Dynamics of birational plane mappings. The Arnold complexity difference equation

Dynamics of Birational Plane Mappings. The Arnold complexit y difference equation. Konstan tin V. Rerikh ∗ Bogoli ub o v Lab oratory of Th e oretical Physic s , JINR, 14198 0, Dubna, The Mosco w Region, Russian F ederation Keyw ord s: birational mappings, dynamics, algebraic geometry , d ynamica l systems, finite difference equa- tions. MSC: 14E05, 14E07, 37F10. Abstract W e consider a dyna mics of a g e neric bira tional plane ma p Φ n : CP 2 → CP 2 , CP 2 -image of the birational mapping (inv er se map is a lso rationa l) F n : C 2 → C 2 and its such imp ortant characteris- tic as the Arnold complexity C A ( k ), which is pr oportio na l d ( k ) = deg(Φ k n )- a deg ree o f k − iteration of the map Φ n , on the basis on alg ebraic-geometrical prop erties of such maps. Additional impo r- tance of this characteris tic follo ws from the V eselov conjectur e ab out the p olynomial b oundedness of the g ro wth of d ( k ) for integrable dynamical sy s tems with a discr ete time defined by birational plane maps. The autonomous linear difference equation with int e g er co efficient s for d ( k ) is ob- tained. This equation is fully defined by σ 1 nonnegative integers m 1 , · · · , m σ 1 that are deter mined by relations: Φ − m i n ( O α i ) = O ( − 1) β i , i ∈ (1 , 2 , · · · , σ 1 ), where Φ − m i n is m i -iteration of inv e rse map, O α i , O ( − 1) β i , α i , β i ∈ (1 , 2 , · · · , σ ) are indetermina cy p oin ts of the direct a nd inverse maps, σ 1 ≤ σ and σ is a num b er of indeterminacy p oints of Φ n , Φ − 1 n . If σ 1 is equal to zero that d ( k ) = n k , other- wise the g ro wth of d ( k ) is fully defined b y a ro ot spectrum of the s ecular eq uation asso ciated with the difference equation for d ( k ). The V eselov conjecture corresp onds to the r oot sp ectrum consisting of v alues b eing equal to mo dulo one. The author do esn’t s uppose that the r eader has a cquain tance with the alg e br aic geometry (AG) in CP 2 and the dynamical systems theory (DST) o r the functiona l equations since in the pap er there ar e given all needed definitions of used co ncepts of AG or DST and theorems. 1 In tro duction. S et of the prob lem and Main Result. Let consider the system of bir a tional f unctional equations (BFEs) for f u nctio n s 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 ab o ve BFEs are a dynamical system with discrete 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 o lyn omia ls in y , deg F n ( y ) = max N +1 i =1 { deg( f i ( y )) } = n, is a giv en birational m ap of the group of all automorphisms of C N → C N ( the Cr emo n a grou p or Bir ( C N )) with th e co efficie n ts fr om C. A p reliminary in vesti gation of the dynamics of the m a p ping F n is im p ortant in the con text of consid- eration of the in tegrabilit y p roblem of dynamical systems or the BFEs of the form (1). Su c h consideration is more con venien t and effectiv e to r ealize in CP N . Let us giv e the definition of the map p ing Φ n , the ∗ e-mail:rerikh@thsun1.jinr.ru 1 image of th e map F n in CP N , at N = 2 s i nce b elo w we shall mainly consider the dynamics of plane birational mappin gs. A t the transition to CP 2 y 7→ z : y i = z i /z 3 , i = (1 , 2) the maps F n , F − 1 n transform in to the maps Φ n , Φ − 1 n : Φ n : z 7→ z ′ , z ′ 1 : z ′ 2 : z ′ 3 = φ 1 ( z ) : φ 2 ( z ) : φ 3 ( z ) , z , z ′ ∈ CP 2 , (2) φ i ( z ) = z n 3 f i ( z l /z 3 ) , i ∈ (1 , 2 , 3) , l = (1 , 2) , (3) and φ i ( z ) are h o m ogeneous p olynomials in z without any common factors. The map Φ − 1 n : z ′ 7→ z , z 1 : z 2 : z 3 = φ ( − 1) 1 ( z ′ ) : φ ( − 1) 2 ( z ′ ) : φ ( − 1) 3 ( z ′ ) , z , z ′ ∈ CP 2 (4) is defined analogously . In the abstract and ab o v e we used suc h familiar concepts as ”in tegrabilit y”, ”in tegrable maps ” , ”in tegrable d ynamical systems”, and ”in tegrable functional equ ations”. In order to a v oid differen t u nder- standing of these terms, w e shall b e low giv e our definition of these co ncepts. The int egrability pr o b le m for dynamical systems and f unctional equations is s o lved if we obtain for th e BFEs (1) the family of first in tegrals I ( y ( w ) , w ) of dimension of 1 ≤ m ≤ N − 1 of the form : I ( y ( w ) , w ) = c ( w ) , c ( w ) : C → C m , c ( w + 1) = c ( w ) , (5) I ( y , w ) : (C N ⊗ C) → C m , (6) where arbitrary p eriod ic fu nctio n s c ( w ) in w parameterize the leve l lines of fi rst in tegrals. In addition to these m fi rst integ r al s w e alw a ys h a ve one m o r e fi rst transitiv e in tegral parameterized b y the p erio dic function w → w + β ( w ) , β ( w ) : C → C , β ( w + 1) = β ( w ) . (7) If in teger m = N − 1, we can sp ea k ab out full in tegrabilit y , otherwise p artia l integrabilit y of equation (1) at 1 ≤ m < N − 1. W e ha ve a general solution of equations BFEs (1) if w e obtain the solution of BFEs (1) in the explicit form y = Y ( w , c ( w )) with c ( w ) : C → C N − 1 . If I ( y , w ) is a rational function of ( y , w ) or a r a tional function of y and f r a ction-linear one in v ariables τ ( w ) where τ ( w ) are v ariables of the f o r m λ w , th en we can sp eak ab out the algebraic in tegrabilit y of BFEs (1). If I ( y , w ) ∈ H ol ( y , w ), i.e. I ( y , w ) is a holomorph ic function of v ariables, then w e can sp eak a b out non-alge b raic integrabilit y of BFEs (1). If the algebraic integ r a b ili ty is the sub j e ct for using algebra-geometrical metho ds, then non-algebraic in tegrabilit y is the sub ject for usin g classical metho ds an d theorems from the theory of dyn a m ic al systems due to H. Poincare, C.L. Siegel, G.D. Birkhoff, A.N. Kolmogoro v, V.I. Arnold, J. Moser, D.V. Anoso v and others (see Arnold and Il’y ashen k o (1988), Anoso v et al. (1988)), and also using classical results of the n u m b er theory and the tran s c en d en tal n u m b er theory (see, for example, (Moser, 1994), (Rerikh, 1995a), (Rerikh, 1997), (Rerikh, 1998b) as examples of using classical resu lt s of A. Baker (Bak er , 1990 ), (Bak er , 1966), (Bak er , 1967a,b, 1968), (Bak er , 1971), (Bak er and W ¨ ustholz, 1993 ) and N.I. F eldm an (F eldman, 1982), (F eldman, 1968)). In the pap er, we sh a ll not discuss the in tegrabilit y problem in more detail. There are also other definitions of the concept int egrabilit y . The A.P . V eselo v defin ition as applied to the dyn amical systems of th e form (1) acting in the plane (bip olynomial m ap s , (V eselo v , 1989), (V eselo v , 1991)) is as f o llows: ”The map F n is inte grab le if there exist another map Ψ for w hic h Ψ m 6 = F k n ∀ k , m ∈ Z, w here Ψ m is the m th iteration of the map Ψ but F k n is the k th iteration of th e map F n , commuting with F n : Ψ ◦ F n ≡ F n ◦ Ψ ”. The Moser definition means the existence of a holomorphic map H ( y ) : u = H ( y ) , C N → C N that transforms the maps F n to its linear part H ◦ F n ◦ H − 1 ≡ A def = ∂ F n ∂ y | y = y 0 , where y 0 is a fixed p oin t of the map F n but the matrix A d e fi nes the linear part of the map F n . S uc h a definition is natur al for the theory of dynamical systems. (see examples of non-algebraica lly in tegrable dynamical systems (Rerikh, 1992), (Rerikh, 1995b) (Rerikh, 1995a), (Rerikh, 1997), (Rerikh, 1998a)). This Moser d efinition of inte grability is in fact a lo ca l concept in th e neigh b ourho od of a fixed p oin t of a map as we ll as a concept of a lo cal n o n -i n tegrabilit y in a n eig hb ourho od of a fi xed p oi nt of a map . (See 2 the Moser example (Moser, 1960) of n o n -in tegrable cub ic bip olynomial map in the neigh b ourho o d of th e zero elliptic fi xed p oin t.) V.I. Arnold in pap ers (Arnold, 1990b), (Arn o ld , 1990a) introdu c ed and inv estigated suc h a c harac- teristic of a dynamical system as the top olog ical c omplexity of the in tersection of a sub manifold X of manifold M , m o ved b y a dynamical system, a s mooth mapp ing A : M → M with the other giv en compact smo oth subm an if old Y of M : Z k = ( A k X ) ∪ Y . In the simplest case for plane mappings Φ the complexity C Φ A ( k ) ≡ Z k can b e defined (V eselo v , 199 2 ) as the n umb e r of int ersection p o ints of a fixed curve Γ 1 with the image of another curv e Γ 2 under the k th iteration of Φ: C Φ A ;Γ 1 Γ 2 ( k ) = #  Γ 1 ∩ Φ k (Γ 2 )  . If the mapping Φ is a birational one from the group BirCP 2 and the curv es Γ 1 , Γ 2 are algebraic curv es in CP 2 , then it is easy to see that the growth of C Φ A ;Γ 1 Γ 2 ( k ) will in general b e as follo ws: C Φ A ;Γ 1 Γ 2 ( k ) = deg(Γ 1 )deg(Γ 2 ) d Φ ( k ) ≤ deg(Γ 1 )deg(Γ 2 )(degΦ) k , where d Φ ( k ) = d eg(Φ k ) is th e degree of the mapping Φ k = Φ ◦ Φ ◦ · · · ◦ Φ, wh ic h agrees well with ge n eral Arnold’s results for smo oth mappings and diffeomorphisms (Arnold, 1990b), (Arnold, 1990a ). The Arnold complexit y w as found to b e an imp o r ta nt charact eristic in the con text of th e in tegrabilit y of suc h d ynamica l systems. A.P . V eselo v int r oduced in (V eselo v , 19 91 ), (V eselo v , 1992) the conjecture ab out a p olynomial growth of th e Arnold complexit y d ( k ) with k for inte grable plane birational mapp ings and p ro ved it for in tegrable b ipolynomial ones in (V eselo v , 1989) ( d ( k ) is b ou n ded by a constan t). T o b e more exac t we reform ulate th e A.P . V eselo v co n ject u re as ” all inte grable bir a tional mappings ha ve a p olynomially b ounded gro wth of the Arn old complexit y d ( k ) on k ”. The v alidit y of the A.P . V eselo v conjecture wa s also co n firmed for many concrete integrable mapp ings b y d iffe r en t researc hers so that it is actual to p ro ve it in a general ca s e. Th is pap er is the first step in th is direction. Th e aim of th is pap er is to discuss the d ynamics of generic birational p la ne mapp i ngs in the frames of their algebraic-geo metrical pr o p erties and obtain the autonomous linear d ifference equation for the Arn o ld complexit y d ( k ). The main r esu lt of the pap er is the obtained autonomous linear difference equation for d ( k ). This equation is fu ll y defined by σ 1 nonnegativ e inte gers m 1 , · · · , m σ 1 that are determined by relations: Φ − m i n ( O α i ) = O ( − 1) β i , i ∈ (1 , 2 , · · · , σ 1 ), where Φ − m i n is the m i -iteration of the in verse map, O α i , O ( − 1) β i , α i , β i ∈ (1 , 2 , · · · , σ ), are indeterminacy p oin ts of the direct and in verse maps, σ 1 ≤ σ and σ is a n umb er of indeterminacy p oin ts of Φ n , Φ − 1 n . If σ 1 is equal to zero, then d ( k ) = n k , otherw ise the growth of d ( k ) is fully defined b y a ro ot sp ec trum of the secular equation associated with the difference equation for d ( k ). The A.P . V eselo v conjecture corresp onds to the ro ot sp ectrum consisting of v alues b eing equal to mo dulo one. Thus, this equation gi ves the possib il ity to pr e s e nt all sets of n umbers m 1 , · · · , m σ 1 corresp onding to int egrable mapp ings if the A.P . V eselo v conjecture is true. In the follo w ing section, we shall p erform a br ief excursus in to the theory of the Cremona trans- formations in th e plane follo wing (Hudson, 1927), Snyder et al. (1970), (Isk ovskikh and Reid, 1991), (Shafarevic h , 1977), (Coble, 1961). In Section 3 we int r oduce a new notion –the decomp osit ion of sets of the indeterminacy p oin ts of direct Φ n and in verse Φ − 1 n maps. Then in Section 4 w e obtain the main equa- tions of the d y n amic s of a generic b irati onal map and the difference equ a tion for the Arn o ld complexit y d(k). Differen t examples f o r illustration of Sections 2- 4 are set in App endices A, B. 2 Brief excursus in to the algebraic geometry Let z = ( z 1 , z 2 , z 3 ) b e a p oin t of the pro jectiv e plane CP 2 . Let us consider a general curve of d eg r e e µ f µ ( z ) defined b y the equation f µ ( z ) = X | l | = µ c l z l = 0 , l def = ( l 1 , l 2 , l 3 ) , | l | def = l 1 + l 2 + l 3 , (8) 3 whic h has, in general, µ ( µ +3) 2 free parameters. 2.1 Linear systems of curves Definition 1 Let P = ( z ∗ 1 , z ∗ 2 , z ∗ 3 ) b e a p oint of the cu rv e (8) and let z ∗ 3 b e the coordinate of the p o int P w h ic h is n on zero, but, therefore, w e can assign z ∗ 3 = 1 as a r e sult of th e c hange P 7→ P /z ∗ 3 . The p oin t P is called an r -fo ld one of the cur ve (8) if f µ ( z ) h as the f o llowing f orm in th e system of co ordinates z ′ : z ′ 1 = z 1 − z ∗ 1 z 3 , z ′ 2 = z 2 − z ∗ 2 z 3 , z ′ 3 = z 3 f µ ( z ) = f ′ µ ( z ′ ) = µ X k = r z ′ 3 µ − k u k ( z ′ 1 , z ′ 2 ) , (9) r def = mult( f µ ( z )) | z = z ∗ , mult def = multiplic ity , (10) where u k ( z ′ ) are homogeneous p olynomial s of degree k in v ariables z ′ 1 , z ′ 2 , but the first function u r ( z ′ ) in expansion (9) defines r tangents for the cur v e at the p oin t P . An r -fold p oin t imp oses r ( r +1) 2 = P r − 1 i =0 ( i + 1 ) conditions ensu r ed for a cu rv e (8) of the form (9) and is called a simple, doub le , triple one, if r is equal 1 , 2 , 3 and so on. ⊳ Let us giv e a definition of a linear sys t em of cur v es which is imp ortan t in what follo ws. Definition 2 (F or more details see Snyder et al. (1970) and also all references therein on results and notions r evi ewed here.) The system of plane cur ves f µ of degree µ is rep r ese n ted b y an equation of the form f µ = k +1 X i =1 c i f i ( z ) = 0 , (11) where the f unctions f i ( z ) are homoge n eo u s p olynomials from z = ( z 1 , z 2 , z 3 ) of th e same order µ and are linearly ind ep endent, is a linear s yste m of cur v es (LSC) of d imension k . Definition 3 A p oint B ( r j ) j whic h is at least an r i -fold one for eac h curve of the system is calle d an r i -fold basis p oin t (see Definition 1) b ut a join of all b asis p o ints is called a b a sis set or a base of the LSA C B = S j B ( r j ) j , where j ∈ 1 , 2 , · · · , N B . Thus, the base is fully defi ned by t wo sets: the set of b asis p oin ts and the set of theirs multiplicit ies on the LSA C which are linked w ith eac h other, as it is set ab o v e. Count in g separately the conditions imp ose d b y all basis p o ints necessary for reduction from the general curv e (8) to th e LSA C (11), w e ha ve the virtu a l dimension K = µ ( µ + 3 ) 2 − N B X j =1 r j ( r j + 1) 2 . In certain cases the conditions are not indep enden t so that the effectiv e d imension is k = K + s, (12) where s is th e n u m b er of ind epend e nt relations among the linear conditions imp osed b y the base B on curv es of order µ . A system for which s = 0 is said to b e regular, otherw ise irr eg u la r with irregularit y ( sup erabu n dance) s . The effectiv e genus p of a general curve of the irr ed ucible system coincides with vir tual P and is p = P = ( µ − 1)( µ − 2) 2 − N B X j =1 r j ( r j − 1) 2 . (13) 4 F or reducible curves the effectiv e gen u s p equals p = P + c − 1 , (14) where c is a num b er of comp onen ts of a redu ci ble curve . Th e n umb er of v ariable intersecti ons of t wo curv es of the system is the grade D D = µ 2 − N B X j =1 r 2 j . (15) The num b ers K, D and P satisfy the relation K = D − P + 1 , so that k = D − p + s + 1 for an irreducible system, where s is ≤ p , b e cause D ≥ ( k − 1). The num b ers D , p and k are inv arian t under birational mapp ings. A linear system of dimension k = 1 , 2 , 3 is called a p encil, a net and a we b , resp ectiv ely (see also Remark 1). ⊳ Remark 1 Definition 2 can b e extended to the case of the linear systems of curv es of w hic h the system of the b asis r -fold p oin ts includes some non-ordinary (extra ordinary) r -fold ones (see Defin ition 4 b elo w). Definition 4 The r -fold basis p oin t of th e linear system of cur v es is called a non -ord inary one if at this p oin t the linear system of curves satisfies some additional tangency conditions as the existence of r 1 , 1 ≤ r 1 ≤ r , common tangen ts ( r -fold p oin t of a simp le cont act) or the existence of some fi xe d curv e touc hing up on these common tangent s and osculating with eac h cur ve of the system ( r -fold p oin t of higher con tact). Eac h r -fold non-ordinary p oin t can b e repr ese nted by the system of in fi nitely near ordin a r y p oin ts and b e resolv ed using th e tec hnics of r eso lu ti on of sin gu larities of plane curves (see (Hudson, 1927), Chapter VI I and also b elo w Section 2.5 ). Let us giv e a definition of a bir a tional mapping Φ n : CP 2 → CP 2 . 2.2 Definition of B ir at ional Map, No ether t heorem and Quadratic maps Definition 5 Bir ational map. A mapping Φ n : z 7→ z ′ , z , z ′ ∈ CP 2 in (2), where φ i are h o m o geneous p olynomials in z , i = (1 , 2 , 3), of degree n , is called a birational mapping if it assigns one-to-one cor- resp ondence b et ween z and z ′ , wh il e the in v erse mapping is giv en by (4 ) and it is also rational (gen us p = 0), φ ′ i b eing also homogeneous p olynomials in z ′ , moreo ver, φ i and φ ( − 1) i ha ve no common factors. Asso cia ted with Φ n and Φ − 1 n the linear systems of curv es φ, φ ( − 1) of dimension k = 2, gen us p = 0 and grade D = 1 φ = c 1 φ 1 + c 2 φ 2 + c 3 φ 3 , (16) φ ( − 1) = c ( − 1) 1 φ ( − 1) 1 + c ( − 1) 2 φ ( − 1) 2 + c ( − 1) 3 φ ( − 1) 3 (17) (for c i , c ( − 1) i ∈ C) are fu lly given b y theirs bases B , B − 1 (for b a s es of LSA Cs asso ciated with maps w e shall use sy mb ols B def = O , B − 1 def = O − 1 , r α def = i α , r ( − 1) β def = i ( − 1) β , α ∈ (1 , 2 , · · · , σ def = N B ) , β ∈ (1 , 2 , · · · , σ ( − 1) def = N B − 1 ) , ( σ = σ ( − 1) ) . ) defin e the first and second r a tional n e ts which are images of n e ts of lines. The basis p oin ts O α , O − 1 β are indeterminacy ones f o r the maps Φ n , Φ − 1 n and are called fu ndamen tal ones (or F-p oin ts). The equalit y of genus p to zero is a necessary and sufficient condition for the b irat ionalit y of the rational m ap Φ n (2). ⊳ 5 Theorem 1 (M. No ether) Ev ery C remona plane mapp in g Φ n (2) can b e resolv ed in to quadratic mappings Φ n = C ◦ Q 1 ◦ Q 2 · · · ◦ Q j , where C is a collineation (linear mapp ing in CP 2 ), but mapp ings Q 1 , · · · , Q j are quadr a tic ones. . ⊳ A t the end , we s hould giv e the definition of the main ob ject– the generators of the Cremona group, namely , birational qu adratic mappings. Definition 6 Any generic quadratic Cremona mapping is generated by a comp ositio n Φ 2 ≡ B − 1 ◦ I s ◦ B 1 , (18) where B : z 7→ j ( − 1) = B z , B 1 : z 7→ j = B 1 z (19) are generic linear mapp ings from the PGL(2 , C) group and I s is an inv olution, th e standard Cremona mapping with th r ee simple F -p o ints O α ∈ { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } and th ree principal lines J α = ( z : j α ( z ) = 0) ∈ { ( z 1 = 0) , ( z 2 = 0) , ( z 3 = 0) } : I s : z 7→ z ′ z ′ 1 : z ′ 2 : z ′ 3 = z 2 z 3 : z 1 z 3 : z 1 z 2 . (20) I s : J β → O ( − 1) β , O ( − 1) β ∈ ((1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1)) , (21) I ( − 1) s : z ′ → z z 1 : z 2 : z 3 = z ′ 2 z ′ 3 : z ′ 1 z ′ 3 : z ′ 1 z ′ 2 , (22) I ( − 1) s : J ( − 1) α → O α , O α ∈ { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } . (23) In the triangular frame of reference (19) mapping (18) take s a v ery simple form Φ 2 : j ( z ) 7→ j ( − 1) ( z ′ ) j ( − 1) 1 ( z ′ ) : j ( − 1) 2 ( z ′ ) : j ( − 1) 3 ( z ′ ) = j 2 ( z ) j 3 ( z ) : j 1 ( z ) j 3 ( z ) : j 1 ( z ) j 2 ( z ) . (24) The mapp ing Φ 2 (18) is sp ecializ ed if t wo or three F -p oin ts are adjacen t or infi nite ly near (Isko vs kikh and Reid, 1991) and has, resp ectiv ely , the follo wing forms: Φ 2 a ≡ B − 1 ◦ I a ◦ B 1 , I a : z 7→ z ′ z ′ 1 : z ′ 2 : z ′ 3 = z 2 2 : z 1 z 2 : z 1 z 3 , (25) Φ 2 b ≡ B − 1 ◦ I b ◦ B 1 , I b : z 7→ z ′ z ′ 1 : z ′ 2 : z ′ 3 = z 2 1 : z 1 z 2 : ( z 2 2 − z 1 z 3 ) , (26) moreo v er, in volutions I a , I b from (25), (26) can b e resolved as a comp osit ion of t wo or four, b ut not fewer, general mappings (18), r esp ectiv ely (see (Hudson, 1927 ), c hapter I I I, p p. 35,37). An y t wo memb ers of the net (25) touc h one another an d ha v e a fixed common tange nt j 1 ≡ z 1 = 0, bu t ones of the net (26) ha ve a fi xed common tangen t j ≡ z 1 and osculate a fixed conic z 2 2 − z 1 z 3 . Th ese tangency conditions are sim ulated by t wo or thr e e infin ite ly near p oint s, so as equations (31)-(35) remain correct. 2.3 Prop erties of Birational Mapping Definition 7 Prop erties. T he one-to-o n e corresp ondence for direct Φ n and inv erse Φ − 1 n mappings do es not hold only at indeterminacy or fundamental p oin ts ( F -p oin ts) O α ∈ O , O ( − 1) β ∈ O ( − 1 ) , α, β = (1 , 2 , . . . , σ ) , i.e., co m mon b asis p oin ts of m ultiplicities i α , i ( − 1) β for functions φ k ( z ) , φ ( − 1) k ( z ) , k = (1 , 2 , 3) , and the asso cia ted linear systems φ (16) and φ ( − 1) (17), resp ectiv ely , and on p rincipal or exceptional curv es or exceptional divisors J β , J ( − 1) α , α, β = (1 , 2 , . . . , σ ), J β def = { z : j β ( z ) = 0 } , J ( − 1) α def = { z : j ( − 1) α ( z ) = 0 } , α, β = (1 , · · · , σ ) , (27) 6 where j β , j ( − 1) α are homogeneous p olynomials in z of degrees i ( − 1) β , i α , resp ectiv ely , moreo v er, the p oin ts O α , O ( − 1) β blo w up into the curv es J ( − 1) α , J β of degrees i α , i ( − 1) β and the curves J ( − 1) α , J β blo w d own int o the p oin ts O α , O ( − 1) β , O α → ← J ( − 1) α , deg J ( − 1) α = i α , (28) O ( − 1) β → ← J β , deg J β = i ( − 1) β , (29) resp ectiv ely (see the concept of σ -pro cess of b lo wing up of sin g u la rities in the theory of ordinary different ial equations (Arnold, 1988) and the Ko daira theorem in the algebraic geometry (Griffiths and Harris, 1978)). ⊳ Definition 8 A fu ndamen tal p oin t is called ordinary if at this p oin t there are no any additional tangency conditions. In sp ecial cases of non-ordinary (extra-ordinary) F-p oin ts (see Definition 4), tange n cy condi- tions of an y t wo mem b ers of the asso ciat ed linear s yste m s are expr e ssed as m u lt ip li cities of infin itely near p oin ts (Isko vskikh and Reid, 1991), or adjoin t p oin ts in the termin o logy of (Hudson, 1927). Eac h r -fold non-ordinary p oin t can b e repr e s e nted by the system of infi n ite ly near ordin a r y p oin ts and b e transformed in to ordin a r y ones using the tec h nics of r eso lu ti on of singularities of plane cur v es (see (Hudson, 19 27 ), Chapter VI I, Theorem 3 b elo w and examples : 1, 2 in App endices A, B). Theorem 2 Jacobian. T he Jacobia n J of the mapping Φ n equals J =     ∂ φ k ∂ z i     ∼ σ Y α =1 j α . d eg J = 3 n − 3 (30) The formula corresp o n ds to a birational map with ord inary F − p oints but in the case of non-ordinary ( infi nitel y near) p oin ts it r emains correct if we assign to the infinitely near F − p oin ts the same f actors j α with m u lt ip li cities in accordance with a c haracteristic of the map (see b elo w Definition 2 and also examples 1, 2 in App endix A). The d et erm inat ion of the Jacobian is a v ery simple w ay to find the pr incipal curves. The prin ci p al cur v es J α ( J ( − 1) β ) intersec t eac h other only at f u ndamen tal p oin ts O α ( O ( − 1) β ). Remark 2 Characteristic. The set of num b ers char (Φ n ) = { n ; i 1 , i 2 , . . . , i σ } , , i 1 ≥ i 2 ≥ · · · ≥ i σ , where i α are the m u lt ip li cities of all indeterminacy p oin ts O α of the mapping Φ n , including infin itely near ones, is called the c haracteristic of mapping Φ n . W e shall denote the infin ite ly near F − p oint s b y the star: i ∗ α and O ∗ α . Next in s im p lic ity after quadratic birational map w ith char (Φ 2 ) = { 2; 1 , 1 , 1 } is a cubic map w ith char (Φ 3 ) = { 3; 2 , 1 , 1 , 1 , 1 } and then t wo quartic maps with char (Φ 4 ) = { 4; 2 , 2 , 2 , 1 , 1 , 1 } and { 4; 3 , 1 , 1 , 1 , 1 , 1 , 1 } . The general mapp ing with a giv en c h a r a cteristic d e p end s on 2 σ + 8 p a rameters. ⊳ Remark 3 Characteristic n umbers. Let i ( − 1) β α b e the multiplicit y of curve J ( − 1) α at p oin t O ( − 1) β and i αβ b e that of curv e J β at O α . Then w e hav e the equalit y i αβ = i ( − 1) β α and the follo wing relatio n s b et ween n u m b ers i α , i ( − 1) β , i αβ , expressing certai n geometrical facts (summing in the left column o v er α an d in the righ t one ov er β from 1 to σ ): X i α = 3( n − 1) , X i ( − 1) β = 3( n − 1) , (31) X i 2 α = n 2 − 1 , X i ( − 1) β 2 = n 2 − 1 , (32) X i αβ = 3 i ( − 1) β − 1 , X i αβ = 3 i α − 1 , (33) X i α i αβ = i ( − 1) β n, X i ( − 1) β i αβ = i α n, (34) X i αβ i αγ = i ( − 1) β i ( − 1) γ + δ β γ , X i αβ i γ β = i α i γ + δ αγ . (35) 7 The conditions (31), (3 2 ) mean that the asso c iated linear systems (16), (17) hav e the grade D = 1, the gen us p = 0, the d ime nsion k = 2, and th e sup erabund ance s = 0. The cond itions (33), (35) p ro vid e rationalit y of th e curve s J β , J ( − 1) α (27), and that their d egrees are i ( − 1) β and i α , resp ectiv ely . In the case of n on-o r dinary F -p oin ts the total n umb er of distinct F -p oin ts need n o t b e the same for th e dir ect (2) and inv erse (4) mappings. In the sp ecial cases, if at s ome β in the left parts of equations (33) and (3 5) (at β = γ ) j β breaks u p in to 1 ≤ ν ≤ i ( − 1) β comp onen ts, then the left parts of th e se equ a tions must b e replaced by X i αβ = 3 i ( − 1) β − ν , X i 2 αβ = i ( − 1) 2 β + ν . (36) Analogous c hanges in the righ t parts of equations (33) and (35) at some α m u st b e made. The up p er limit f o r σ of the total num b er of F -p oin ts is giv en b y the follo win g form ula: σ ≤ 2 n − 1 , if n > 1 . ⊳ 2.4 Beha viour of algebraic curves and LSA C under the action of the birational map (2) Remark 4 Consider prop erties of a general curv e f µ ( z ′ ) = 0 of degree µ und er the mapping (2). By map (2), the curve f µ ( z ′ ) is mapp ed in to the cur v e f µ ( φ ( z )) = f ′ µ ′ ( z ) of degree µ ′ = µ n ; moreov er, eve ry p oin t O α whic h is i α -fold on φ ( z ) is µi α -fold on f ′ µ ′ . If f µ ( z ′ ) has multiplicities γ ( − 1) β at p oin ts O ( − 1) β , then (deg ( j β ) ≡ i ( − 1) β ) f µ ( z ′ ) = f ′ µ ′ ( z ) σ Y β =1 j γ ( − 1) β β , µ ′ = µ n − σ X β =1 γ ( − 1) β i ( − 1) β ; (37) moreo v er, f ′ µ ′ has multiplic ities γ ′ α at O α (see the meaning of i αβ in Remark 3): γ ′ α = µi α − σ X β =1 i αβ γ ( − 1) β . (38) If f µ ( z ) = 0 is a general curv e of a linear system of curves of dimens io n k but f ′ µ ′ is its image under the map Φ n (2), that genus p and dimension k of the LSAC are inv arian ts. 2.5 Birational equiv alence and resolution of singularities of plane curves Let us in tro duce the defin iti on of b ir a tionally equiv alen t mapping in CP 2 . Definition 9 A mapping Φ n ∈ BirCP 2 , z → z ′ ∼ φ ( z ) , is birationally equiv alen t (or conjugated in terminology of Hudson’s b o ok) to a mapping U : y → y ′ , y ′ ∼ u ( y ) y , y ′ ∈ CP 2 , if th e r e exists a birational mapping V m : z → y , y ∼ v ( z ) of degree m s u c h that Φ n ≡ V − 1 m ◦ U ◦ V m . Due to the stand a r d metho d of resolution of singularities of p la n e curves (see (Hudson, 1927), chapter VI I, p.129) the follo w ing p roblems can b e solved by applying a comp osition of the corresp onding Cr e m o n a quadratic mapp ings: 1. to tr an s form an y n on-o r dinary m ultiple p oint in to a net of s imple p o ints; 2. to r eso lve any non-ordinary m u lt iple p oin t int o an equiv alen t set of ordin ary multiple p oin ts; 3. to tr an s form an y algebraic curve in to one ha v in g ord inary m ultiple p oin ts only; 4. to tr an s form an y lin e ar system of algebraic cur v es into one having ordinary base p oint s only . 8 As the consequence of the standard method of resolution of singularities of plane cur v es the follo wing theorem is r e p resen ted to b e v alid. Theorem 3 Any mapp ing Φ n (2) with non-ordinary F -p oints (see Definition 4) b y the corresp ondin g transformation of birational equiv alence (see Definition 9) is transformed into some mapp ing Φ n ′ = V − 1 ◦ Φ n ◦ V with only ordinary F -points w here degree of this mapping n ′ ≥ n but the mapping V is a comp ositio n of a necessary (for resolution of all infi nitely near p oin ts) num b er of quadr atic mappings. (see Definition 6 ) ( see example in App endix B) ⊳ F or illustration of this Section w e set the examples of quadratic and cub ic maps with ordinary and non-ordinary ind et erm inac y p oin ts in App endix A but the example of using Theorem 3 is in App end ix B. Belo w w e shall deal with mappings ha vin g only ordin a r y indeterminacy p oin ts sup p osing that maps with non-ordin a r y ones was previously r e placed b y b irat ionally equiv alent maps w it h the help of Theorem 3. 3 Decomp osition of the set of indeterminacy p oin ts Let us consider orbits of ind etermin acy p oin ts O α ∈ O an d O ( − 1) β ∈ O ( − 1) relativ e to the ac tion of the in verse Φ − 1 n (4) and the d irect map Φ n (2), resp ectiv ely , and let us in tro duce the follo wing defin itions. Definition 10 The orbit O z of a p oin t z w it h resp ect to the mapp ing Φ − 1 n (4) is the s e t of p oin ts O k z = Φ − k n ( z ) = (Φ − 1 n ) k ( z ) , k ∈ Z + , where Z + is the set of n on-nega tive in tegers. The orb it O ( − 1) z of a p oin t z with r espect to Φ n (2) is defined analogously , O ( − 1) k z = Φ k n ( z ) = (Φ n ) k ( z ) and Φ k n ( z ) def = Φ n (Φ n ( . . . ( z ) . . . )) , Φ − k n ( z ) def = Φ − 1 n (Φ − 1 n ( . . . ( z ) . . . )) (see, for example, (Arnold and Il’ya s henk o, 1988), (Arnold, 1988)). Definition 11 If the n u m b er k of the p oints of the orbit O z of the p oin t z with resp ect to the mapping Φ − 1 n (4) is fi nite, where non-negativ e int eger k is a minimal integer defin ed by the condition O k z = Φ − k n ( z ) = z , k ∈ Z + , then the p erio dic p oin ts  Φ − m n ( z )  , m =  0 , 1 , · · · k − 1  form the set O ( cy c le ) z – a cycle of index k or p eriod k of the mapp ing Φ − 1 n ( z ), but z is a fixed p oin t of the mapping Φ − k n ( z ). A cycle of index k of the mapping Φ n ( z ) (2 ) is defin ed similarly with help of the changes: O z 7→ O ( − 1) z and Φ − m n ( z ) 7→ Φ m n ( z ) , (see (Arnold and I l’ yashenk o, 1988)). Let us in tro duce the notion of a tail of th e cycle. Definition 12 Let us call a subset O ( tail ) z of the set O z a tail of the length l of the cycle O ( cy c le ) y where non-negativ e in teger l is a minimal intege r defined b y th e condition O ( tail ) z of length l : { y = Φ − l n ( z ) , w h ere y ∈ O ( cy c le ) y } , l ∈ Z + , so that the p oin t z is th e b eginning of the tail, but the p oint y is the b eg in n ing of the cycle and do es not b elong to the tail. Definition 13 Decomp osition Let Φ n (2) b e a mapping of c haracteristic n ; i 1 , i 2 , · · · , i σ and Φ − 1 n (4) b e the in ve r se mapping (see Definition-Theorem 5, Remark 2). Define (Rerikh , 1998a) the decomp osition of the sets O , O ( − 1 ) of fundamental p oints O α , O ( − 1) β of these mapp ings as follo ws: O ≡ O ( res t ) ∪ O ( int ) , O ( − 1 ) ≡ O ( − 1 ) ( res t ) ∪ O ( − 1 ) ( int ) , (39) O ( res t ) ≡ O ( cy c le ) ∪ O ( tails ) ∪ O ( inf ) , O ( − 1 ) ( res t ) ≡ O ( − 1 ) ( cy c le ) ∪ O ( − 1 ) ( tails ) ∪ O ( − 1 ) ( inf ) . (40) 9 Here: O ( inf ) is a su bset of fundamental p oints O α with infin it e orbits O ( inf ) : [Φ − k n ( O ) ∩ [ O ( − 1) ∪ O ]] = ∅ at ∀ k 1 ≤ k < ∞ ; (41) O ( cy c le ) is a su bset of fundamental p oin ts O α ha ving cyclic orb its O m z , z ∈ O , of ind e x m α ; O ( tails ) is a subset of fund ame ntal p oin ts O α b elonging to the tails of the orbits of the subset O ( cy c le ) , to the tails of the orbits of the su bset O ( int ) and to th e tails of the orbits of the subs e t O ( inf ) . The sub set s O ( − 1)( int ) and O ( int ) of the sets O ( − 1 ) and O are defin ed b elo w. Definition 14 The subsets O ( − 1 ) ( int ) and O ( int ) of the sets O ( − 1 ) and O are under construction in the follo wing manner . Let Φ − k n ( O ( k )) , Φ k n ( O ( − 1 ) ( k )) , k ≥ 0 , Φ − k n | k =0 ≡ id b e k th − iterations of p u nctual sets O ( k ) , O ( − 1 ) ( k ) , O (0 ) ≡ O , O ( − 1 ) (0) ≡ O ( − 1 ) under the ac tion of in verse and d irec t maps Φ − 1 n , Φ n . Let us in tro duce the pu nctual set O ( int ) ( k ) def = Φ − k n ( O ( k )) \ O ( − 1) ( k ) , O ( − 1)( int ) ( k ) def = Φ k n ( O ( − 1 ) ( k )) \ O ( k ) , k ≥ 0 (42) and defin e the construction of the set O ( k ) and O ( − 1) ( k ) at k ≥ 1 in the follo win g manner: O ( k ) def = O ( k − 1) / O ( int ) ( k − 1) , O ( − 1) ( k ) def = O ( − 1) ( k − 1) / O ( − 1)( int ) ( k − 1) , k ≥ 1 . (43) Then subsets O ( int ) and O ( − 1)( int ) are defin ed with the help of (42) and (43) by O ( int ) def = k = m [ k =0 O ( int ) ( k ) , O ( − 1)( int ) def = k = m [ k =0 O ( − 1)( int ) ( k ) , (44) where p ositiv e integer m is defined by the condition O ( m + 1) / O ( res t ) ≡ ∅ , O ( − 1) ( m + 1) / O ( − 1)( rest ) ≡ ∅ , where there su bsets O ( res t ) , O ( − 1)( rest ) are defin ed ab o ve by (4 0 ). Let b e # O ( int ) = σ 1 . (45) The ab o ve construction establishes the one-to- on e corresp ondence b et ween σ 1 pairs of equiv alen t inde- terminacy p oin ts of the su bsets O ( int ) and O ( − 1)( int ) constructed ab o v e as follo w s: Φ − m j ( O α j ) ≡ O ( − 1) β j , j = (1 , · · · , σ 1 ) , (46) O α j ∈ O ( int ) , O ( − 1) β j ∈ O ( − 1)( int ) , α j and β j ∈ (1 , · · · , σ ) , (47) m j ∈ ( m 1 , · · · , m σ 1 ) 0 ≤ m 1 ≤ · · · , ≤ m σ 1 , (48) where nonnegativ e in tegers m j are lengths of the orb its of p oint s O α j but inte ger m in equation (44 ) is equal to m σ 1 . ⊳ R emark 5 Since the coord inate s of the indeterminacy p oin ts O α j , O ( − 1) β j , j = (1 , 2 , · · · , σ 1 ) are functions of 2 σ + 8 parameters, equations (46)-(48) define in th e s p ac e of (2 σ + 8) parameters 2 σ 1 subv arieties of dim en sio n 2 σ + 8 − 2 σ 1 . If the A.P . V eselo v conjecture is tr ue, inte grable mappings corresp ond to these su bv arieties. ⊳ 10 4 Dynamics of a generic birational mapp ing. Difference equation for the Arnold complexit y . Theorem 4 d efines th e dynamics of birational mapping and the difference equ a tion for d ( k ). Theorem 4 Let d ( k ) b e the degree of the mapp ing Φ k n Φ k n : z → z ′ z ′ 1 : z ′ 2 : z ′ 3 = φ ( k ) 1 ( z ) : φ ( k ) 2 ( z ) : φ ( k ) 3 ( z ) , (49) the k th iteration of the mapping Φ n (2) of charact er istic char (Φ n ) = { n, i 1 , · · · , i σ } , O α j and O ( − 1) β j , j = (1 , · · · , σ 1 ) b e σ 1 pairs of equiv alen t in dete rminacy p oin ts of the subsets O ( int ) and O ( − 1)( int ) defined b y r elations (46), (47) and (48 ) (see Section 3, Definitions 13 and 14). Let also γ α j ( k ) b e common multiplicitie s of the curves { φ ( k ) i ( z ) = 0 , i = (1 , 2 , 3) } and the general curv e of the linear system φ ( k ) µ ( z ) = { P i =3 i =1 c i φ ( k ) i ( z ) = 0 , ∀ c i ∈ C , } of degree µ = d ( k ) at ind e termin a cy p oin ts O α j of the d irec t mappin g (2) (w e assume that all F-p oin ts are already ord inary after th e birational equiv alence transf o r mat ion –see Theorem 3). Then the dyn amics of the mapping Φ n (2) (see Definition 5, Remarks 2 and 3) is completely determin ed b y the follo w ing set of difference equations: d ( k ) = nd ( k − 1) − σ 1 X l =1 i ( − 1) β l γ α l ( k − m l − 1) , (50) γ α j ( k ) = i α j d ( k − 1) − σ 1 X l =1 i α j β l γ α l ( k − m l − 1) , j = 1 , · · · , σ 1 , (51) γ α ( k ) = i α d ( k − 1) − σ 1 X l =1 i αβ l γ α l ( k − m l − 1) , α 6 = α j , (52) moreo v er, d (0) = 1 , d (1) = n , γ α (1) = i α , γ α ( k ) = 0 for k ≤ 0 . (53) The secular equation corresp onding to the set of difference equations (50 )- (51 ) is det(Λ) = λ m + m − 1 X i =0 a i λ i = 0 , (54) where inte ger m is m = m 1 + m 2 + · · · + m σ 1 + σ 1 + 1 , (55) in tegers a i are co efficien ts of expansion in p o wer series of det(Λ) in λ and the matrix Λ is Λ =       λ − n, i ( − 1) β 1 , · · · , i ( − 1) β σ 1 − i α 1 , ( λ m 1 +1 + i α 1 β 1 ) , · · · , i α 1 β σ 1 . . . . . . . . . . . . . . . . . . − i α σ 1 , i α σ 1 β 1 , · · · , ( λ m σ 1 +1 + i α σ 1 β σ 1 )       . (56) The linear difference equation for d ( k ) corresp onding to the secular equation (54 ) h a s the form d ( k + m ) + m − 1 X i =0 a i d ( k + i ) = 0 , (57) where a i are the same integ ers as in equation (54). 11 According to a general th eory of linear difference equ a tions with constan t co efficien ts (Gel’fond, 1971) (Chapter V), th e solution of equation (57) h a s the f o r m d ( k ) = l X i =1 λ k i ( s i − 1 X j =0 c ij k j ) , (58) where λ 1 , ..., λ l are multiple r oots of equation (54) with m u lti p lic ities s 1 , ..., s l , s 1 + s 2 + · · · + s l = m, and c ij are arb it r a ry constants to b e d e term in ed f rom m initial v alues d (1) = n, d (2) , ..., d ( m ) obtained with the help of equations (50)- (53). It is obvious that the cond it ion | λ i | = 1 ∀ i ∈ (1 , 2 , · · · , l ) (59) is sufficient for the p olynomial b ounded ness of the gro wth of d ( k ) with k . Remark 6 If some intege r s m i are equal and, moreov er, are fulfilled corresp onding conditions for th e co e ffi c ients at the terms γ α i ( k ) in equations (50)-(51), we can decrease the order of the system of difference equations (50)-(51) and, as result, the ord er of the matrix Λ. ⊳ Pro of. Let us p ro ve the theorem by an induction method . Let us consider the map Φ k n (49)-the k th iteration of the mapping Φ n (2) as an iteration of the map Φ k − 1 n and let us consider the tr an s fo rmation of a general curve of a linear system of the cu r v es φ ( k − 1) µ ( z ) = P i =3 i =1 c i φ ( k − 1) i ( z ) = 0 of degree µ = d ( k − 1) b y the action of the mapp i ng Φ n (2). Let γ ( − 1) β ( k − 1) b e common m ultiplicities of the curves { φ ( k − 1) i ( z ) = 0 , ∀ i ∈ (1 , 2 , 3) } and of the general curv e of the linear system { φ ( k − 1) µ ( z ) = 0 } at the indeterminacy p oin ts O ( − 1) β of the inv erse map Φ ( − 1) n (4). T hen, according to Su bsectio n 2.4, Remark 4, (37) and (38), w e ha ve φ ( k − 1) µ ( φ ( z )) = φ ( k ) µ ′ ( z ) σ Y β =1 j γ ( − 1) β ( k − 1) β ( z ) , (60) µ ′ = µn − σ X β =1 i ( − 1) β γ ( − 1) β ( k − 1) , (61) γ α ( k ) = µi α − σ X β =1 i αβ γ ( − 1) β ( k − 1) , (62) where φ ( k ) µ ′ ( z ) = P i =3 i =1 c i φ ( k ) i ( z ) is a general curve of a linear system of curves of degree µ ′ = d ( k ) asso cia ted with the map Φ k n but γ ( k ) α are its multiplici ties at the p oi n ts O α . Since the linear system of the curv es { φ ( k − 1) µ ( z ) = 0 } is completely defined by its basis set, the d iffe r e n ce of v alues γ ( − 1) β ( k − 1) from zero means that the set O ( k − 1) T O ( − 1) 6 = ∅ where O ( k − 1) is the set of indeterminacy p oint s the mapping Φ k − 1 n . It is ob vious that the s e t O ( k − 1) is equal to (O (1) ≡ O ) O ( k − 1) = l = k − 2 [ l =0 Φ − l n (O( l )) , (63) where the set O( l ) is d e fi ned b y equation (43) (see Section 3). Let us decomp ose the set O ( k − 1) in to tw o subsets O ( k − 1) = O ( k − 1)( int ) [ O ( k − 1)( r est ) (64) 12 related with the subs e ts O ( int ) and O ( res t ) in equation (39). Then the s u bset O ( k − 1)( int ) is equal to O ( k − 1)( int ) = j = σ 1 [ j =1 l =min( m j ,k − 2) [ l =0 Φ − l n ( O α j ) , (6 5) according to Section 3, Defin it ions 13, 14 and (46), (47) and (48). Since, according to Section 3, the in tersection of the sub sets O ( k − 1)( r est ) and O ( − 1) is empt y , then O ( k − 1) T O ( − 1) = O ( k − 1)( int ) T O ( − 1) and O ( k − 1) \ O ( − 1) = ( S j = σ 1 j =1 Φ − m j n ( O α j ) = S j = σ 1 j =1 O ( − 1) β j , ∀ ( k − 2) ≥ m j , ∅ ∀ ( k − 2) < m j . ) (66) Then, according to (66), we ha ve for γ ( − 1) β ( k − 1) = m ult( φ ( k − 1) µ ( z ) | z = O ( − 1) β γ ( − 1) β ( k − 1) =  0 , ∀ β 6 = β j , ( β , β j ) ∈ (1 , · · · , σ ) γ α j ( k − 1 − m j ) , β ≡ β j , j ∈ (1 , · · · , σ 1 )  , (67) where γ α ( k ) = 0 , k ≤ 0 , γ α (1) def = i α . (68) A t last, su bstituting (67) in to equations (60)-(62) and taking into accoun t (68) and d (0 ) = 1 , d (1) = n , w e ha v e equati on s (50)-(53). Equations (50)-(53 ) hold at k = 1 and, therefore, at ∀ k > 1. L et us p ro ve the v alidit y of equations (54)-(59). There are tw o wa ys of obtaining them. Let us transf orm the system of σ 1 + 1 equations (50) and (51) at α j = ( α 1 , α 2 , ..., α σ 1 ) b y changing k → k + m 1 + 1 to the follo win g form ( j = 1 , · · · , σ 1 ): σ 1 X j =1 i ( − 1) β j γ α j ( k + m 1 − m j ) = nd ( k + m 1 ) − d ( k + m 1 + 1) (69) γ α j ( k + m 1 + 1) + σ 1 X l =1 i α j β j γ α l ( k + m 1 − m l ) = d ( k + m 1 ) i α j . (70) The fi rst w a y is to obtain a linear homogeneous difference equation (57) for d ( k ) excepting from the homogeneous system of σ 1 + 1 difference equatio n s (69), (70) for un kno wn d ( k ) , γ α j ( k ) , j ∈ (1 , 2 , · · · , σ 1 ) step by step γ α 1 ( k ) , γ α 2 ( k ), · · · and so on, until w e d o n o t obtain equation (57) for d ( k ). Ho we ver, we can pr ese n t a more dir e ct metho d of obtaining equation (57) for the f unction d ( k ) through finding the c h a racteristic or secular equation immediately fr om th e system of equ a tions (69)-(70) p erforming substitution in them accordingly to d ( k ) = b 0 λ k , γ α j ( k ) = b j λ k , j ∈ (1 , 2 , · · · , σ 1 ) (71) where b 0 , b j , j ∈ (1 , 2 , · · · , σ 1 ) are un kno w n constants. After the su bstitution (71) the system of equations (69)-(70) has the follo wing matrix f orm : Λ D B = 0 , D = diag ( λ m 1 , λ m 1 − m 2 , · · · , λ m 1 − m σ 1 ) , B = ( b 0 , b 1 , · · · , b σ 1 ) , (72) where the matrix Λ is defined by equation (56). The compatibilit y condition of the homogeneous system (72) with r espect to unkno w n parameters b 0 , b j , j ∈ (1 , 2 , · · · , σ 1 ) is a secular equation (54) w h ere intege r s a i , the same as in eq. (57), are the co e ffi c ients of expans io n in p o w er series of det(Λ) in λ but intege r m is defi ned b y (55). Since the difference equ a tion for d ( k ) (57) and the secular equation (54) are in one-to-one corresp ondence by the substitution d ( k ) = λ k , we can uniquely reconstru c t (57) f rom (54). A t the end, according to the general th e ory of linear difference equations with constan t co efficien ts (see (Gel’fond, 1971), c hapter V), the general solution of equation (57) is completely defi ned by the sp ectrum of eigen v alues of the c haracteristic secular equation (54) an d this solution has the f orm (58). Remark that the metho d of obtaining a general solution of the system of difference equations (50), (51) offered ab o ve is fully analogous to usual pr a ctice of solving a sys t em of linear differen tial equations of an order more one with constant co effic ients ( see, for example, (Arnold, 1984), the chapter 3, § 25) . ⊳ 13 5 Conclusion W e b eliev e in th at all mappings with the Arnold complexit y defined b y equation (58) and the sp ectrum (59) of the secular equation (54) of the order m (55) with the matrix Λ (56) are algebraically in tegrable ones and are in tend e d to pr o ve this theorem. Theorem 4 giv es us to p ossibilit y to generate, for example, all integ r a b le families of map s of d e gree n = 2 in the parameter sp ac e of dimension 2 σ + 8 − 2 σ 1 = 14 − 2 σ 1 b eing stratified on algebraic su b v arieties in this space. W e can presen t here some different interesting sets f o r n = 2 , σ 1 = 3 , α 1 = β 1 , α 2 = β 2 , α 3 = β 3 : m 1 = 0 , m 2 6 = m 3 6 = 0 , d et (Λ ) = ( λ − 1) 2 ( λ m 2 + m 3 +2 − 1); (73) m 1 = 1 , m 2 = 2 , m 3 = 3 , det(Λ) = ( λ − 1) 3 ( λ + 1) ( λ 9 + 1) ( λ 3 + 1) ; (74) m 1 = 1 , m 2 = 2 , m 3 = 4 , det(Λ) = ( λ − 1) 3 [ ( λ 15 + 1)( λ + 1) ( λ 5 + 1)( λ 3 + 1) ]; (75) m 1 = 1 , m 2 = 2 , m 3 = 5 , det(Λ) = ( λ − 1) 3 ( λ + 1)( λ 5 − 1)( λ 3 − 1); (76) m 1 = 0 , m 2 = m 3 = m ≥ 1 , det(Λ) = ( λ − 1) 2 ( λ m +1 + 1); (77) m 1 = 1 , m 2 = m 3 = 2 , d et (Λ ) = ( λ − 1) 2 [ λ 6 + 1 λ 2 + 1 ]; (78) m 1 = m 2 = m 3 = 1 , d et (Λ ) = ( λ − 1) λ 3 + 1 λ + 1 ; (79) m 1 = m 2 = m 3 = 2 , det(Λ) = ( λ − 1) 3 ( λ + 1) . (80) It isn’t d ifficult to obtain p olynomially ab ounded dep endence d ( k ) for sets (73)–(80) One presents itself interesti ng also to give classification of all int egrable cubic m a p s. 6 Ac kno wledgemen ts The author is grateful to V.A. Isk ovskikh, V.V. Kozlo v, A.N. Pa rshin, I.R. Shafarevich and D.V. T reschev for useful discussions and interest to the p aper. 7 App endix A. Examples of maps and F Es. 1. FE of the pap er. y ( w + 2) = y ( w + 1)( λy ( w + 1) + dy ( w )) y ( w ) , (81) Supp osing y ( w + 2) = y ′ 2 , y ( w + 1) = y ′ 1 = y 2 , y ( w ) = y 1 w e hav e the map y 7→ y ′ : C 2 → C 2 and then c h a n gi n g y 7→ z : y i = z i /z 3 w e obtain the m a p z 7→ z ′ : CP 2 → C P 2 : Φ 2 : z ′ 1 : z ′ 2 : z ′ 3 = z 2 z 1 : z 2 ( λz 2 + dz 1 ) : z 1 z 3 , (82) Φ ( − 1) 2 : z 1 : z 2 : z 3 = λz ′ 1 2 : z ′ 1 ( z ′ 2 − dz ′ 1 ) : z ′ 3 ( z ′ 2 − dz ′ 1 ) , (83) J ac (Φ 2 ) = 2 λz 1 z 2 2 , J ac (Φ ( − 1) 2 ) = 2 λz ′ 1 2 ( z ′ 2 − dz ′ 1 ) , (84) O 1 = (1 , 0 , 0) , O ∗ 2 = O ∗ 3 = (0 , 0 , 1) , J 1 : ( z 1 = 0) , J 2 = J 3 : ( z 2 = 0) , (85) O ( − 1) 1 = (0 , 1 , 0) , O ( − 1) 2 ∗ = O ( − 1) 3 ∗ = (0 , 0 , 1) , J ( − 1) 1 : ( z ′ 2 − dz ′ 1 = 0) , J ( − 1) 2 = J ( − 1) 3 : ( z ′ 1 = 0) . (86) 14 2. the FE (90) from (Rerikh, 199 2 ) F ( w + 1) = 3 F ( w ) − F ( w − 1)+ F ( w ) F ( w − 1) 1+ F ( w ) . Omitting change s (see previous example) we ha ve Φ 2 : z ′ 1 : z ′ 2 : z ′ 3 = z 2 ( z 2 + z 3 ) : 3 z 2 z 3 − z 1 z 3 + z 1 z 2 : z 3 ( z 2 + z 3 ) , (87) Φ ( − 1) 2 : z 1 : z 2 : z 3 = 3 z ′ 1 z ′ 3 − z ′ 2 z ′ 3 − z ′ 1 z ′ 2 : z ′ 1 ( z ′ 3 − z ′ 1 ) : z ′ 3 ( z ′ 3 − z ′ 1 ) , (88) Jac(Φ 2 ) = 2( z 3 − z 2 )( z 2 + z 3 ) 2 , Jac(Φ ( − 1) 2 ) = 2( z ′ 1 + z ′ 3 )( z ′ 3 − z ′ 1 ) 2 , (89) O 1 = ( − 3 / 2 , − 1 , 1) , O ∗ 2 = O ∗ 3 = (1 , 0 , 0) , J 1 : ( z 3 − z 2 = 0) , J 2 = J 3 : ( z 3 + z 2 = 0) , (90) O ( − 1) 1 = (1 , 3 / 2 , 1) , O ( − 1) ∗ 2 = O ( − 1) ∗ 3 = (0 , 1 , 0) , J ( − 1) 1 : ( z ′ 1 + z ′ 3 = 0) , J ( − 1) 2 = J ( − 1) 3 : ( z ′ 3 − z ′ 1 = 0) (91) 3. Mapping of the pap er. Φ 2 : z ′ 1 : z ′ 2 : z ′ 3 = [ z 1 z 3 − p 2 z 2 1 + ( q 2 + q 3 ) 2 z 1 z 2 + ( q 2 − q 3 ) 2 12 p 2 z 2 2 ] : − z 2 [ z 3 + 2 p 2 z 1 + q 2 + q 3 2 z 2 ] : z 2 3 + 3 2 ( q 2 + q 3 ) z 2 z 3 − p 2 2 z 2 1 + p 2 ( q 2 + q 3 ) z 1 z 2 + 1 12 (5 q 2 2 + 14 q 2 q 3 + 5 q 2 3 ) z 2 2 , (92) Φ − 1 2 = Λ ◦ Φ 2 ◦ Λ , Λ = diag ( − 1 , 1 , 1) . (93) The mapping (92) f o llows from generic quadratic map (18) if w e s upp ose B 1 = B Λ , Λ = diag ( − 1 , 1 , 1) and B =   p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3   , (94) where p 1 = − 2 p 2 , p 3 = p 2 , q 1 = ( q 2 + q 3 ) / 2 , r 1 = r 2 = r 3 = 1 . (95) In accordance with form ulaes (18), (19) and (24)w e hav e thr ee p rincipal lines J i , J ( − 1) i and three F -points O i , O ( − 1) i , O i = ( j j = 0) ∩ ( j k = 0) , O ( − 1) i = ( j ( − 1) j = 0) ∩ ( j ( − 1) k = 0) , i 6 = j 6 = k , i, j, k ∈ (1 , 2 , 3): J i : ( j i = − p i z 1 + q i z 2 + r i z 3 = 0) , J ( − 1) i : ( j ( − 1) i = p i z 1 + q i z 2 + r i z 3 = 0) , (96) O i = { q j − q k , p j − p k , p k q j − p j q k } , O ( − 1) i = Λ O i , (97) O 1 = ( 1 p 2 , 0 , 1) , O 2 =  − q 3 − q 2 p 2 (5 q 3 + q 2 ) , − 6 5 q 3 + q 2 , 1  , O 3 =  q 3 − q 2 p 2 (5 q 2 + q 3 ) , − 6 5 q 2 + q 3 , 1  . (98) 4. FE F ( w + 1) = 4+2 F ( w ) F ( w − 1)+ F ( w − 1) − 14 F 2 ( w ) − 4 F ( w − 1) F 2 ( w ) 1 − 2 F ( w ) − 2 F ( w − 1) − 4 F 2 ( w ) of pap er (Rerikh, 1995b) 15 Let u s consider th e cubic b irati onal mapp ing Φ 3 : C P 2 7→ C P 2 asso ci ated with the ab o v e fu nc- tional equation from (Rerikh, 1995b) (see eq. 23 on p. 67 and eq. 30 on p. 68) Φ 3 : z ′ 1 : z ′ 2 : z ′ 3 = z 2 ( z 2 3 − 2 z 1 z 3 − 2 z 2 z 3 − 4 z 2 2 ) : (4 z 3 3 + z 1 z 2 3 + 2 z 1 z 2 z 3 − 14 z 2 2 z 3 − 4 z 1 z 2 2 ) : z 3 ( z 2 3 − 2 z 1 z 3 − 2 z 2 z 3 − 4 z 2 2 ) , (99) Φ ( − 1) 3 : z 1 : z 2 : z 3 = − (4 z ′ 3 3 − z ′ 2 z ′ 2 3 + 2 z ′ 1 z ′ 2 z ′ 3 − 14 z ′ 2 1 z ′ 3 + 4 z ′ 2 z ′ 2 1 ) : z ′ 1 ( z 2 3 + 2 z ′ 1 z ′ 3 + 2 z ′ 2 z ′ 3 − 4 z ′ 2 1 ) : z ′ 3 ( z 2 3 + 2 z ′ 1 z ′ 3 + 2 z ′ 2 z ′ 3 − 4 z ′ 2 1 ) , (100) where y 1 = F ( w − 1) , y ′ 1 = y 2 = F ( w ) , y ′ 2 = F ( w + 1) and y i = z i z 3 . These maps (char= { 3;2 ,1,1,1,1 } ) ha v e the follo wing indeterminacy p oint s and p rincipal curv es: O 1 = (1 , 0 , 0) , O 2 = ( − 1 / 2 , 1 / 2 , 1) , O 3 = ( − 5 / 2 , − 3 / 2 , 1) , O 4 = (1 / 2 , − 1 / 2 , 1) , O 5 = ( − 1 1 / 2 , 3 / 2 , 1) , (101) O ( − 1) 1 = (0 , 1 , 0) , O ( − 1) 2 = (3 / 2 , 5 / 2 , 1) , O ( − 1) 3 = ( − 1 / 2 , 1 / 2 , 1) , O ( − 1) 4 = ( − 3 / 2 , 11 / 2 , 1) , O ( − 1) 5 = (1 / 2 , − 1 / 2 , 1) , (102) J 1 : ( z 2 3 − 2 z 1 z 3 − 2 z 2 z 3 − 4 z 2 2 = 0) , J 2 : (2 z 2 − 3 z 3 = 0) , J 3 : ( z 3 + 2 z 2 = 0) , J 4 : (2 z 2 + 3 z 3 = 0) , J 5 : ( z 3 − 2 z 2 = 0) , (103) J ( − 1) 1 : ( z ′ 2 3 + 2 z ′ 1 z ′ 3 + 2 z ′ 2 z ′ 3 − 4 z ′ 2 1 = 0) , J ( − 1) 2 : (2 z 1 − z 3 = 0) , J ( − 1) 3 : (2 z 1 + 3 z 3 = 0) , J ( − 1) 4 : (2 z 1 + z 3 = 0) , J ( − 1) 5 : (2 z 1 − 3 z 3 = 0) . (104) It is n o t difficult to obtain i α,β : i α,β =       1 1 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0       . (1 05) 8 App endix B. Examples to Sections 3 and 4 Belo w w e r et u rn again to Examples 1-4 (see Ap pend ix A ) for illus tr a tion of Sections 3 an d 4. Firstly , we giv e the set O ( int ) deriv ed with the help of the d e comp osition pr ocedure, Section 3 (39), (40), an d then giv e the equations of the dynamics (50), (51) of maps 1-4, an d fin ish with a set of difference equation for the Arn o ld complexit y and its solution. 1. Th is map (82) has t wo in fi nitely near p oi nts O ∗ 2 and O ∗ 3 . ( O ∗ 2 is merged with O ∗ 3 in the direction J 1 The p oint O 1 b elongs O ( inf ) : Φ ( − k ) 2 ( O 1 ) = ( λ k , − d λ k − 1 λ − 1 , 0) , but, sa y , the p oint O ∗ 3 ∈ O ( int ) : O ∗ 3 = O ′∗ 3 . W e can ascertain that the p oin t O ∗ 2 do es not b elong to O ( int ) : if w e consider a map Φ 2 ,ǫ b eing a small d eform a tion of in it ial map (82) and ha vin g three different indeterminacy p oin ts ( O 1 = (1 , 0 , 0) , O 2 = (0 , ǫ/λ, 1) , O 3 = (0 , 0 , 1)) Φ 2 ,ǫ : z ′ 1 : z ′ 2 : z ′ 3 = z 2 z 1 : z 2 ( λz 2 − ǫz 3 + dz 1 ) : z 1 z 3 , (106 ) Since it is difficult to p oint out a general metho d of constructing a map coinciding with a given map at a small parameter ǫ = 0 and having on ly ord in a ry indeterminacy p oin ts, we c h a n g e the giv en map 16 b y a birationally equiv alen t map with ordinary ind etermin acy p oin ts. The map Φ 2 : z 7→ z ′ , z , z ′ ∈ CP 2 (82) is bir a tionally equiv alent to the map Φ 3 : u 7→ u ′ , u ′ ∈ CP 2 , Φ 3 = Ψ ( − 1) 2 ◦ Φ 2 ◦ Ψ 2 Φ 3 : u ′ 1 : u ′ 2 : u ′ 3 = [( λ − d ) u 1 + du 2 ][ − u 3 (( λ − d ) u 1 + du 2 ) + ( u 2 − u 1 )( u 3 + u 2 )] : [(1 + d ) u 2 − (1 + d − λ ) u 1 ][ − u 3 (( λ − d ) u 1 + du 2 ) + ( u 2 − u 1 )( u 3 + u 2 )] : u 3 [( λ − d ) u 1 + du 2 ][(1 + d ) u 2 − (1 + d − λ ) u 1 ] , (107) Φ ( − 1) 3 : u 1 : u 2 : u 3 = [ u ′ 1 ( d + 1) − du ′ 2 ][2 u ′ 1 u ′ 3 + u ′ 1 u ′ 2 − u ′ 2 u ′ 3 ] : [ u ′ 1 ( d + 1 − λ ) − ( d − λ ) u ′ 2 ][2 u ′ 1 u ′ 3 + u ′ 1 u ′ 2 − u ′ 2 u ′ 3 ] : u ′ 3 ( u ′ 2 − u ′ 1 )[ u ′ 1 ( d + 1 − λ ) − ( d − λ ) u ′ 2 ] , (108) where the map Ψ 2 : u 7→ z is c hosen so that tw o indeterminacy p oin ts of the map Ψ ( − 1) 2 ma y coincide with the p oin ts O 1 , O ∗ 2 from (85), bu t a dir e ction of the second p r incipal curve (line) for Ψ ( − 1) 2 through the p oin t O ∗ 2 do es not coincide with J 1 from (85): Ψ 2 : u 7→ z z 1 : z 2 : z 3 = ( u 2 u 3 − u 1 u 3 ) : u 1 u 3 : ( − u 1 u 3 + u 1 u 2 ) . (10 9) The maps Φ 3 , Φ ( − 1) 3 ha ve c har = { 2 , 1 , 1 , 1 , 1 } and the follo wing indeterminacy p oin ts: O 1 = (0 , 0 , 1) , O 2 = (1 , 1 , 0) , O 3 = (1 , 0 , 0) , O 4 = ( − d d − λ , − 1 , 1) , O 5 = ( − 2(1 + d ) 1 − λ + d , − 2 , 1) , O ( − 1) 1 = (0 , 0 , 1) , O ( − 1) 2 = (1 , 0 , 0) , O ( − 1) 3 = (0 , 1 , 0) , O ( − 1) 4 = ( − d − λ − 1 d − λ + 1 , − d − λ − 1 d − λ , 1) , O ( − 1) 5 = ( − 1 , − 1 , 1) . Making the decomp ositio n we obtain O 1 , O 3 ∈ O ( int ) , Φ ( − 1) 3 O 2 = O 2 , O 2 ∈ O ( cy c le ) , O 4 , O 5 ∈ O ( inf ) , O ( − 1) 1 , O ( − 1) 2 ∈ O ( − 1)( int ) , O ( − 1) 3 , O ( − 1) 4 ∈ O ( − 1)( inf ) , O ( − 1) 5 ∈ O ( − 1)( cycl e ) . F ollo wing Theorem 4 we ha v e ( i αβ is the same one as in example 4 (105)) d ( k ) = 3 d ( k − 1) − 2 γ 1 ( k − 1) − γ 3 ( k − 1) , (110) γ 1 ( k ) = 2 d ( k − 1) − γ 1 ( k − 1) − γ 3 ( k − 1) , (111) γ 3 ( k ) = d ( k − 1) − γ 1 ( k − 1) . (112) F ollo wing Theorem 4 we obtain d ( k + 2) − 2 d ( k + 1) + d ( k ) = 0 , d ( k ) = 2 k + 1 . (113) 2. Th e map Φ 2 (87) is birationally equiv alen t to the map Φ ∗ 2 = Ψ ( − 1) 2 ◦ Φ 2 ◦ Ψ 2 with ordinary in- determinacy p oin ts ( conditions for the c h oi ce of Ψ 2 : O ( − 1) 1 (Ψ ( − 1) 2 ) = O 2 , 3 (Φ 2 ) , O ( − 1) 2 (Ψ ( − 1) 2 ) = O ( − 1) 2 , 3 (Φ ( − 1) 2 ), J ( − 1) 1 , J 1 for Φ 2 from (90), (91) m u st not coincide with J i for Ψ 2 ) Φ ∗ 2 : u ′ 1 : u ′ 2 : u ′ 3 = u 1 (3 u 2 − u 3 + u 1 ) : u 3 (3 u 2 − u 3 + u 1 ) : u 2 ( u 1 + u 3 ) , (114) Φ ∗ ( − 1) 2 : u 1 : u 2 : u 3 = u ′ 1 ( u ′ 2 + u ′ 1 − 3 u ′ 3 ) : u ′ 3 ( u ′ 1 − u ′ 2 ) : u ′ 2 ( u ′ 2 + u ′ 1 − 3 u ′ 3 ) , (115) 17 but the map Ψ 2 is Ψ 2 : z 1 : z 2 : z 3 = u 1 u 3 : u 1 u 2 : u 2 u 3 . W e ha v e the ind etermin acy p o ints f o r Φ ∗ 2 : O 1 = ( − 1 , 2 3 , 1) , O 2 = (1 , 0 , 1) , O 3 = (0 , 1 , 0) , O ( − 1) 1 = ( 3 2 , 3 2 , 1) , O ( − 1) 2 = ( − 1 , 1 , 0) , O ( − 1) 3 = (0 , 0 , 1). The d e comp osition of the s e ts O , O ( − 1) giv es: O 1 , O 2 ∈ O ( inf ) , O ( − 1) 1 , O ( − 1) 2 ∈ O ( − 1)( inf ) , Φ ∗ ( − 1) 2 ( O 3 ) = O ( − 1) 3 , O 3 ∈ O ( int ) , O ( − 1) 3 ∈ O ( − 1)( int ) . W e ha ve for the Ar nold complexit y d ( k ) = 2 d ( k − 1) − γ 3 ( k − 2) , γ 3 ( k ) = d ( k − 1), d ( k + 3) − 2 d ( k + 2) + d ( k ) = 0 , d ( k ) = − 1 + ( λ k +3 + ( − 1) k λ − ( k + 3) ) / √ 5 , where λ = √ 5+1 2 . 3. Th e decomp osition of the sets O , O ( − 1) of ind et erm inac y p oin ts O α , O ( − 1) β (97), (98) of the mapp ings Φ 2 (92) and Φ ( − 1) 2 (93) giv es Φ − 1 2 ( O 1 ) = {∞ , 0 } , Φ − 2 2 ( O 1 ) = O ( − 1) 1 , Φ − 1 2 ( O 2 ) = { q 3 − q 2 3 p 2 ( q 3 + q 2 ) , − 2 q 3 + q 2 } , Φ − 2 2 ( O 2 ) = O ( − 1) 3 , Φ − 1 2 ( O 3 ) = {− q 3 − q 2 3 p 2 ( q 3 + q 2 ) , − 2 q 3 + q 2 } , Φ − 2 2 ( O 3 ) = O ( − 1) 2 , (11 6) and, consequent ly , O ≡ O ( int ) , O ( − 1) ≡ O ( − 1)( int ) , m j = 2 ∀ j ∈ (1 , 2 , 3) , α j = (1 , 2 , 3) , β j = (1 , 3 , 2) . Due to T heorem (4) and Remark (6 ) we ha ve from (49)- (54) d ( k ) = 2 d ( k − 1) − S ( k − 3) , (117) S ( k ) = 3 d ( k − 1) − 2 S ( k − 3) , (118) where S ( k ) = P 3 α =1 γ α ( k ). In corresp ondence with Th e orem 4 w e obtain the d iffe r e nce equation for d ( k ) d ( k + 4) − 2 d ( k + 3) + 2 d ( k + 1) − d ( k ) = 0 (11 9) and its general solution d ( k ) = 3 4 k 2 − 1 8 ( − 1) k + 9 8 . (120) 4. Th e decomp osition of the s e ts O , O ( − 1) of indeterminacy p oin ts O α , O ( − 1) β (101), (102) of the map- pings Φ 3 (99) and Φ ( − 1) 3 (100) give s: O 2 = O ( − 1) 3 , α 1 = 2 , β 1 = 3 , m 1 = 0 , O 4 = O ( − 1) 5 , α 2 = 4 , β 2 = 5 , m 2 = 0 , Φ ( − 1) 3 ( O 1 ) = O ( − 1) 1 , α 3 = 1 , β 3 = 1 , m 3 = 1 , (121) O ( int ) = { O 2 , O 4 , O 1 } , O ( − 1)( int ) = { O ( − 1) 3 , O ( − 1) 5 , O ( − 1) 1 } , O ( inf ) = { O 3 , O 5 } , O ( − 1)( inf ) = { O ( − 1) 2 , O ( − 1) 4 } . 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