A Novel Integrable Fourth-Order Difference Equation Admitting Three Invariants
69
If there exist N − 1 functionally independent invariants I l , l = 1, . . . , N − 1,
then it is possible to reduce the difference equation (2) to first-order one by solving
the relations:
I j = κ j ,
(5)
where κ j are the value of the invariants on a set of initial data. In such case we say
that the difference equation (2) is integrable.
This definition of integrability is very general. If some additional structures, like
Poisson or symplectic structures, are present, then the number of invariants needed
for integrability can be significantly lowered: this is the content of the discrete
Liouville–Poisson theorem [6, 28, 41].
In general to search for invariants is difficult procedure, see remark 1. For this
reason, several integrability indicators, that is, necessary conditions for integrability, have been introduced. A well-known integrability indicator, which is also an
equivalent definition of integrability, is the algebraic entropy [5, 13, 42]. Algebraic
entropy is defined for bi-rational maps of the complex projective space CP
N to
itself. Rational difference equations (2) which can uniquely solve with respect to
x n , that is, which are fractionally linear in x n , are equivalent to such maps. To see
this first notice that computing the orbit of such an equation is equivalent to iterate
the complex map Φ : C n → C n defined as follows:
Φ (x N −1 , . . . , x 0 ) = (Q (x n+N −1 , . . . , x n ) , x N −1 , . . . , x 1 ) .
(6)
The condition of unique solvability with respect to x 0 of (2) ensures us that the
map Φ has a rational inverse Ψ . Then, introducing the homogeneous coordinates
[X N −1 : . . . , X 0 : T ] by
(x N −1 , . . . , x 0 ) =
X N −1
T
, . . . ,
X 0
T
,
(7)
we have that the map (6) can be lifted to a rational map ϕ : CP
N
→ CP
N . Lifting
the inverse map Ψ to ψ : CP
N
→ CP
N we conclude that the map we obtain is
actually a bi-rational map of the complex projective space CP
N to itself.
Given a bi-rational map, we can take as measure of its complexity, in the sense
of Arnol’d [4], the growth of the number of intersections of the successive images
of a straight line with a generic hyperplane in complex projective space [42]. This
actually corresponds to compute the degrees of its iterates with respect to a generic
initial condition:
d k = deg ϕ
k , k ∈ N.
(8)
69
If there exist N − 1 functionally independent invariants I l , l = 1, . . . , N − 1,
then it is possible to reduce the difference equation (2) to first-order one by solving
the relations:
I j = κ j ,
(5)
where κ j are the value of the invariants on a set of initial data. In such case we say
that the difference equation (2) is integrable.
This definition of integrability is very general. If some additional structures, like
Poisson or symplectic structures, are present, then the number of invariants needed
for integrability can be significantly lowered: this is the content of the discrete
Liouville–Poisson theorem [6, 28, 41].
In general to search for invariants is difficult procedure, see remark 1. For this
reason, several integrability indicators, that is, necessary conditions for integrability, have been introduced. A well-known integrability indicator, which is also an
equivalent definition of integrability, is the algebraic entropy [5, 13, 42]. Algebraic
entropy is defined for bi-rational maps of the complex projective space CP
N to
itself. Rational difference equations (2) which can uniquely solve with respect to
x n , that is, which are fractionally linear in x n , are equivalent to such maps. To see
this first notice that computing the orbit of such an equation is equivalent to iterate
the complex map Φ : C n → C n defined as follows:
Φ (x N −1 , . . . , x 0 ) = (Q (x n+N −1 , . . . , x n ) , x N −1 , . . . , x 1 ) .
(6)
The condition of unique solvability with respect to x 0 of (2) ensures us that the
map Φ has a rational inverse Ψ . Then, introducing the homogeneous coordinates
[X N −1 : . . . , X 0 : T ] by
(x N −1 , . . . , x 0 ) =
X N −1
T
, . . . ,
X 0
T
,
(7)
we have that the map (6) can be lifted to a rational map ϕ : CP
N
→ CP
N . Lifting
the inverse map Ψ to ψ : CP
N
→ CP
N we conclude that the map we obtain is
actually a bi-rational map of the complex projective space CP
N to itself.
Given a bi-rational map, we can take as measure of its complexity, in the sense
of Arnol’d [4], the growth of the number of intersections of the successive images
of a straight line with a generic hyperplane in complex projective space [42]. This
actually corresponds to compute the degrees of its iterates with respect to a generic
initial condition:
d k = deg ϕ
k , k ∈ N.
(8)