Extremal polynomials in stratified groups

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.


Introduction
In any stratified nilpotent Lie group, we compute explicitly the solutions to the adjoint equations for extremal curves related to the natural left invariant horizontal distribution.In the Hamiltonian formalism, a normal extremal (γ, λ), where γ is a curve in the group and λ is the dual curve in the cotangent bundle, solves the system of equations where H is the Hamiltonian function.After fixing a basis of the Lie algebra of the group inducing exponential coordinates of the second type, we integrate the second equation, namely the equation λ = −∂H/∂γ (see the precise formula in Theorem 3.1), and we express λ as a function of γ.In other words, we compute n prime integrals of the Hamiltonian system (1.1),where n is the dimension of the group.The solutions are expressed in terms of a family of polynomials, called extremal polynomials, that satisfy a set of remarkable structure formulas (see (1.3) below).These formulas involve the structure constants of the Lie algebra of the group, in fact, of its Tanaka prolongation.Extremal polynomials can also be used to give an algebraic characterization of abnormal extremals.This is the main motivation of our research.
Let G be a stratified nilpotent Lie group of dimension n and rank r.The Lie algebra g = Lie(G) has the stratification g = g 1 ⊕ • • • ⊕ g s , where s is the step of the algebra, g i = [g i−1 , g 1 ] for i = 2, . . ., s, and g i = {0} for i > s.The rank of g is r = dim(g 1 ).Let X 1 , . . ., X n be a basis of g adapted to the stratification.We identify the group G with R n via exponential coordinates of the second type induced by the basis X 1 , . . ., X n , and we identify g with the corresponding Lie algebra of left invariant vector fields in R n .
Let Prol(g) = k≤s g k be the Tanaka prolongation of g, see [15].Even though this is not strictly needed in our argument, an explicit construction is briefly recalled in Section 2. We extend X 1 , . . ., X n to a basis {X j } j≤n of the prolongation and to each j ≤ n we assign the degree d(j) = k if and only if X j ∈ g k .Then we assume that the basis is adapted to the graduation: this means that i < j implies d(i) ≤ d(j).When the prolongation is finite dimensional, the index j ranges in a finite set, m ≤ j ≤ n for some m ∈ Z.With abuse of notation, we denote the basis {X j } m≤j≤n by {X j } j≤n , as in the infinite dimensional case.Let c k ij ∈ R be the structure constants of Prol(g) associated with the basis {X j } j≤n .Namely, for all i, j ∈ Z with i, j ≤ n we have The sum is always finite, because each stratum g k of Prol(g) is finite dimensional and In this paper, we introduce a family of extremal polynomials P v j (x), j ≤ n and x ∈ R n , associated with the basis {X j } j≤n of Prol(g).They depend linearly on a parameter v ∈ R n , see Definition 2.1.Extremal polynomials satisfy the following structure formulas.
Theorem 1.1.For any v ∈ R n , i = 1, . . ., n, and j ∈ Z with j ≤ n there holds (1.3) Modulo the value at x = 0, extremal polynomials are uniquely determined by the family of identities (1.3).These structure formulas are the core of the paper and of our main technical result, Theorem 2.2.They first appeared in [8], but only in the case of free groups.In [8], the formulas were obtained a posteriori, as a consequence of certain algebraic identities, however only for j = 1, . . ., n with no reference to the prolongation and only for the coordinates related to a Hall basis.
Our interest in extremal polynomials origins in the regularity problem of sub-Riemannian length minimizing curves, one of the main open problems in the field (see [10], [2], [12]).Let M be a differentiable manifold and D ⊂ T M a bracket generating distribution.A Lipschitz curve γ : [0, 1] → M is horizontal if γ(t) ∈ D(γ(t)) for a.e.t ∈ [0, 1].Fixing a quadratic form on D, one can define the length of horizontal curves.Length minimizing curves may be either normal extremals or abnormal extremals: while normal extremals are always smooth, abnormal ones are apriori only Lipschitz continuous.Abnormal extremals depend only on the structure (M, D): they are precisely the singular points of the end-point mapping.
In Section 3, we use extremal polynomials and Theorem 1.1 to give an algebraic characterization of abnormal extremals in stratified nilpotent Lie groups (Carnot groups).This is of special interest because, by Mitchell's theorem, Carnot groups are the infinitesimal model of equiregular sub-Riemannian structures.
Let ϑ 1 , . . ., ϑ n be a basis of 1-forms of g * , the dual of g.
See Section 3 and Theorem 3.1 for more details.In Theorem 3.4, we use the structure formulas (1.3) to integrate the system of adjoint equations (1.4).The solutions are . Thus, we can prove the following theorem (see Theorem 3.8 for the complete statement).
Theorem 1.2.Let G = R n be a stratified nilpotent Lie group and let γ : [0, 1] → G be a horizontal curve with γ(0) = 0.Then, the following statements are equivalent: (A) The curve γ is an abnormal extremal.
(B) There exist v ∈ R n , v = 0, such that P v i (γ(t)) = 0 for all t ∈ [0, 1] and for all i ≤ r.
Abnormal extremals are precisely the horizontal curves lying inside algebraic varieties defined via extremal polynomials.This result extends [8,Theorem 1.1] because it applies to nonfree Carnot groups.It also improves that result because abnormal curves are shown to be in algebraic varieties smaller than those considered in [8].Notice that in (B) also indexes i ≤ 0 related to the Tanaka prolongation are involved.
The role of Tanaka prolongation in our theory is rather subtle.We searched for an integration algorithm inverting the differentiation process that now is established by the structure formulas (1.3).For i = 1, . . ., r and j = 1, . . ., n, consider the integrals We may then further integrate B v ij against γk , etc.The structure of the Tanaka prolongation tells us, however only from a deductive point of view, when the functions B v ij (t) and their iterative integrals are polynomials of the coordinates γ 1 (t), . . ., γ n (t).Part of statement (B) in Theorem 1.2 is that an abnormal extremal γ satisfies P v i (γ) = 0 for some v = 0 and for all i = 1, . . ., r.When the above integration process succeds, we get new polynomials vanishing along the curve γ.More details on this point of view can be found in the example studied in Section 4.
In the final part of the paper, we show two applications of the theory.
In Section 4, we develop a technique to construct a nontrivial algebraic set containing all abnormal extremals passing through one point.This is related to the problem of estimating the size of the set of regular values of the end-point mapping (see [2] and [10,Section 10.2]).In sub-Riemannian manifolds with a distribution of corank 1, the image of the set of length minimizing abnormal extremals starting from one point has zero Lebesgue measure; on the other hand, independently from corank, the image of strictly abnormal length minimizing extremals has empty interior (see Corollary 3 in [14] and [1]).
Our technique seems to work when the prolongation is sufficiently large.In this case, for each abnormal curve γ there is at least one parameter v ∈ R n and many indexes j such that P v j (γ) = 0.It is then possible to find a polynomial Q independent of v such that Q(γ) = 0 for any abnormal curve γ passing through one fixed point.We describe the technique in detail in the case of the free nilpotent Lie group of rank 2 and step 4.However, it can be implemented in many other examples and it is likely to work in any nonrigid group.
Finally, in Section 5 we construct a 64-dimensional Lie group possessing a spirallike Goh extremal whose tangents at the singular point are all lines.This example points out a limitation of the shortening technique introduced in [9] and developed in [11] and [13].It is also interesting in relation to the examples of nonrectifiable spiral-like rigid paths studied in [18] and [17], whereas our spiral-like extremal has finite length.
Acknowledgements.It is a pleasure to thank Ben Warhurst and Alessandro Ottazzi for many illuminating discussions on Tanaka prolongation.We also thank Igor Zelenko for some discussions on a preliminary version of the paper.

Structure formulas for extremal polynomials
Let X 1 , . . ., X n be a basis of g = Lie(G) adapted to the stratification.We identify the group G with R n via exponential coordinates of the second type induced by the basis X 1 , . . ., X n .Namely, for any x = (x 1 , . . ., x n ) ∈ R n we have (2.5) x Above, exp : g → G is the exponential mapping, • is the group law in G = R n , and γ(t) = e tX (x), t ∈ R, is the solution of the Cauchy Problem γ = X(γ) In simply connected nilpotent groups, the exponential mapping is a global diffeomorphism.After the identification (2.5), the Lie algebra g is isomorphic to a Lie algebra of vector fields in R n that are left invariant with respect to the group law •.
We recall the construction of the Tanaka prolongation.First, we define the vector space of all strata-preserving derivations of g: Recall that g i ⊗ g * i is canonically isomorphic to End(g i ).The direct sum vector space s i=0 g i is a graded Lie algebra with the bracket [φ, X] = −[X, φ] = φ(X), for all φ ∈ g 0 and X ∈ g, and with the natural bracket on g and g 0 .By induction, assume that we have a vector space s i=1−k g i for some k ≥ 1 and assume that the bracket [φ, X] = φ(X) is already defined for all φ ∈ s i=1−k g i and X ∈ g.Then we define the vector space of all derivations φ : g → g 1−k ⊕ . . .⊕ g s−k such that φ(g i ) ⊂ g i−k : Recall that g i−k ⊗ g * i is isomorphic to Hom(g i ; g i−k ).As above, the direct sum vector space s i=−k g i is a graded Lie algebra with the bracket [φ, X] = −[X, φ] = φ(X), for all φ ∈ g −k and X ∈ g, with the natural bracket on g, and with the bracket This inductive construction may or may not end after a finite number of steps, i.e., we have either g −k = {0} for some k ≥ 1 or g −k = {0} for all k ≥ 1.In both cases, we let (2.6) Prol(g) = k≤s g k .
Prol(g) is a graded Lie algebra, called Tanaka prolongation of g.Namely, we have (2.7) [g i , g j ] ⊂ g i+j , for all i, j ∈ Z with i, j, i + j ≤ s.This is the unique property that we need in the proof of the structure theorem of extremal polynomials, Theorem 2.2.In fact, we need (2.7)only for j = 1, . . ., s.
We do not specifically need the Tanaka prolongation but only a graded Lie algebra extending g and satisfying (2.6) and (2.7).Among all such extensions Prol(g) is the largest one.
In general, the explicit computation of Prol(g) is difficult.When g is a free nilpotent Lie algebra of step s ≥ 3, then we have Prol(g) = g 0 ⊕ g, with the exceptional case of the step 3 and rank 2 free Lie algebra (see [16]).
Here, only elements X 1 , . . ., X n of the basis of g are involved.We agree that [•, X (0,...,0) ] = Id.The generalized structure constants c k iα ∈ R, with α ∈ I = N n and i, k ∈ Z such that i, k ≤ n, are defined via the relation The above sum is always finite.In fact, letting For α ∈ I and i ∈ Z with i ≤ n, we define the linear mapping φ iα ∈ Hom(R n ; R) extremal polynomial of the Lie algebra Prol(g) with respect to the basis {X j } j≤n .
For any finite set of multi-indexes A ⊂ I, consider the polynomial where c α ∈ R for all α ∈ A. The homogeneous degree of the polynomial is d(P ) = max d(α) : α ∈ A such that c α = 0 .We say that the polynomial is homogeneous of degree k ≥ 0 if d(α) = k for all α ∈ A such that c α = 0. Extremal polynomials P v i are indeed polynomials.In fact, if The following theorem is the main result of the paper.Theorem 2.2.For any v ∈ R n , i = 1, . . ., n, and j ∈ Z with j ≤ n there holds (2.13) Moreover, the polynomials {P v j } j≤n are uniquely determined by (2.13) for i = 1, . . ., r and P v j (0) = v j for j ≤ n.
The identity P v j (0) = v j is proved in (2.34) below.The uniqueness follows from this observation: if f is a smooth function on G = R n such that X i f = 0 for all i = 1, . . ., r, then f is constant.Now, let {Q v j } j≤n be a family of polynomials satisfying (2.13) for i = 1, . . ., r, and Q v j (0) = v j for j ≤ n.Then, for d(j) = s we have Assume by induction that P v j = Q v j for d(j) ≥ ℓ + 1 and take j ≤ n such that d(j) = ℓ.From (2.13), we have and thus Before starting the proof of (2.13), we need three preliminary lemmas.
We clearly have G 1 = G.The following lemma is well-known and we omit the proof.
Lemma 2.3.Let X 1 , . . ., X n be vector fields in R n satisfying (2.5).Then: (i) For all i = 1, . . ., n and for all x ∈ G i we have In the following two lemmas, we prove formula (2.13) in two simplified situations that will serve as base for a long induction argument.
Lemma 2.4.For any i = 1, . . ., n, j ∈ Z with j ≤ n, and v ∈ R n there holds (2.14) where we let On the other hand, for x ∈ G i and v ∈ R n we have (2.17) By (2.16) and (2.17), proving the claim (2.14) is equivalent to show that for all ℓ ≤ n and for all β ∈ I i we have Since {X ℓ } ℓ≤n is a basis, this proves (2.18).
Lemma 2.5.For all j ∈ Z with j ≤ n, i = 1, . . ., n such that d(i) = s, and v ∈ R n there holds (2.20) Proof.When j = 1, . . ., n, the left and right hand sides of (2.20) are both identically 0. Recall that we have d(P v j ) ≤ s − d(j) < s and thus X i P v j = 0. We prove the claim (2.20) for any j ≤ n, in particular for j ≤ 0. As d(i) = s, by (ii) in Lemma 2.3 we have X i = ∂/∂x i on G = R n .Then we have formula (2.16) with β ∈ I replacing β ∈ I i .We also have formula (2.17), again with β ∈ I replacing β ∈ I i .Then we are reduced to check identity (2.18) for all β ∈ I.
We claim that, for any β ∈ I, we have We prove (2.21) by induction on the length of β and we assume without loss of generality that i = n.Notice that the vector field X n commutes with any vector field Assume that β k = 0 and β h = 0 for h > k.Then, by the Jacobi identity, by [X k , X n ] = 0, and by formula (2.21) for β − e k replacing β (and with i = n), we have and we are finished.Now, using (2.21) the proof can be concluded as in (2.18)-(2.19).
Proof of Theorem 2.2.The proof of (2.13) is a triple nested induction.We fix the vector v ∈ R n and we let P v j = P j .The claim in Theorem 2.2 reads The first induction is descending on i = 1, . . ., n and the base of induction is This statement holds by Lemma 2.5, because d(n) = s.
The first inductive assumption is the following (2.22)X h P j = k≤n c k hj P k on G for all h > i and for all j ≤ n.
Our goal is to prove identity (2.22) also for h = i.The proof of this claim is a descending induction on j ≤ n.The base of induction for j = n is the following This identity holds because we have X i P n = 0 (P n is constant), and c k in = 0 for any i ≥ 1 and for all k.
The second inductive assumption is the following (2.23) on G ℓ for all h > j and for all ℓ = 1, . . ., n.
The goal is to prove identity (2.23) also for h = j.The proof is a descending induction on ℓ = 1, . . ., i.The base of induction for ℓ = i is the following The third inductive assumption is (2.24) Our goal is to prove identity (2.24) on G ℓ .
and, moreover, x p ℓ y α 0 = x α+pe ℓ .Thus, Our goal is to prove that the quantity We compute the coefficient χ m,β When is proved as soon as we verify the following (2.29) Contracting with X m the left hand side of (2.29) we find m,k≤n (2.30) Contracting with X m the first term in the right hand side of (2.29) we find k,h,m≤n (2.31) Contracting the second term in the right hand side of (2.29) we find (2.32) We used β 1 = . . .= β ℓ = 0.In (2.30), (2.31), and (2.32), commutators have the same tail X β−β ℓ e ℓ .Thus the claim (2.29) follows provided that we prove the following identity: We prove (2.33) by induction on β ℓ ≥ 1.The base of induction for and this holds by the Jacobi identity.
By induction, we assume that (2.33) holds for β ℓ and we prove it for β ℓ + 1.We have: This finishes the proof of Theorem 2.2.
In the next proposition, we list some elementary properties of extremal polynomials that are used in Section 3.
(i) For all i = 1, . . ., n and v ∈ R n , we have Proof.(i) For any i = 1, . . ., n, we have (2.34) because c k i0 = δ ik , the Kronecker symbol.(ii) This follows as in (i) from the agreement that v k = 0 for k ≤ 0. (iii) Assume that P v k = 0 when d(k) = 1.We claim that P v k = 0 for all k = 1, . . ., n.Because of (i), this will imply v = 0.The proof is by induction on d(k).Assume that P v k = 0 for all k = 1, . . ., n such that d(k) ≤ ℓ, where ℓ < s.Take k = 1, . . ., n such that d(k) = ℓ + 1.By the stratification, there are constants and thus we have the identity (2.35) From P v j = 0 for d(j) = ℓ, it follows that X i P v j = 0. Now, from Theorem 2.2 and (2.35) we obtain 0 = This finishes the proof.

Algebraic characterization of abnormal extremals
In this section, we prove the theorems about the algebraic characterization of abnormal extremals.The Lie group G is identified with R n via exponential coordinates as in (2.5), where X 1 , . . ., X n is a basis of g = Lie(G) adapted to the stratification.The vector fields X 1 , . . ., X r are a basis for g 1 (and thus generators for g).The distribution of r-planes The functions h are called controls of γ and, when γ : [0, 1] → R n is given by the coordinates γ = (γ 1 , . . ., γ n ), we have h j = γj , j = 1, . . ., r.This follows from the structure of the vector fields X 1 , . . ., X r described in Lemma 2.3.
Theorem 3.1 is a version of Pontryagin Maximum Principle adapted to the present setting.Equations (3.37) are called adjoint equations.We refer to [3,Chapter 12] for a proof of Theorem 3.1 in a more general framework.The version (3.37) of the adjoint equations is derived in [8], Theorem 2.6.The curve λ is called a dual curve of γ.Definition 3.2.We say that a horizontal curve γ : [0, 1] → G is an extremal if there exist λ 0 ∈ {0, 1} and a curve of 1-forms λ ∈ Lip([0, 1]; g * ) such that i), ii), and iii) in Theorem 3.1 hold.
We say that γ is a normal extremal if there exists such a pair (λ 0 , λ) with λ 0 = 1.
We say that γ is an abnormal extremal if there exists such a pair with λ 0 = 0. We say that γ is a strictly abnormal extremal if γ is an abnormal extremal but not a normal one.Remark 3.3.If γ is an abnormal extremal with dual curve λ, then we have This follows from condition ii) of Theorem 3.1 with λ 0 = 0.
Next, we define the notion of corank for an abnormal extremal.For any h ∈ L 2 ([0, 1]; R r ), let γ h be the solution of the problem , is called the end-point mapping with initial point x 0 ∈ G.A horizontal curve γ starting from x 0 with controls h is an abnormal extremal if and only if there exists v ∈ R n , v = 0, such that Here, dE(h) is the differential of E at the point h.
Definition 3.6.Let G = R n be a stratified nilpotent Lie group of rank r.For any v ∈ R n , v = 0, we call the set For linearly independent vectors v 1 , . . ., v m ∈ R n , m ≥ 2, we call the set Z v 1 ∩ . . .∩ Z vm an abnormal variety of G of corank m.
Remark 3.7.The abnormal variety Z v is the intersection of the zero sets of all the polynomials P v j with j ≤ r.In this intersection, also indexes j ≤ 0 are involved.With this respect, Definition 3.6 differs from the analogous definition in [8].
From the above argument, we conclude that if corank(γ) ≥ m then there exist at least m linearly independent vectors v 1 , . . ., v m ∈ R n su that γ(t) ∈ Z v i for all t ∈ [0, 1] and i = 1, . . ., m.
4. The abnormal set in the free group of rank 2 and step 4 In this section, we present a method to compute a set containing all abnormal extremals passing through one point.The method works when each abnormal extremal is in the zero sets of a sufficiently large number of extremal polynomials.This is the case when the prolongation is sufficiently large.In order to make the presentation clear, we focus on the free group of rank 2 and step 4. The key point is to show that a certain polynomial is nontrivial, see Theorem 4.1 below.
Let G be the free nilpotent Lie group of rank r = 2 and step s = 4. Via exponential coordinates of the second type, G is diffeomorphic to R 8 .The extremal polynomials in R 8 associated with the Tanaka prolongation of g = Lie(G) are, for any fixed v ∈ R 8 , where Q jk are the polynomials in R 8 Let v ∈ R 8 be a vector such that v 1 = v 2 = v 3 = 0.The first three polynomials P v 1 , P v 2 , and P v 3 are the following 2 . (4.46) These polynomials can be computed using the structure relations (4.42) and formulas (4.44).
In order to compute the four polynomials P v −3 , P v −2 , P v −1 , and P v 0 associated with the stratum g 0 of Prol(g), we have to choose a basis X −3 , X −2 , X −1 , X 0 of g 0 .We can identify g 0 with End(g 1 ), and g 1 with R 2 via the basis X 1 , X 2 .Hence, we can make For each j = −3, . . ., 3, we define the following 5-dimensional vector of polynomials where Q jk are defined in (4.45).By (4.44), identity (4.49) reads In other words, along the curve γ the seven 5-dimensional vectors Q −3 (x), . . ., Q 3 (x) are orthogonal to the nonzero vector (v 4 , . . ., v 8 ) ∈ R 5 .It follows that the 7×5 matrix    has rank at most 4 along the curve γ.
The polynomials f 1 , . . ., f 21 do not depend on v. Proof.It is enough to show that at least one of the polynomials f 1 , . . ., f 21 in (4.50) is nonzero.Let us consider the polynomial with least degree The homogeneous degree of Q is 14.We highlight the anti-diagonal of the matrix M: Upon inspection of the polynomials P v −1 , . . ., P v 3 in (4.46) and (4.47), we observe the following facts.The variable x 8 appears in the entry M ij only when i = j = 5.When i, j ≤ 4, the variable x 6 appears only in the entry M ij with i = j = 4.When i, j ≤ 3, the variable x 4 appears only in the entry M ij with i = j = 3.When i, j ≤ 2, the variable x 3 appears only in the entry M ij with i, j = 2. Finally, we have where R(x) is a polynomial that does not contain the monomial x 1 x 3 x 4 x 6 x 8 .This proves that Q = 0.
Remark 4.2.Any abnormal curve passing through 0 is in the intersection of the zero sets of the 21 polynomials (4.50) in R 8 .Even though all these polynomials are explicitly computable, the precise structure of this intersection is not clear.For any v ∈ R 8 with v = 0 and v 1 = v 2 = v 3 , the equation P v 3 (x) = 0 determines two (in some exceptional cases four) different abnormal extremals γ v parameterized by arc-length and such that γ v (0) = 0.The mapping v → γ v (1) seems to parametrize a subset of R 8 that is at most 5-dimensional.The abnormal set of G = R 8 is then presumably 6-dimensional, while in Theorem 4.1 abnormal extremals are shown to be in a set with dimension less than or equal to 7.

Spiral-like Goh extremals
The shortening technique introduced in [9] has two steps: a curve with a corner at a given point is blown up; the limit curve obtained in this way is shown not to be length minimizing.This provides an "almost-C 1 " regularity result for sub-Riemannian length minimizing curves.The technique, however, fails when, in the blow up at the singular point, the curve has no proper angle.Here, we show that there do exist Goh extremals of this kind.The example is a generalization of [8,Example 6.4].
Let G be an n-dimensional stratified nilpotent Lie group with Lie algebra g = g 1 ⊕ • • • ⊕ g s , where s ≥ 3 is step of the group.An abnormal extremal γ : [0, 1] → G with dual curve λ = λ 1 ϑ 1 + • • • + λ n ϑ n is said to be a Goh extremal if λ i = 0 for all i = 1, . . ., n such that d(i) = 1 or d(i) = 2.By Theorem 3.4, a horizontal curve γ : [0, 1] → G with γ(0) = 0 is a Goh extremal precisely when there exists v ∈ R n , v = 0, such that (5.51) P v i (γ(t)) = 0, for all t ∈ [0, 1] and d(i) ∈ {1, 2}.By Theorem 2.2, the condition P v i (γ) = 0 for d(i) = 2 is equivalent to (5.51).Let F be the free nilpotent Lie group of rank 3 and step 4 and consider the direct product G = F × F .As F is diffeomorphic to R 32 , then G is a stratified Lie group of rank 6 and step 4 diffeomorphic to R 64 .We fix a basis X 1 , . . ., X 64 of Lie(G) adapted to the stratification and we identify G with R 64 via exponential coordinates of the second type.We reorder and relabel the basis as Y 1 , . . ., Y 32 , Z 1 , . . ., Z 32 , where Y 1 , . . ., Y 32 is an adapted basis of a first copy of Lie(F ) and Z 1 , . . ., Z 32 is an adapted basis of a second copy of Lie(F ).We denote the corresponding coordinates on G by (y, z) ∈ R 64 with y, z ∈ R 32 .
Viceversa, if the limit in the left hand side does exist for some infinitesimal sequence (λ k ) k∈N , then it is of the given form for some α ∈ [0, 2π).