Classification of degree 2 polynomial automorphisms of C3

Publicacions Matem`atiques, Vol 42 (1998), 195–210. CLASSIFICATION OF DEGREE 2 POLYNOMIAL AUTOMORPHISMS OF C3 John Erik Fornæss and He Wu Abstract For the family of degree at most 2 polynomial self-maps of C3with nowhere vanishing Jacobian determinant, we give the following classification: for any such map f, it is affinely conjugate to one of the following maps: (i) An affine automorphism; (ii) An elementary polynomial autormorphism E(x, y, z)=(P(y, z)+ax, Q(z)+by, cz +d), where Pand Qare polynomials with max{deg(P),deg(Q)}=2 and abc 6=0. (iii)            H1(x, y, z)=(P(x, z)+ay, Q(z)+x, cz +d) H2(x, y, z)=(P(y, z)+ax, Q(y)+bz, y) H3(x, y, z)=(P(x, z)+ay, Q(x)+z,x) H4(x, y, z)=(P(x, y)+az, Q(y)+x, y) H5(x, y, z)=(P(x, y)+az, Q(x)+by, x) where Pand Qare polynomials with max{deg(P),deg(Q)}=2 and abc 6=0. 1. Introduction In this note, we will give a classification theorem for the family of degree at most 2 polynomial self-maps of C3with nowhere vanishing Jacobian determinant. Note that any polynomial automorphism has a nowhere vanishing Jacobian determinant. Our Theorem 2.1 implies that any degree at most 2 polynomial self-map of C3with nowhere vanishing Jacobian determinant is a polynomial automorphism. 196 J. E. Fornæss, H. Wu Let Gbe the group of polynomial automorphisms of C2. Let Abe the group of affine automorphisms of C2and let Ebe the group of elementary polynomial automorphisms of C2such that each e∈Eis of the form: e(x, y)=(ax +P(y),by+c) where Pis a polynomial and a,b6= 0. Note that Eis the group of all polynomial automorphisms that carry each line of the form y=constant to a line of the form y=constant0. Then Jung’s Theorem [J] asserts that Gis generated by Aand E. Applying Jung’s Theorem, Friedland and Milnor [FM] classified the polynomial automorphisms of C2:Any polynomial automorphism of C2is affinely conjugate to one of the following types of maps: (i) an affine automorphism; (ii) an elementary polynomial automorphism; (iii) A finite composition of generalized H´enon mappings. Each generalized H´enon mapping is of the form h(x, y)=(P(x)−ay, x) where pis a polynomial of xof degree at least 2and a6=0. It seems to be difficult to extend Jung’s Theorem to Cnfor n≥3. So we cannot follow Friedland and Milnor’s proof to classify polynomial automorphisms in higher dimensions. But if we restrict to polynomials of degree at most 2 in C2, it is not necessary to apply Jung’s Theorem for the classification, see [HO] for a proof. In this paper, we give the classification of degree at most 2 polynomial self-maps of C3with nowhere vanishing Jacobian determinant up to affine conjugation. The organization of this paper is as follows: in Section 2, we give the statement of our classification Theorem 2.1 and we also include some remarks on the dynamical differences between the various classes in our theorem. In Section 3, the proof of Theorem 2.1 is given and in Section 4, we briefly give some discussions of some basic dynamical properties of these maps. 2. The statement of main theorem and some remarks Theorem 2.1. If f:C3→C3is a degree at most 2polynomial selfmap with nowhere vanishing Jacobian determinant, then fis affinely conjugate to one of the following maps: (1) An affine automorphism; (2) An elementary polynomial automorphism E(x, y, z)=(P(y, z)+ax, Q(z)+by, cz +d), Polynomial automorphisms of C3197 where Pis a polynomial of y,zof degree at most 2,Qis a polynomial of zof degree at most 2and abc 6=0. Note that it maps every hyperplane z=kto a hyperplane z=k0and maps every line y=k1,z=k2to a line y=k0 1,z=k0 2; (3)              H1(x, y, z)=(P(x, z)+ay, Q(z)+x, cz +d) H2(x, y, z)=(P(y, z)+ax, Q(y)+bz, y) H3(x, y, z)=(P(x, z)+ay, Q(x)+z,x) H4(x, y, z)=(P(x, y)+az, Q(y)+x, y) H5(x, y, z)=(P(x, y)+az, Q(x)+by, x) where Pand Qare polynomials with max{deg(P),deg(Q)}=2and abc 6=0. Remark 2.2.                                                    H−1 1(x, y, z)=µy−Qµ1 c(z−d) ¶, 1 a·x−Pµy−Qµ1 c(z−d) ¶,1 c(z−d) ¶¸,1 c(z−d)¶ H−1 2(x, y, z)=µ1 a·x−Pµz, 1 b(y−Q(z))¶¸,z,1 b[y−Q(z)]¶ H−1 3(x, y, z)=µz, 1 a[x−P(z,y −Q(z))],y−Q(z) ¶ H−1 4(x, y, z)=µy−Q(z),z, 1 a[x−P(y−Q(z),z)],z¶ H−1 5(x, y, z)=µz, 1 b[y−Q(z)],1 a·x−Pµz, 1 b(y−Q(z))¶¸¶. Remark 2.3. If τ(x, y, z)=(y,z,x), then H4=H3◦τ,H2=H5◦τ, H1=H5◦τ2. Remark 2.4. Some Generic Dynamical differences between the various classes: Assume for simplicity that the constants |a|,|b|,|c|<1. First of all the elementary maps and the class H1distinguish themselves from the other classes by the fact that the maps fix a hypersurface and the orbits of all points outside this hypersurface converge to it. Hence the dynamics reduces to two dimensions. In the case of H1, the 198 J. E. Fornæss, H. Wu maps reduces to a H´enon map on the fixed hypersurface z=α. In the case of the elementary maps, the orbits in the fixed hypersurface z=α converge to the fixed curve y=βon which the maps are automorphisms. In fact both the elementary maps and the maps of class H1are semidirect products over a mapping A(z)=cz +d, i.e. there is a function z→fz(x, y)∈Aut(C2) such that F(x, y, z)=(f z (x, y),A(z)). Hence we need only to find dynamical differences between the classes H2,H3,H4,H5. First we can observe that it is natural to consider the maps H2,H3 together as opposed to the maps H4,H5. There is a dynamical difference in the asymptotic dynamics. For the maps H2,H3, the orbits generically converge to one point at infinity. For example, for the map H3,ifP(x, y)=Ax2+···,Q(x)=Bx2+···, then this is the point [A:B: 0: 0] at infinity in projective coordinates. On the other hand, for the maps H4,H5the generic orbit converges to a complex line at infinity. It remains to distinguish dynamically the maps H2and H3as well as to distinguish the maps H4and H5. Comparing the maps H2and H3, we observe that the map H2is a H´enon map in the last two coordinates, (y,x)→(Q(y)+bz, y). In other words such a map Fis a semi-direct product over a mapping h(y,z)∈ Aut(C2), i.e. there is an analytic function (y,z)→Ay,z ∈Aut(C) such that F(x, y, z)=(A y,z(x),h(y,z)). This sets H2apart from H3. Comparing the maps H4and H5we consider again their behaviour at infinity. We see that there is a P1at infinity which is mapped to itself. For H4this map is a second degree polynomial, while for H5this map is rational of degree 2, i.e. has a more complicated dynamics. 3. The Proof of Theorem 2.1 Let Gbe the family of degree at most 2 polynomial self-map of C3 with nowhere vanishing Jacobian determinant. For any f∈G, we can write fin the following form: f(x, y, z)=(f 1 (x, y, z),f 2(x, y, z),f 3(x, y, z)). Because the degree of fis at most 2, the Jacobian matrix of fis as follows: f0(x, y, z)=  w 1 (x, y, z)w2(x, y, z)w3(x, y, z) w4(x, y, z)w5(x, y, z)w6(x, y, z) w7(x, y, z)w8(x, y, z)w9(x, y, z)   where wj(x, y, z)=a j x+b j y+c j z+d jfor 1 ≤j≤9. Polynomial automorphisms of C3199 Since the determinant of the Jacobian matrix f0(x, y, z) is a nonzero constant, all coefficients of the polynomial det(f0) must be zero except the constant. In particular, the coefficients of x3,y3and z3must be zero, i.e., (1) det(A) = det(B) = det(C)=0 where (2) A=  a1a2a3 a4a5a6 a7a8a9  ,B=   b 1b 2b 3 b 4b 5b 6 b 7b 8b 9  ,C=   c 1c 2c 3 c 4c 5c 6 c 7c 8c 9  . It is easy to see that both b1and a2are the coefficient of xy in f1(x, y, z), this implies that b1=a2. By looking at the coefficients of xy, yz, and xz in f=(f 1 ,f 2,f 3), we obtain the following table: b1=a2c1=a3c2=b3 b4=a5c4=a6c5=b6 b7=a8c7=a9c8=b9 Table (ABC) By using the above table we can write our function fin the following form: (3) f(x, y, z)=(φ 1 (x, y, z)+L 1 (x, y, z),φ 2(x, y, z) +L2(x, y, z),φ 3(x, y, z)+L 3 (x, y, z)) where                                      φ1(x, y, z)=1 2 a 1 x 2+1 2 b 2 y 2+1 2 c 3 z 2+b 1 xy +c2yz +c1xz φ2(x, y, z)=1 2 a 4 x 2+1 2 b 5 y 2+1 2 c 6 z 2+b 4 xy +c5yz +c4xz φ3(x, y, z)=1 2 a 7 x 2+1 2 b 8 y 2+1 2 c 9 z 2+b 7 xy +c8yz +c7xz L1(x, y, z)=d 1 x+d 2 y+d 3 z+e 1 L 2 (x, y, z)=d 4 x+d 5 y+d 6 z+e 2 L 3 (x, y, z)=d 7 x+d 8 y+d 9 z+e 3 . 200 J. E. Fornæss, H. Wu Let’s introduce the following trivial lemma which is useful in our proof of Theorem 2.1. Lemma 3.1. Let F=(f 1 ,... ,f n)be a polynomial self-map of Cnof degree at most 2, with nowhere vanishing Jacobian. Let gand hbe any affine automorphisms of Cn. We denote the degree 2homogeneous part of Fby (φ1,... ,φ n)and the degree 2homogeneous part of g◦F◦hby (ψ1,... ,ψ n). Then the following statements are equivalent: (i) There exist constants α1,... ,α nwith Pn j=1 |αj|6=0such that Pn j=1 αjφj≡0, (ii) There exist constants βjwith Pn j=1 |βj|6=0such that Pn j=1 βjψj=0. Proof: Clear. Remark 3.2. For the map Fin above lemma, if we want to prove that Pn j=1 αjφj≡0, we can simplify JF, the Jacobian matrix of F,by composing constant invertible matrices in both sides of JF. Note that J(g◦F◦h)=Jg(F(h))JF(h)Jh=JgJF(h)Jh, i.e., we have to use the new variables for the Jacobian matrix of F, but this doesn’t matter because Jhis a constant matrix and therefore we may keep the original notation as the new variables. Let’s recall the following result from [HO]: Lemma 3.3. Let f(x, y)=(f 1 (x, y),f 2(x, y))=(P 1 (x, y)+A 1 (x, y), P2(x, y)+A 2 (x, y)) be a polynomial self-map of C2of degree at most 2, with nowhere vanishing Jacobian, where Pj(x, y)is the corresponding degree 2homogeneous polynomial of fjand Aj=fj−Pj. Then the homogeneous polynomials P1and P2are proportional. Lemma 3.4. If for all constants α,βand γwith |α|+|β|+|γ|6=0, we have αφ1+βφ2+γφ36≡ 0, then there exist affine automorphisms g and hsuch that (4) ψ1=x2,ψ 2 =xy, ψ3=y2, where (ψ1,ψ 2,ψ 3)is the degree 2homogeneous part of the map g◦f◦h. Proof: Let φ=[φ 1 :φ 2 :φ 3 ]:P 2→P 2 . Since the Jacobian determinant of φis 0, the rank of φis at most 1. If φ1≡0, then Polynomial automorphisms of C3201 1φ1+0φ 2+0φ 3≡0. So WLOG, we may assume that φj6≡ 0 for j=1,2,3. (i) If the rank of φis 0, i.e., φis constant, then we may assume that [φ1:φ2:φ3] = [1: 0: 0]. In this case, we have 0φ1+1φ 2+0φ 3≡0. (ii) If rank of φis 1, we may assume that φ([1: 0: 0]) = [1: 0: 0] and that φ([x: 0: 1]) is non-constant. Let (φ1,φ 2,φ 3)| [x:0:1] =(a 1 x 2+b 1 x+c 1 ,a 2x 2+b 2x+c 2,a 3x 3+b 3x+c 3). Then since φ([1: 0: 0]) = [1: 0: 0], a16=0,a 2=a 3=0. Hence (φ1,φ 2,φ 3)| [x:0:1] =(a 1 x 2+b 1 x+c 1 ,b 2x+c 2,b 3x+c 3). Since φ([x: 0: 1]) is non-constant, it follows that b2or b36=0. By Lemma 3.1 we may assume that b26= 0 and b1=b3=0. Hence (φ1,φ 2,φ 3)| x:0:1] =(a 1 x 2+c 1 ,b 2x+c 2,c 3). If c3= 0, then we have φ3= 0, so we have 0φ1+0φ 2+1φ 36= 0, which is impossible. Hence c36= 0, so we may assume that (φ1,φ 2,φ 3)| [x:0:1] =(x 2 ,x,1). Hence φ(y=0)=(XZ =Y2). Sice φhas rank 1, φ(P2)⊂(XZ = Y2), so (5) φ1φ3≡φ2 2. (1) If φ1=cφ2for a nonzero constant c, then φ1−c−1φ2+0φ 3≡0. (2) If φ3=cφ2for a nonzero constant c, then we have 0φ1+c−1φ2− φ3≡0. (3) Hence φ16=c1φ2and φ36=c2φ2for any constants c1and c2. Then φ2must be a product of two nonproportional linear factors, φ=L1L2. This implies that φ1=c1L2 1and φ3=c3L2 2or vice versa. Setting L1=x and L2=yand scaling we finish the proof of the lemma. 202 J. E. Fornæss, H. Wu Lemma 3.5. There exist constants α,βand γsuch that |α|+|β|+ |γ|6=0and αφ1+βφ2+γφ3≡0. Proof: If for all constants α,βand γwith |α|+|β|+|γ|6=0,αφ1+ βφ2+γφ36≡ 0, then by Lemma 3.1 and Lemma 3.4 we may assume that ψ1=x2,ψ2=xy,ψ3=y2. In this case, the Jacobian matrix of fis as follows: f0(x, y, z)=  2x+d 1d 2d 3 y+d 4x+d 5d 6 d 72y+d 8d 9  . Since the Jacobian determinant is a nonzero constant, then det   2x+d1d2d3 y+d4x+d5d6 d72y+d8d9  = det   d1d2d3 d4d5d6 d7d8d9  = const. 6=0. But det   2x+d1d2d3 y+d4x+d5d6 d72y+d8d9  =2d 9 x 2−4d 6 xy +2d 3 y 2+··· . This implies that d3=d6=d9= 0, i.e., det   d1d2d3 d4d5d6 d7d8d9  =0. This is a contradiction. Proof of the Theorem 2.1: If deg(f) = 1, then it is easy to see that f is an affine automorphism. If deg(f) = 2, then by Lemma 3.5 we can assume that there exist constants k1and k2such that φ3=k1φ1+k2φ2. Then (6) f=(φ 1+L 1 ,φ 2+L 2,k 1φ 1+k 2φ 2+L 3). Polynomial automorphisms of C3203 Let g1(x, y, z)=(x, y, z −k1x−k2y), then g−1 1(x, y, z)=(x, y, z + k1x+k2y). Then (7) F1(x, y, z):=g 1◦f◦g −1 1(x, y, z) =g1◦f(x, y, z +k1x+k2y) =(φ 4 (x, y, z)+L 4 (x, y, z),φ 5(x, y, z) +L5(x, y, z),L 6(x, y, z)) where φjare degree 2 homogeneous polynomial and Lj(x, y, z) are degree 1 polynomial and written as Lj(x, y, z):=α j x+β j y+γ j z+ρ j .In particular, |α6|+|β6|+|γ6|6=0. We will classify L6into the following 3 cases: Case (i): α66=0. Let g2(x, y, z)=(α 6 x+β 6 y+γ 6 z+ρ 6 ,y,z), then g−1 2(x, y, z)= ³1 α 6 (x−β 6 y−γ 6 z−ρ 6 ),y,z´. Then (8) F2(x, y, z):=g 2◦F 1◦g −1 2(x, y, z) =(φ 7 (x, y, z)+L 7 (x, y, z),φ 8(x, y, z)+L 8 (x, y, z),x) where φjare degree 2 homogeneous polynomials and Lj(x, y, z) are degree 1 polynomials and written as Lj(x, y, z):=α j x+β j y+γ j z+ρ j . Then the Jacobian matrix of F2has the following form: (9) F0 2(x, y, z)=  (φ 7+L 7 ) 0 x(φ 7+L 7 ) 0 y(φ 7+L 7 ) 0 z (φ 8+L 8 ) 0 x(φ 8+L 8 ) 0 y(φ 8+L 8 ) 0 z 100   . Then since det(F0 2(x, y, z)) is a nonzero constant, det µ(φ7+L7)0 y(φ7+L7)0 z (φ8+L8)0 y(φ8+L8)0 z¶=anonzero constant. Then for any fixed x, we may consider Fx(y,z)=(φ 7+L 7 ,φ 8+L 8) as a degree 2 polynomial self-map of C2with nowhere vanishing Jacobian determinant. We can write φ7+L7=a7x2+b7y2+c7z2+d7yz +e7xy +f7xz +··· φ 8+L 8=a 8x 2+b 8y2+c 8z2+d 8yz +e8xy +f8xz +··· 210 J. E. Fornæss, H. Wu C2for any fixed y.IfPand Qare degree 2 polynomials, we believe that the map H5has interesting dynamics and different from the dynamics of H´enon map in C2. There are also some uninteresting examples like this: H3(x, y, z)= (2xz, 2y−x2,x). It is easy to check this map has only one periodic point which is the fixed point at the origin. Remark 4.8. For every map of Hjand E, the degree of its inverse polynomial could be 3 or 4 if the deg(P) = deg(Q) = 2. But it must be 2 if either deg(P)≤1 or deg(Q)≤1. Remark 4.9. The detailed discussion of the dynamical properties of our maps Hjand Ewill appear in our forthcoming papers. References [F] S. Friedland, Two problems in complex dynamics, ?. [FM] S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems ? (1989), 67–99. [HO] J. Hubbard and W. Oberste-Vorth,H´enon mappings in the complex domain II, IHES Publ. Math. 79 (1994), 5–46. [J] H. W. E. Jung, Uber ganze birationale transformationes der Ebene, J. Reine Angew. Math. 184 (1942), 161–174. Mathematics Department The University of Michigan 2072 EH, 525 E. University Ave. Ann Arbor, MI 48109-1109 U.S.A. Rebut el 29 de maig de 1997