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

General Solution of F unctional Equations Defined b y Generic Linear-fractional Mappings F 1 : C N → C N and b y Generic Maps birationall y equiv alen t to F 1 . Konstan tin V. Rerikh ∗ Bogoli ub o v Lab oratory of Theoretical P h ysics, JINR, 141980, Du b na, The Mosco w Region, Russian F ederation Keyw ord s: b ir a tional mappings, discrete dynamical systems, functional equations, finite difference equa- tions; MSC: 14E05, 14E07, 37F10, 39A99, 39B99. Abstract W e consider a system o f birational functional equa tio ns (BFEs) (or finite-difference equa tio ns at w = m ∈ Z ) fo r functions y ( w ) of the fo rm y ( w + 1) = F n ( y ( w )) , y ( w ) : C → C N , n def = deg F n ( y ) , F n ∈ Bir ( C N ) , where the map F n is a given birational o ne of the g roup of all auto morphisms of C N → C N . The relation o f the B FE s with ordinar y differential equations is discussed. W e pr esen t a genera l solution of the ab o ve BFEs for n = 1 , ∀ N and of the ones with the map F n birationally equiv alent to F 1 : F n ≡ V ◦ F 1 ◦ V − 1 , ∀ V ∈ Bir ( C N ) . 1 In tro duction. Set of the Problem. By this pap er we start a discu ssio n of a general problem of int egrabilit y of birational functional equations for functions y ( w ) : C → C N in one complex v ariable w of the form y ( w + 1) = F n ( y ( w )) , y ( w ) : C → C N , w ∈ C , F n ∈ Bir ( C N ) . (1) F or w = m ∈ Z the abov e BFEs are a dynamical system with a d isc rete time or cascade. Here the map F n : y 7→ y ′ = F n ( y ) = f i ( y ) f N +1 ( y ) , i = (1 , 2 , . . . , N ), f i ( y ) for ∀ i are p olynomial s in y , deg F n ( y ) = max N +1 i =1 { deg( f i ( y )) } = n, is a give n birational one of the group of all automorphisms of C N → C N with co e fficien ts from C . By the w a y , the all said ab o ve is v alid and for the coefficient s from any algebraically closed field K . T h is f a ct considerably extends the frames of p ossible applications and using of BFEs (1). The BFEs (1) are equiv alen t to BFEs of more general form with a c hange ( w + 1) → ψ ( w ) where the map ψ is a giv en one from the li near-fractional group Aut ( C ). Actually , the c h a nge w 7→ τ ( w ) = ln  ( w − w 1 ) / ( w − w 2 )  / ln( λ 1 /λ 2 ), wh e re the terms w i , λ i , i = (1 , 2) are the fixed p oin ts and the eigen v alues of the map ψ ( w ) at these p oint s transform the map ψ into: ψ ( τ ) : τ 7→ τ ′ = τ + 1. Note also that discretization of a standard autonomous differential equation corresp onding to a ve ctor field (1) ( . x def = dx/dτ , τ ∈ C ) . x = v ( x ) , x ∈ U ⊂ V , (2) according to . x 7→ x ( τ + h ) − x ( τ ) h and τ 7→ hw , x ( hw ) 7→ y h ( w ) give s us a fun c tional equation for y ( w ): y h ( w + 1) = F n ( y h ( w )) , where F n ( y h ( w )) def = y h ( w ) + hv ( y h ( w )) . ∗ rerikh@thsun1.jinr.ru 1 Th us, if w e limit our s e lv es to the v ector fi e ld v ( y ) def = F n ( y ) − y h where F n ( y ) ∈ Bir ( C N ), then w e deriv e the BFEs (1).(Of course, there are and other views of discretization.) The dynamical sys t ems with a discrete time ( at w = m ∈ Z) and BFEs of the type (1 ) are an ob ject of many in ve stigatio ns of p roblem of their integrabilit y b oth algebraic (see (2)-(12 ) and others) and non-algebraic (see (13) -(17)). The algebraic integ rabilit y of BFEs (1) and dyn amic al systems of this type at N = 2 and f o r ∀ n ≥ 2 will b e a sub ject of another pap er. The problem of in tegrabilit y of the BFEs (1) for n = 1 and for any N is fully solv ed by the follo w ing theorem. 2 Main Result Let us p erform a transition f rom the mapping F n in C N to the mapp ing Φ n in CP N . Definition 1 Birational maps Φ n , Φ − 1 n are the imag es of the maps F n , F − 1 n in CP N y 7→ z : y i = z i /z N +1 , i ∈ (1 , 2 , · · · , N ) and are defin ed b elo w: Φ n : z 7→ z ′ , z ′ 1 : · · · : z ′ N +1 = φ 1 ( z ) : · · · : φ N +1 ( z ) , z , z ′ ∈ CP N , (3) φ i ( z ) = z n N +1 f i ( z l /z N +1 ) , i ∈ (1 , 2 , · · · , N + 1) , l ∈ (1 , 2 , · · · , N ) , (4) and φ i ( z ) are homogeneous p ol ynomials in z without any co m mon fact ors. The map Φ − 1 n : z ′ 7→ z ∼ φ ( − 1) ( z ′ ) is defin ed analogously . Theorem 1 The system of BFEs (1 ) at n = d eg F ( y ) = 1 , φ ( z ) = Az , where A is a complex matrix ( N + 1) × ( N + 1), has a general solution rationally d ep endin g on w and linear-fractionally on N p erio d ic arbitrary functions I j ( w ) , j ∈ (1 , · · · , N ) of w . W e assume that the matrix A, det( A ) 6 = 0 , is pr e liminary reduced to the n o rmal Jord a n form (see (18)): D = U AU − 1 , D = d ia g( D (1) , · · · , D ( r ) ) , r ≥ 1 , dim( D ( i ) ) = k i , (5) D ( i ) s,t = λ i δ s,t + δ s,t − 1 , s, t ∈ (1 , · · · k i ) , r X i =1 k i = ( N + 1) . (6) Then this solution has the form: y i ( w ) = P N +1 l =1 U − 1 i,l Y l ( w ) P N +1 l =1 U − 1 N +1 ,l Y l ( w ) , i = (1 , · · · , N ) , (7) Y ( w ) = { Y (1) 1 ( w ) , · · · , Y (1) k 1 ( w ) , · · · , Y ( r ) 1 ( w ) , · · · , Y ( r ) k r ( w ) } , Y ( i ) l ( w ) = ( λ i λ r ) w k i X m =1 C ( i ) l,m ( w ) I ( i ) m ( w ) , where (8) I ( i ) l ( w + 1) = I ( i ) l ( w ) , l = (1 , · · · , k i ) , C ( i ) l,m = Γ( w + 1) λ − ( m − l ) i Γ( w + 1 − m + l )Γ( m − l + 1) , where (9) C ( i ) m,m ≡ 1 , C ( i ) l,m ≡ 0 for l > m. In (7)-(8) th e fun c tions Y r k r ( w ) , I r k r ( w ) are identica lly equal to 1. Pro of: Let us consider the equ a tion for z ( w ) , z ( w ) : C → CP N , z ( w + 1) ∼ U − 1 ◦ D ◦ U z ( w ) . (10) 2 Then supp osing Y ( w ) = Y 1 ( w ) , Y 2 ( w ) , · · · , Y r ( w ) = U z ( w ) we h a v e equations for the function Y ( w ) : C 7→ CP N and the fun ct ion Y i ( w ) : C 7→ CP k i : Y ( w + 1) ∼ D Y ( w ) , Y i ( w + 1) ∼ D i Y i ( w ) . (11) Remark that the symb o l ∼ m eans a pro jectiv e similarit y of vec tors z ( w + 1) , Y ( w + 1) , Y i ( w + 1) to v ectors in the right-hand side of equations (10), (11). T hen the substitution Y ( i ) ( w ) fr o m (8) transforms equation (11) int o id e n tity . The fun c tions Y ( r ) k r ( w ) and I ( r ) k r ( w ) are normalized to 1 due to a homogeneous dep endence of the n u merato r and the denomin ator of expression for y i ( w ) (7) from the fu nctio ns Y ( w ). ⊳ W e can easily generalize this result. Let us introd uce the follo wing definition. Definition 2 Let call birational mapp in g F n birationally equiv alen t to another birational map F n ′ if there exists suc h a birational m ap p ing V suc h that the follo wing equalit y holds: F n = V ◦ F n ′ ◦ V − 1 . (12) ⊳ The follo wing theorem is v alid. Theorem 2 Let BFE (1) b e giv en b y the mappin g F n birationally equiv alen t to the mapp ing F 1 from Theorem 1: F n = V ◦ F 1 ◦ V − 1 . T hen the general solution of (1) for the function e y ( w ) is equ al to e y ( w ) = V ( y ( w )) , w here y ( w ) is giv en by formulae (7)-(9). The pro of is ob v ious . Remark 1 Let F k n = F n ◦ · · ·◦ F n b e a k -ite ration of the map F n and m = deg V b e a d eg r ee of the map V from Theorem 2. Then it is ob vious that a b oun d edness of deg F k n is a necessary condition for a birational equiv alence of the m ap F n to the map F 1 since F k n = V ◦ F k 1 ◦ V − 1 , deg F k 1 = 1 , i.e. deg F k n ≤ m 2 . 3 Ac kno wledgemen ts The author is grateful to R.I. Bogdano v, V.A. Isko vskikh, V.V. Kozlo v, V.S. Kuliko v, A.N. P ars h in, A.G. Sergeev, I.R. Shafarevich, D.V. T r e sc hev, I.V. V olo vic h and V.S. Vladimiro v for useful discussions and inte r est in the pap er. 3 References [1] V.I. Arnold, Y u.S. Il’y ashenk o, Ord inary Differen tial E quati ons, in Dyn amical S ystems, V ol. 1, eds. D.V. Anoso v and V.I. Arnold (Encyclopaedia Math. 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