of C is equal to the smallest integer s such that T q C 2 L s (q) for all q in C. This corollary is a particular case of a result of Gromov [0, p. 04]: for a submanifold N in an equiregular manifold, the Hausdor dimension is equal to dim H N = max q2n where L s N (q) is the linear subspace Ls (q) \ T q N. s s dim? L s N (q)=ls? N (q) ; (4)

13 3.4 Example : the car control system 3 Theorem 6. With the notations of Theorem 4, for all path C generic at singular points, we have M (C; ") (C; ") Moreover, for any path C, le < t Z T 0 k _q(t)k " dt: < < t s? < T be the parameters of the singular points for C for which the property (H) is not satised (see page 2). Set also = 0 and t s = T. Then there exists a positive non-zero constant such that, for " small enough, s i= (Ci " ; ") (C; ") s i= (C i ; "); where the paths C i = q([t i? ; t i ]) and C " i = q([t i? + "; t i? "]) are generic at singular points. For a path not generic at singular points, we can not know directly if complexity and entropy are equivalent or not. For instance we give in 3.6 a path for which they are not equivalent. However both inequalities of Theorem 6 grouped together provides a sucient condition: if Z ti t i? k _q(t)k " dt Z ti?" t i?+" k _q(t)k " dt for every i; 3.4 Example : the car control system then M (C; ") (C; "): The classical example of nonholonomic control system is the car (see Laumond [7] or Jean [3]). It is represented as two driving wheel 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 angle of the car. In the manifold M = IR 2 S, the control system is _q = u + u 2 2 ; with cos sin A ; 2 = 0 L 2 (q) is 3-dimensional at every point q of M. Thus we have J = f; 2; (; 2)g and there is only one family I, which corresponds to the basis, 2, [ ; 2 ]. Let C be a path in M and q(t) a parameterization of C. We dene '(t) as the angle between _q(t) and the plane generated by (q(t)) and 2 (q(t)). The coordinate of _q(t) on [ ; 2 ](q(t)) is sin '(t). According to Theorem 6 we have (C; ") " 2 Z T 0 0 max("; j sin '(t)j)dt: If the path is a trajectory, then ' 0, and the complexity equals length(c)=". On the other hand if the path is always perpendicular to the direction of the car, the complexity is equivalent to =" 2. So we show that to reverse into a parking place needs more maneuvers than going in a straight line! 3.5 Example 2: nilpotent systems in IR 3 Let us consider now the system of vector elds in IR ; ; 0 0 A :

14 4 3 PRINCIPAL RESULT where r 2 is an integer. The Lie algebra L( ; 2 ) is nilpotent of order r (if r = 2, it is the Lie algebra associated with the Heisenberg group). The degree of nonholonomy is two everywhere except on the plane fx = 0g, where it equals r. We are interested in paths with singular points. We consider for instance the paths fx = z p ; y = 0; 0 x; z g where either p or =p is an integer and - if p, then p is the multiplicity of the intersection between C and the singular locus fx = 0g; p = means that C is included in the singular locus; - if =p, then =p is the multiplicity of the intersection between C and the plane fz = 0g (that is the plane f (0); 2 (0)g); =p = (or p = 0) means that C is 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. Some easy, but tedious manipulation yields to the following entropy estimates (an example of computation is shown page in the case r = 3). If p = 0 (C is a trajectory of the system): M (C; ") " and dim H (C) = : It is the only case where the Hausdor dimension equals the topological one. If 0 < p < =(r? 2) (C is tangent to the distribution a): M (C; ") " 2 and dim H (C) = 2: The path is tangent to the distribution f ; 2 g at the singular poin, but not at regular points. The leading term in M (C; ") 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 when r = 2, only this case or the previous one can occur. If p = =(r? 2) (C is transverse to the singular locus a and, if r 4, tangent to the distribution a): M (C; ") " 2 log( " ): In this case, the entropy dimension is dim e C = 2 although M (C; ")" 2 tends to innity. For the Hausdor dimension we have dim H C 2. However the equality dim H C = 2 is easy to show (C contains a path, which is not a trajectory, included in the regular locus). If =(r? 2) < p < (C may be either transverse or tangent to the singular locus): M (C; ") " r?=p and dim H (C) = r? =p: The Hausdor dimension is greater than 2, the Hausdor dimension of a path included in the regular set, but less than r, the one of a path included in the singular locus (see next case). Notice that it can be not an integer when p is greater than one (in this case r? dim H C < r). If p = (the path is included in the singular locus): M (C; ") " r and dim H (C) = r:

15 3.6 Complexity and entropy can be non equivalent Complexity and entropy can be non equivalent Let us consider the system of vector elds in IR ; + (x9? xz : The singular locus is the set f9x 8 = z 2 g. The degree of nonholonomy equals 0 on the subset fx = z = 0g, 3 on the remainder of the singular locus, and 2 elsewhere. We choose a path tangent to fx = z = 0g a but which does not go through the singular locus, say C = f(0; y; y 2 ); y 2 [?; ]g. Notice that the origin is a singular point for C and do not satisfy (H). Thus C is not generic at singular points. A short calculation gives the "-norm k _q(y)k " = ( max? "? ; 2 9! y"?0 if jyj (9!) =4 " 2 ; 2y?3 "?2 if (9!) =4 " 2 jyj : On the other hand we have d(q("); q(?")) = 2". It is then easy to show that (C; ") (q([?;?"]); ") + (q(["; ]); "). By applying Theorems 4 and 6, we obtain (C; ") Z " k _q(y)k " dy and M (C; ") Using the expression of the "-norm, this yields to Z M (C; ") "?6 and (C; ") "?4 : Thus entropy and complexity are not equivalent. 0 k _q(y)k " dy: 4 Proof of Theorems 4 and 6 We show Theorems 4 and 6 in the same way. The proof is divided in three steps. We rst describe the shape of sub-riemannian balls in function of their radius in Theorem 7. It is a general result extending Ball-Box Theorem. It can be read on its own. We present thus only a sketch of the reasoning in 4. and postpone intermediate proofs to the appendix. In a second step ( 4.2), we use this result to obtain some properties on a reduced class of paths. One of the properties is an inequality on the integral of k _q(t)k " depending on the distance between points of the path. In the last step ( 4.3), we link the previous inequality with entropy and complexity and then prove Theorems 4 and Shape of sub-riemannian balls We consider the sub-riemannian manifold (M; d) attached to a system ; : : :; m. B (or B d ) denotes a ball for the distance d. Given p 2 M and a family I of multi-indices, we set B I (p; ") = fp exp(u n [ In ]) exp(u [ I ]); ju i j < " jiij ; i ng: Remark. Here and in the sequel we note on the right the action of dieomorphisms: p exp(t) results of the action of exp(t) on point p. This notation is consistent with the notation for Lie group (used in the proof of Lemma 8): dieomorphisms which come from ows of left-invariant vector elds are dened by right multiplication.

16 6 4 PROOF OF THEOREMS 4 AND 6 Theorem 7. Let M be a compact set and r the maximum of the degree of nonholonomy on. We denote by J the set of multi-indices I with jij r. There exist a constant 0 > 0 and functions ; K = O(), 0 < () < K(), such that: for all p 2, " and 0, if I 2 J n is a family associated with (p; "), then, Remarks. B I (p; ()") B(p; ") B I (p; K()"): (5) This theorem extends Ball-Box Theorem (page 5). Indeed, let p be a point in. For " less than some "(p), a family I associated with (p; ") denes coordinates (u ; : : :; u n ) on B I (p; ") which are privileged at p (see B., page 30). For these values of " and = 0 xed, (5) is equivalent to the estimate of Ball-Box Theorem. Hence the estimate of Ball-Box Theorem holds for balls of radius less than "(p), though (5) holds for radius less than 0, independent of p. We say then that this estimate of sub-riemannian balls depends uniformly of their radius. In estimate (5) the family I used to construct B I depends on ". That's why we have introduced : it allows to change the radius of the ball B(p; ") without changing I. Proof. To show this theorem we rst construct locally an extended manifold (f M; e d) for which we are able to estimate the balls. Then we project the estimates onto M and use the compactness to make the result global on. Local estimate in a lifted space Lemma 8. Let p 2. There exist a manifold M f = M IR en?n ; a neighborhood U e M f of (p; 0); a neighborhood U M of p, U f0g U; e coordinates (y; z) on U; e and vector elds on U e such that, i (y; z) = i (y) + the system ( ; : : :; m ) is controllable on e U; j b ij j (6) every point eq 2 e U is regular for ( f M; e d), the sub-riemannian manifold attached to ( ; : : :; m ); denoting the canonical projection from f M onto M, we have, for q 2 U and " > 0 such that B e d ((q; 0); ") e U, B(q; ") =? B e d ((q; 0); ") : The proof is postponed to Appendix A. Thanks to this lemma we can reduce locally the study to sub-riemannian manifold with no singular points. But for such a manifold Ball-Box Theorem may give an estimate of the distance varying continuously with the points. Reducing eventually e U we have indeed the following result (shown in Appendix B.).

17 4. Shape of sub-riemannian balls 7 Lemma 9. There exist functions C (eq); C 2 (eq) and " (eq) > 0 continuous on e U and a family of commutators ([ J ]; : : :; [ Jen ]) such that, for all eq 2 e U, if " < " (eq), then where e B(eq; ") is the set eb(eq; C ") B e d (eq; ") e B(eq; C 2 ") eb(eq; ") = feq exp(x en [ Jen ]) exp(x [ J ]); jx i j < " jjij ; i eng: Projection of M f onto M To come back to the original manifold M, we have to project the sets B(eq; e "). Using (6), we write a commutator [ J ] as [ J ](y; z) = [ J ](y) + j b Jj j : (7) Now, if eq 0 = eq exp(x en [ Jen ]) exp(x [ J ]) belongs to e B(eq; ") and if eq = (q; 0), then the projection of eq 0 onto M is (eq 0 ) = q exp(x en [ Jen ]) exp(x [ J ]): Denoting by B(q; ") the projection of the set e B((q; 0); "), we have B(q; ") = fq exp(x en [ Jen ]) exp(x [ J ]); jx i j < " jjij ; i eng: As U f0g e U, it follows from Lemmas 8 and 9 that, for q 2 U and " " (q; 0), B(q; C (q)") B(q; ") B(q; C 2 (q)"): (8) Remark. We have chosen [ J ]; : : :; [ Jen ] such that their values at every point eq 2 e U form a basis of T eq f M adapted to the ag (see Appendix B.) f0g = e L 0 (eq) e L (eq) e L r (eq) = T eq f M: This yields to the following properties for ([ J ]; : : :; [ Jen ]): ([ J ](q); : : :; [ Jen ](q)) generate T q M for all q 2 U; for each pair of multi-indices J i and J j, [ Ji ; Jj ](q) = where c k (q) is a smooth function on U. jj kjjj ij+jj jj c k (q) [ Jk ](q) From local to global For every p in, the construction above gives a neighborhood U p and continuous functions C (q), C 2 (q) and " (q) such that inclusions (8) hold. Since is compact, there exists a nite covering of with compact sets E pl U pl and constants C, C 2, " > 0 such that (8) holds on each E pl for " ". The dierence between the E pl 's lies in the family ([ J ]; : : :; [ Jenl ]) used to dene the sets B. To summarize we have a nite covering of with compact sets E l, some multi-indices sets J l, and constants C, C 2 and " > 0 such that, for each l, ([ J ]; J 2 J l ) satisfy both property of the remark above (page 7); 8q 2 E l and " < ", B(q; C ") B(q; ") B(q; C 2 ") (9) where the set B is dened with the family ([ J ]; J 2 J l ).

18 8 4 PROOF OF THEOREMS 4 AND 6 Projection of balls Consider now one of the compact sets E l = E given above. The integer en and the set J of multi-indices J ; : : :; J en are xed on E. Fix a point p in M and a family I in J n. Let I : IR n! M be the application I (u ; : : :; u n ) = p exp(u n [ In ]) exp(u [ I ]): We dene B I (p; ") as the image by I of the box Q(") = fju i j < " jiij ; i ng, that is B I (p; ") = fp exp(u n [ In ]) exp(u [ I ]); ju i j < " jiij ; i ng: Lemma 0. There exist a constan < 0 < and a function K = O() such that, if p 2 E, " and if I 2 J n is a family associated with (p; "), then, for all 0, B I (p; ") B(p; ") B I (p; K()"): Moreover the following properties are satised. (i) There is a constant C > 0, independent of p, " and I, and such that for any q in the ball B I (p; K( 0 )"), 2 I (p) I (q) 2I (p) j I (q)j" D(I) C max K2J n j K(q)j" D(K) : (ii) I is a local dieomorphism in a neighborhood of every point of Q(K( 0 )"). (iii) Given > 0, we can nd 2]0; 0 ] such that, if q 2 B I (p; K()"), then, denoting by ( ; : : :; n ) the local inverse application of I, [Ij ]: i (q) " jiij?jijj if j 6= i; 2 [Ii ]: i (q) 2: The proof is postponed to Appendix C. We see that J is used only as the multi-indices set in which an associated family is chosen. We can then add to J some multi-indices which are never involved in an associated family. Reminding the construction of J we can increase it in such a way that the [ I ]'s, with I 2 J, generate L s (as a linear subspace of the space of vector elds on M) for a given s r. We construct J as follow. Let r be the maximum of the degree of nonholonomy on. We choose J as the set of multi-indices I such that jij r. The set J is then the same for every compact E l. Combining then (9) with Lemma 0 we obtain an estimate of balls on the whole, which ends the proof of Theorem 7. Remark. Properties (i)(iii) will be used in the next section. 4.2 Estimate of k _q(t)k " along a path In this section we restrict ourselves to paths for which (H) is satised at every point, except maybe at the extremities (property (H) is dened page 2). To shorten, such paths are said to satisfy (H) everywhere.

19 4.2 Estimate of k _q(t)k " along a path 9 Theorem. There exist constants, k, k 2 > 0 and k such that, if C is a path satisfying (H) everywhere and q(t) a parameterization of C, then. property (P k ) is veried: there exists 0 > 0 such that, for q( ) and q(t ) 2 C, d(q( ); q(t )) < 0 ) d(q( ); q(t)) kd(q( ); q(t )) 8t 2 [ ; t ]; 2. if 2 [0; T ] and " 0, then d(q( ); q(t )) = " implies k Z t k _q(t)k " dt k 2 : (0) Remark. Constants, k, k 2 and k are independent of the path C. On the contrary, 0 depends not only on C but also on the parameterization of C. After some notations, we will estimate coordinates? I along the path. Using these estimates and Theorem 7 we prove successively both properties of Theorem Notations Let C a path and q(t), t 2 [0; T ], a parameterization of C. In the whole proof we assume that C satises (H) everywhere, that is: For each t in ]0; T [, if _q(t) belongs to L s (q(t)), then there exist [ I ]; : : :; [ Il ] 2 L s which values at q(t) form a basis of L s (q(t)) and such that, for near t, _q( ) 2 span [ I ](q( )); : : :; [ Il ](q( )) : We make use below of notations and denitions introduced in 3.. Moreover, for I 2 J n and i = ; : : :; n, we set det I i [ (t) = det I ](q(t)); : : :; _q(t) ; : : :; [ In ](q(t)) ; I i (t) = deti i (t) I (q(t)) {z} i?th if I (q(t)) 6= 0: This is consistent? with notations of 3.: when dened, the I i (t)'s are the coordinates of _q(t) in the basis [ I ](q(t)); : : :; [ In ](q(t)), that is n _q(t) = i= I i (t)[ I i ](q(t)): () We use this expression to see _q(t) as the restriction on C of a vector eld on M. Let F be the set of analytic functions I (q(t)) and det I i (t), for I 2 J n and i 2 f; : : :; ng. Zeroes of functions in this set dene on [0; T ] a nite number of real numbers 0 = T 0 < T < < T N = T such that, on ]T j ; T j+ [, a function in F is either identically zero or everywhere non zero. In particular a function non identically zero on ]T j ; T j+ [ has a constant sign on this interval. There is also a constant r 0, 0 < r 0 < 2 max j(t j+? T j ), such that : for all j 2 f0; Ng, t 2 [T j? r 0 ; T j + r 0 ], and f and g 2 F, if f(t j ) 6= 0 and g(t j ) = 0; then f(t) g(t): (2) T 0 ; : : :; T N and r 0 are needed only in the proof of Proposition 3.

20 20 4 PROOF OF THEOREMS 4 AND Preliminary results We rst construct coordinates dened on the path C (this construction is also used in Proposition 27, page 4). Let [ ; t ] [0; T ]. We assume that q([ ; t ]) is included in B I (q( ); K( 0 )) for some < and a family I associated with (q( ); ) (K and 0 are dened in Theorem 7). The application I (dened page 8) is a local dieomorphism on Q(K( 0 )). So there is an unique absolutely continuous application : [ ; t ]! IR n such that (t) 2 Q(K( 0 )) and I ((t)) = q(t), for all t 2 [ ; t ]. But q(t) and are one-to-one on [ ; t ] and I is a local dieomorphism on Q(K( 0 )). Therefore I is a global dieomorphism on a neighborhood of ([ ; t ]). It follows that the inverse application? I is dened on this neighborhood. We denote by ( (t); : : :; n (t)) the application I? (q(t)) for t 2 [; t ]. Proposition 2. We consider 2 [0; T [, t 2 [ ; + r 0 ], and a family I associated with (q( ); ). Let p be an integer such that Z t I p (s)?ji pj ds = max in Z t If q([ ; t ]) is included in B I (q( ); K( 0 )), then I i (s)?ji ij ds: j p (t )j 4 Z t j i (t)j 2n jiij Z t As a consequence we have We show rst the following result. I p (s) ds; (3) I p (s)?ji pj ds; 8t 2 [ ; t ]; 8i 2 f; : : :; ng: (4) j i (t)j 8n jiij?jipj j p (t )j: Proposition 3. The function I p(t) has a constant sign on [ ; t ]. This implies that, for all t 2 [ ; t ] and i 2 f; : : :; ng, Z t I i (s)?ji ij ds Z t I p (s)?ji pj ds = Z t I p (s)?jipj ds : Remark. In the proof of this proposition and below in those of Propositions 2, 4 and 5 the family I is xed. We then write i and det i instead of I i and deti i. Proof. The parameter belongs to some interval [T i ; T i+ [, say [T 0 ; T [. As r 0 < T 2?T, we have [ ; t ] [T 0 ; T 2 ]. We distinguish two cases according to the position of t with respect to T. - If t T : each i and then p has a constant sign on [ ; t ]. - If t > T : q(t ) belongs to B I (q( ); K( 0 )). It follows from Lemma 0(i) that I (q(t )) 6= 0, and so that the sign of I (q(t)) is constant on [ ; t ]. On the other hand, since q(t ) satises (H), there exists l such that We have then two inequalities: det l (T ) 6= 0 and; if ji i j > ji l j; det i 0 on [T 0 ; T ]:

21 4.2 Estimate of k _q(t)k " along a path 2 - if ji i j < ji l j, for small enough and t 2 [ ; t ], we have j i (t)j?jiij j l (t)j?jilj ; - if ji i j = ji l j and det i (T ) = 0, from denition (2) of r 0, we have j i (t)j j l (t)j for t 2 [ ; t ]. This implies that p has to be such that ji p j = ji l j and det p (T ) 6= 0. Thus p has a constant sign on [ ; t ]. Proof (of Proposition 2). Let t 2 [ ; t ] and i 2 f; : : :; ng. The application? I dened at every point q(s) with s 2 [ ; t]. We can write is Z t i(t) = d i dt (s)ds: Thanks to formula () we see _q(s) as the value at q(s) of a vector eld on M. The derivative of i (s) is then, so d i n dt (s) = _q: i(s) = l (s)[ Il ]: i (s): We integrate from to t and obtain i(t) = n l= Z t l= l (s)[ Il ]: i (s)ds: According to Lemma 0(iii), reducing eventually 0, we have, Z t Z t i (s)[ Ii ]: i (s)ds 2 i (s) ds; Z t Z l (s)[ Il ]: i (s)ds t 4n jiij?jilj l (s) ds; for l 6= i: (5) Using Proposition 3 we majorize the integral of j l j in function of the integral of j p j. We obtain and inequality (4) is proved. On the other hand we have p (t ) Z t i (t) n Z t 2 jiij?jilj jilj p (s)?ji pj ds l= p (s)[ Ip ]: p (s)ds? Z t l6=p l (s)[ Il ]: p (s)ds : But p (s) has a constant sign (Proposition 3). The function [ Ip ]: p (s) has also a constant sign since j[ Ip ]: p (s)j =2 (Lemma 0(iii)). This observation and inequality (5) applied to i = p yield to p (t ) 2 Z t We obtain then inequality (3). p (s) ds? l6=p 4n jipj?jilj jilj Z t p (s)?ji pj ds: