General solution of functional equations defined by generic linear-fractional mappings F_1: C^N to C^N and by generic maps birationally equivalent to F_1

By this paper we start a discussion of a general problem of integrability of birational functional equations for functions y(w) : C → C N in one complex variable w of the form y(w + 1) = F n (y(w)), y(w) : C → C N , w ∈ C, F n ∈ Bir(C N ).

(

For w = m ∈ Z the above BFEs are a dynamical system with a discrete time or cascade. Here the map F n : y → y ′ = F n (y) = f i (y) f N+1 (y) , i = (1, 2, . . . , N ), f i (y) for ∀ i are polynomials in y, degF n (y) = max N +1 i=1 {deg(f i (y))} = n, is a given birational one of the group of all automorphisms of C N → C N with coefficients from C. By the way, the all said above is valid and for the coefficients from any algebraically closed field K. This fact considerably extends the frames of possible applications and using of BFEs (1).

The BFEs (1) are equivalent to BFEs of more general form with a change (w + 1) → ψ(w) where the map ψ is a given one from the linear-fractional group Aut(C). Actually, the change w → τ (w) = ln (w-w 1 )/(w-w 2 ) / ln(λ 1 /λ 2 ), where the terms w i , λ i , i = (1, 2) are the fixed points and the eigenvalues of the map ψ(w) at these points transform the map ψ into: ψ(τ

Note also that discretization of a standard autonomous differential equation corresponding to a vector field (1) ( .

according to .

x → x(τ +h)-x(τ ) h and τ → hw, x(hw) → y h (w) gives us a functional equation for y(w):

, where F n (y h (w)) def = y h (w) + hv(y h (w)). * rerikh@thsun1.jinr.ru

Thus, if we limit ourselves to the vector field v(y) def = Fn(y)-y h where F n (y) ∈ Bir(C N ), then we derive the BFEs (1).(Of course, there are and other views of discretization.)

The dynamical systems with a discrete time ( at w = m ∈ Z) and BFEs of the type ( 1) are an object of many investigations of problem of their integrability both algebraic (see ( 2)-( 12) and others) and non-algebraic (see ( 13) -( 17)).

The algebraic integrability of BFEs (1) and dynamical systems of this type at N = 2 and for ∀n ≥ 2 will be a subject of another paper.

The problem of integrability of the BFEs (1) for n = 1 and for any N is fully solved by the following theorem.

Let us perform a transition from the mapping F n in C N to the mapping Φ n in CP N .

n are the images of the maps

) and are defined below:

and φ i (z) are homogeneous polynomials in z without any common factors. The map Φ -1 n : z ′ → z ∼ φ (-1) (z ′ ) is defined analogously. Theorem 1 The system of BFEs (1) at n = degF (y) = 1, φ(z) = Az, where A is a complex matrix (N + 1) × (N + 1), has a general solution rationally depending on w and linear-fractionally on N periodic arbitrary functions I j (w), j ∈ (1, • • • , N ) of w. We assume that the matrix A, det(A) = 0, is preliminary reduced to the normal Jordan form (see (18)):

Then this solution has the form:

, where

l,m ≡ 0 for l > m.

In ( 7)-( 8) the functions Y r kr (w), I r kr (w) are identically equal to 1.

Proof: Let us consider the equation for z(w), z(w) :

Then supposing

Remark that the symbol ∼ means a projective similarity of vectors z(w + 1), Y (w + 1), Y i (w + 1) to vectors in the right-hand side of equations ( 10), (11). Then the substitution Y (i) (w) from ( 8) transforms equation (11) into identity. The functions Y kr (w) are normalized to 1 due to a homogeneous dependence of the numerator and the denominator of expression for y i (w) (7) from the functions Y (w).

We can easily generalize this result. Let us introduce the following definition.

Definition 2 Let call birational mapping F n birationally equivalent to another birational map F n ′ if there exists such a birational mapping V such that the following equality holds:

The following theorem is valid.

Theorem 2 Let BFE (1) be given by the mapping F n birationally equivalent to the mapping F 1 from Theorem 1:

Then the general solution of (1) for the function y(w) is equal to y(w) = V (y(w)), where y(w) is given by formulae ( 7)-( 9).

The proof is obvious.

a k-iteration of the map F n and m = degV be a degree of the map V from Theorem 2. Then it is obvious that a boundedness of degF k n is a necessary condition for a birational equivalence of the map F n to the map