Casale, G., and J. Roques. (2008) “Dynamics of Rational Symplectic Mappings and Difference Galois Theory,” International Mathematics Research Notices, Vol. 2008, Article ID rnn103, 23 pages. doi:10.1093/imrn/rnn103 Dynamics of Rational Symplectic Mappings and Difference Galois Theory Guy Casale 1 and Julien Roques 2 1 IRMAR UMR 6625, Universit´ e de Rennes 1, Campus de Beaulieu 35042 Rennes Cedex, France and 2 ´ Ecole Normale Sup ´ erieure, D ´ epartement de Math ´ ematiques et Applications UMR 8553, 45, rue d’Ulm, 75230 Paris Cedex 05, France Correspondence to be sent to: [email protected]In this paper, we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales–Ramis type, ensuring the nonintegrability of a rational symplectic map in the noncom- mutative sense (Mishchenko–Fomenko). As a particular case, we obtain a complete discrete analogue of Morales–Ramis Theorems for nonintegrability in the sense of Liouville. 1 Introduction and Organization The problem of deciding whether a given continuous dynamical system is integrable is an old and difficult problem. Many examples arise in classical mechanics and physics, the three body problem being one of the most famous. In 1999, after anterior works by a number of authors among whom Kowalevskaya, Poincar´ e, Painlev´ e and more recently Received April 4, 2008; Revised April 4, 2008; Accepted August 1, 2008 Communicated by Alexei Borodin C The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. at Bibliothèque IRMAR on November 19, 2010 imrn.oxfordjournals.org Downloaded from
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Casale, G., and J. Roques. (2008) “Dynamics of Rational Symplectic Mappings and Difference Galois Theory,”International Mathematics Research Notices, Vol. 2008, Article ID rnn103, 23 pages.doi:10.1093/imrn/rnn103
Dynamics of Rational Symplectic Mappings and DifferenceGalois Theory
Guy Casale1 and Julien Roques2
1IRMAR UMR 6625, Universite de Rennes 1, Campus de Beaulieu 35042Rennes Cedex, France and 2Ecole Normale Superieure, Departement deMathematiques et Applications UMR 8553, 45, rue d’Ulm, 75230 ParisCedex 05, France
where C (p, n) stands for the set of length p increasing sequences i1 < · · · < ip in {1, . . . , n}and dr∧I stands for dri1 ∧ · · · ∧ drip. The lifting up of ω to R1V is the
(np
)-uple of functions:
hJω
( − ri − rε( j)i − ) =
∑wI r
JI ,
where J ⊂ {1, . . . , n} with #J = p and rJI stands for the determinant of the matrix (rε( j)
i ) i∈Ij∈J
.
Easy computations give
L Xω =∑
XwI dr∧I +∑
(−1)�+1wI∂ap
∂rkdrk ∧ dr∧I−{i�},
R1 X =∑
ai∂
∂ri+
∑ ∂ai
∂rkrε( j)
k
∂
∂rε( j)i
,
and (R1 X)(hJω) =
∑XwI r
JI +
∑wI
∂ai
∂rkrε( j)
k
∂rJI
∂rε( j)i
.
Thus one has hJL Xω = (R1 X)(hJ
ω).
For instance, starting from a X-invariant function h on V , one gets n + 1 order 1
differential invariants: h itself and h1dh, . . . , hn
dh.
Rational vector field: Let v be a rational vector field on V : v = ∑vi
∂∂ri
in local coordi-
nates. An order 1 frame determines a basis e1, . . . , en of TrV . The coordinates of v(r) in
this basis determine n functions hiv : R1V ��� C.
Higher order forms: A p-form ω on RqV determines functions on R1 RqV . The induced
function on Rq+1V are the pull-back by the natural inclusion Rq+1V ⊂ R1 RqV .
Theorem 4.1. The Lie algebra of Malgrange groupoid of an integrable (in the isotropic
way) symplectic map is commutative. �
Proof. Vector fields of the Lie algebra of Malgrange groupoid must preserve n + � in-
dependent functions H1, . . . , Hn+�: XHi = 0. Because they preserve Hi and the symplectic
form ω, they must preserve the symplectic gradients Xi.
On the subvarieties Hi = ci the integrable map preserves a parallelism given by
XH1 , . . . , XHn−�. This parallelism is commutative: [XHi , XHj ] = XHj Hi = 0 for all 1 ≤ i ≤ n + �
and 1 ≤ j ≤ n − �. Because the Lie algebra of vector fields preserving a commutative
parallelism is commutative (see for instance [8]), the theorem is proved. �
Van der Put and Singer proved in [23] that such an extension exists and that it is
unique up to isomorphism of difference rings. Furthermore, the field of constants of the
Picard–Vessiot extension is equal to the fields of constants of the base field.
Let us give the general lines of the proof of the existence of the Picard–Vessiot
extension. We denote by X = (Xi, j)1≤i, j≤n a matrix of indeterminates over k and we extend
the difference operator φ to the k-algebra
U = k[(Xi, j)1≤i, j≤n, det(X)−1]
by setting (φXi, j)1≤i, j≤n = A(Xi, j)1≤i, j≤n. Then, for any maximal difference ideal I (which
is not necessarily a maximal ideal) of U , the couple (R = U/I, φ) is a Picard–Vessiot ring
for the difference system (2).
Definition 5.3. The Galois group G of the difference system (2) is the group of difference
automorphisms of its Picard–Vessiot extension over the base field (k, φ). It is a linear
algebraic group over the field of constants C . �
If A“takes its values” in a particular linear algebraic group, then the Galois group
of (2) is a subgroup of this algebraic group. For later use, let us reprove this statement
in the symplectic case.
Lemma 5.1. Assume that k = C(z) and that A ∈ Gln(C(z)) is sympletic: t AJ A = Id where
J denotes the matrix of the symplectic form. Then there exists R a Picard–Vessiot exten-
sion of (2) containing a symplectic fundamental system of solutions of φY = AY; that is,
there exists Y a fundamental system of solutions of φY = AY with entries in R such thattYJY = Id. The corresponding Galois group belongs to the symplectic group. �
Proof. As above, we denote by X = (Xi, j)1≤i, j≤n a matrix of indeterminates over k and we
extend the difference operator φ to the k-algebra
U = k[(Xi, j)1≤i, j≤n, det(X)−1]
by setting (φXi, j)1≤i, j≤n = A(Xi, j)1≤i, j≤n. Remark that the ideal I of U generated by entries
of t X J X − Id is a difference ideal. Indeed this ideal is made of the linear combinations
Let us consider a rational embedding ι : P1(C) ��� V . The set of rational functions
on V , whose polar locus does not contain ι(P1(C)), is denoted by C[V ]ι.
Definition 6.4. The generic valuation νι(H ) of a function H ∈ C[V ]ι along ι is defined by:
νι(H ) = min{k ∈ N | Dk H (ι) �≡ 0 over P1(C)}.
It extends to H ∈ C(V ) by setting νι(H ) = νι(F ) − νι(G), where H = F/G with F , G ∈ C[V ]ι.
�
Definition 6.5. Let us consider H ∈ C[V ]ι and Tι = ι∗TV . We define H ◦ι : Tι ��� C, the
generic junior part of H along ι, by
H ◦ι = Dνι(H ) F (ι).
We extend this definition to H ∈ C(V ) by setting H ◦ = F ◦/G◦ for any F , G ∈ C[V ]ι
such that H = F/G. �
The reader will easily verify that the above definitions do not depend on a
particular choice of F , G.
Remark . Junior parts of F at points p ∈ ι(P1(C)) in the sense of [13] coincide with the
generic junior part over a Zariski-dense subset of P1(C). �
The interest of the generic junior part in our context consists in the fact that it
converts a first integral of f into a first integral of the variational equation.
Lemma 6.1. Let us consider H ∈ C(V ) a first integral of f . Then H ◦ι is a first integral of
the variational equation of f along ι. �
Proof. Let us consider F , G ∈ C[V ]ι such that H = F/G. The assertion follows by differ-
entiating vι(F ) + vι(G) times the equality (F ◦ f ) · G = (G ◦ f ) · F . �
Therefore, if H1, . . . , Hn+� are n + � first integrals of f , then (H1)◦ι , . . . , (Hn+�)◦ιare n + � first integrals of the variational equation of f along ι. However, in general,
these function are not functionally independent, even if H1, . . . , Hn+� are functionally
7.2 First proof of the main theorem: Picard–Vessiot approach
Lemma 7.1. Let (k = C(z), φ) be a difference field with field of constants C. Let H be a first
integral of the difference system with coefficients in C(z) : φY = AY. Let R be a Picard–
Vessiot extension for this difference system, which is supposed to be a domain. Let
Y ∈ Gln(R) be a fundamental system of solutions with coefficients in R. Then the function:
YH : Cn → C
y �→ H (z, Yy)
is invariant under the action of the Galois group G.
�
The action of the Galois group is given by the following: for (σ , y) ∈ G × Cn, we
set σ · y = C (σ )y where C (σ ) ∈ Gln(C) is such that σY = YC (σ ).
Proof. The first point to check is that, for any y ∈ Cn, H (z, Yy) ∈ C. A priori H (z, Yy) be-
longs to Frac(R), the Picard–Vessiot field, but φH (z, Yy) = H (φz, (φY)y) = H (φz, (A(z)Y)y) =H (φz, A(z)(Yy)) = H (z, Yy) so that H (z, Yy) is φ-invariant: H (z, Yy) belongs to the field of
contant of the Picard–Vessiot field, that is to C. Since H (z, Yy) lies in the field of constants
C, it is invariant by the Galois group: σ H (z, Yy) = H (z, Yy). But, since H has rational co-
efficients in z and that C(z) is invariant by the action of the Galois group, we also have,
for all σ ∈ G, σ H (z, Yy) = H (z, σYy) = H (z, YC (σ )y) = H (z, Y(σ · y)). �
Lemma 7.2. Suppose that the symplectic difference equation φY = AY with coefficients
in C(z) is integrable and that its Picard-Vessiot ring is a domain. Then the Lie algebra of
its Galois group is abelian. �
Proof. Let H1, . . . , Hn+� be n + � first integrals insuring integrability of the differ-
ence equation as in Definition 6.3. Lemma 5.1 ensures that there exists a Picard–
Vessiot extension R containing a symplectic fundamental system of solution Y ∈ Sp2n(R).Because
∑j
∂YHi
∂yjdyj =
∑j,k
∂ Hi
∂YkYk, jdyj
and invertibility of Y independence of the n + � forms∑
k∂ Hi∂Yk
dYk implies independence
the n + � differentials dYHi. Moreover, from Lemma 7.1, these functions are invariant
under the action of the Galois group.
Let X be a linear vector field on C2n in the Lie algebra of the Galois group.
By the assumption of integrability, all vector field preserving YH1, . . . , YHn+� are
combinations of the n − � first symplectic gradients:
X =n−�∑i=1
ci XY Hi
with ci ∈ C(y1, . . . , y2n).
Because the Galois group is a subgroup of Sp2n(C), if it preserves a function, it
preserves its symplectic gradient too. Thus one gets [XY Hj , X] = 0 for all 1 ≤ j ≤ n + �
and
dci =n+�∑i=1
cji dYHj
for some cji ∈ C(y1, . . . , y2n). If Y is a second vector field in the Lie algebra of the Galois
group, an easy computation shows that [X, Y] = 0. This proves the lemma. �
Proposition 7.1. If a linear symplectic difference system is integrable then the neutral
component of its Galois group is abelian. �
Proof. Invoking Lemma 5.2, we get that for every integer s the system φY = φs−1 A· · · φA ·AY is symplectic and integrable. The lemma follows from this observation and from
Lemma 5.3 together with Lemma 7.2 (indeed maintaining the notations of Lemma 5.3,
Lemma 7.2 implies that G ′0 is commutative because G ′ is the Galois group of a symplectic
and integrable system having a domain (Ri) as a Picard–Vessiot ring). �
First proof of theorem 7.1. It is a direct consequence of Theorem 6.1 and Proposition
7.1. �
7.3 Second proof using Malgrange groupoid
Let f : V ��� V be a rational map and ι : P1(C) ��� V a φ-adapted curve.
Lemma 7.3. If the Malgrange groupoid of f is infinitesimally commutative then the
Malgrange groupoid of Rq f is infinitesimally commutative too. �
Proof. Differential invariants of f give differential invariants for Rq f as described in
Section 4.2. Because Rq f is a prolongation, it preserves also the vector field Dj on RqV :
(Rq f )∗Dj = Dj. Hence Malgrange groupoid of Rq f must preserve these vector fields. A
direct computation shows that a vector field commutes with all the Dj if and only if it is