Dynamics of (Pseudo) Automorphisms of 3-space: Periodicity versus positive entropy

We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of finite order. This family also contains pseudo-automorphisms of infinite order. One such family consists of completely integrable maps, and another family consists of maps of positive entropy. Both of these families have invariant families of $K3$ surfaces.

is an automorphism. For generic c and c ′ , the surfaces S c and S c ′ are biholomorphically inequivalent, and the automorphisms f 3 | S c and f 3 | S c ′ are not smoothly conjugate.
The surface S 0 is invariant, and the restriction f S 0 is an automorphism which has the same entropy as f . This is smaller than the entropy of the automorphism constructed in [M2,Theorem 1.2] and is thus the smallest known entropy for a projective K3 surface automorphism.
Closely related to the dynamics of f Z is the (1,1)-current T + which is expanded by f * Z , and a current T − for f −1 Z . This is obtained in §7, as well as the invariant (2,2)-current T + ∧T − . The slices of T ± and T + ∧ T − on the surfaces S c give the expanded/contracted currents, as well as the unique invariant measure, for the automorphism f | S c .
The following mappings have quadratic degree growth and complete integrability: Theorem 3. Suppose that β = (0, 1, 0, 0) and either α = (0, 0, ω, 1) or α = (a, 0, 1, 1) where a ∈ C \ {1}, ω = 1, and ω 3 = 1. Then the degree of f n grows quadratically in n. Further, there is a modification π : Z → P 3 such that f Z is a pseudo-automorphism. There is a twoparameter family of surfaces S c , c = (c 1 , c 2 ) ∈ C 2 which are invariant under f 3 . For generic c and c ′ , S c is a smooth K3 surface, and S c ∩ S c ′ is a smooth elliptic curve.
For the mappings in Theorems 1 and 3, f is reversible on the level of cohomology: f * Z is conjugate to (f −1 Z ) * = (f * Z ) −1 . The identity δ 1 (f ) = δ 2 (f ) for such maps is a consequence of the duality between H 1,1 and H 2,2 , so they are not cohomologically hyperbolic, in the terminology of [G2]. For each of these maps, the family of invariant K3 surfaces becomes singular at an invariant 8-cycle R of rational surfaces (see (7.2)). We show that the restriction f | R is not birationally conjugate to a surface automorphism: see Appendix C for the maps in Theorem 1 and Proposition 8.2 for the maps in Theorem 3. By Corollary 1.6, then, we have: Theorem 4. Let f be a map from Theorems 1 and 3. If a = 1, then f is not birationally conjugate to an automorphism.
We note that for birational surface maps, the degree growth of the iterates determines whether the map is birationally conjugate to an automorphism: This occurs if and only if either (i) the degrees are bounded or degree growth is quadratic (see [DiF]), or (ii) if the dynamical degree is a Salem number (see [BC]). Theorem 4 shows that this result does not hold in dimension 3.
We will also determine which mappings f α,β are periodic, or finite order, in the sense that f p = id for some p > 0. In contrast to Theorem 4, it was shown by de Fernex and Ein [dFE] that if f is a rational map of finite order, then there is a modification f X as above, which is an automorphism of X. If f X is periodic, then f * X will also be periodic. In (4.1) and (4.2) we identify conditions which are necessary for f to be periodic and are sufficient for the existence of a space Z = Z α,β such that f Z is a pseudo-automorphism. We show that for a map in (0.1), if f * Z is periodic, then f also turns out to be periodic. The birational map (0.1) may also be considered as a 3-step linear fractional recurrence: given z 0 , z 1 , z 2 , we define a sequence {z n } by z n+3 = α 0 + α 1 z n + α 2 z n+1 + α 3 z n+2 β 0 + β 1 z n + β 2 z n+1 + β 3 z n+2 . (0. 2) The recurrence (0.2) is said to be periodic if the sequence {z n } is periodic for all choices of initial terms z 0 , z 1 and z 2 . Equivalently, f p α,β = id for some p. For all r > 0 there are r-step recurrences of the form (0.2). In [BK2] we determined the possible periods for 2-step linear so that f i 's have no common polynomial factor. We define the degree of f , deg(f ), to be the (common) degree of the f j 's. The indeterminacy locus of f is defined by and is a subvariety of codimension at least 2, and f defines a holomorphic mapping f : P d \ I(f ) → P d . If S is an irreducible subvariety of P d , and S ⊂ I(f ), we define the strict transform, written simply as f (S), to be the closure of f (S − I(f )). We say that an irreducible variety V is exceptional for a rational mapping f if V ⊂ I(f ), and if the dimension of f (V − I(f )) is strictly less than the dimension of V . Following [DO,p. 64], we say that f : X Y is a pseudo-isomorphism if f is birational, and if neither f nor f −1 has an exceptional hypersurface. It follows that if f is a pseudo-isomorphism, then f : X \ I(f ) → Y \ I(f −1 ) is biholomorphic. If X = Y , we say that f is a pseudo-automorphism.
Theorem 1.1. If f : X Y is a pseudo-isomorphism between 3-dimensional manifolds, then the indeterminacy locus has no isolated points.

Proof.
Suppose that there is an isolated point p ∈ I(f ). Since f −1 has no exceptional hypersurfaces, f must blow p up to a curve C ′ ⊂ Y . Now we consider the behavior of f −1 on C ′ . We must have C ′ ⊂ I(f −1 ), for if f −1 is regular at a point q ∈ C ′ , then f −1 must map an open subset of C ′ to p. Thus the jacobian of f −1 must vanish at q. Since the jacobian vanishes on a hypersurface, f −1 would have an exceptional hypersurface containing q. Thus q must be indeterminate. Since the total transform of q under f −1 is given by ǫ>0 (f −1 (B(q, ǫ) − I(f −1 ))), it must be connected, and it must be a curve C containing p. But since p was an isolated point of I(f ), there are nearby points p ′ ∈ C − I(f ). Since f is regular at these points, it must map them to q, and thus f must have an exceptional hypersurface. By this contradiction, we see that I(f ) has no isolated points.
For a rational map f : X X, we consider the iterates f j = f • · · · • f , j > 0. If Σ is an irreducible hypersurface, then Σ ⊂ I(f j ) for reasons of dimension, so we may consider the sequence of varieties V j := f j (Σ), for j > 0. Since we will be interested in knowing to what extent the iterates of f behave like a pointwise-defined dynamical system, we note: If S ⊂ I(g) is irreducible and if g(S) ⊂ I(f ), then S ⊂ I(f • g), and f (g(S)) = (f • g)(S). We may also define f at points of indeterminacy. Let γ f = {(x, y) ∈ (P d − I) × P d : y = f (x)} denote the graph of f at its regular points, and we let Γ denote the closure of γ f inside P d × P d . It follows that Γ is an irreducible variety of dimension d, and there are holomorphic projections π j : Γ → P d , j = 1, 2, onto the first and second factors, respectively, and we have f = π 2 • π −1 1 on P d − I. For a point p ∈ P d , we define the total transform to be f * p := π 2 (π −1 1 p), and then we define f * (S) := p∈S f * p. It is easily seen that we have: If Σ is an irreducible hypersurface, then f * (g(Σ)) ⊃ (f • g)(Σ).
Proposition 1.2. Suppose that f : X X is rational, and suppose that for each exceptional hypersurface E and for m > 0, we have f m (E − I) ⊂ I . If follows that (f * ) n = (f n ) * on H 1,1 (X).
Proof. It is sufficient to show that (f * ) 2 = (f 2 ) * on P ic(X). If D is a divisor, then f * D is the divisor on X which is the same as f −1 D on X − I. Since I has codimension at least 2, we also have (f 2 ) * D = f * (f * D) on X − I − f −1 (I). By our hypothesis f −1 (I) has codimension at least 2. Thus we have (f 2 ) * D = (f * ) 2 D on X.
In a similar way, we may define f * : H p,q (X) → H p,q (X). That is, if β is a (p, q) form on X, then the pullback π * 2 β is a smooth form on Γ. We may let [π * 2 β] denote the reinterpretation of the form as a current, and we may push it forward to obtain a current f * β = π 1 * [π * 2 β] on X. This pulls smooth forms back to currents and is well defined at the level of cohomology classes. If α ∈ H p ′ ,q ′ is an element of the dual cohomology group, then we have α, f * β = π * 1 α, π * 2 β . Now if f is birational and g = f −1 , then If we have (f n ) * = (f * ) n on H p,q for n ≥ 0 , then this gives us (g n ) * = (g * ) n on H p ′ ,q ′ . The following is proved along standard lines: From this we get the following: Proposition 1.4. If f : X X is a pseudo-automorphism on a d-dimensional manifold, then for all n ∈ Z we have (f * ) n = (f n ) * on both H 1,1 and H d−1,d−1 . Further, the characteristic polynomials of f * on H 1,1 and H d−1,d−1 are the same and therefore the first and the d − 1 st dynamical degree are the same.
Remark. Suppose that γ ′ and γ ′′ are curves in P 3 which intersect transversally at points {p 1 , . . . , p N }. We have local coordinate systems for 1 ≤ j ≤ N so that p j is the origin, and γ ′ (resp. γ ′′ ) coincides with the x-axis (resp. the y-axis) in a neighborhood of p j . Since the operation of blowing up the axes is local near p j , we may construct a blowup space π : W → P 3 in which γ ′ and γ ′′ are both blown up, and over each p j we are free to choose whether γ ′ or γ ′′ was blown up first, independently of the choices over p k for k = j.
Theorem 1.5. Let f be a birational map of X. Let X 0 ⊂ X be a hypersurface such that the strict transform is f (X 0 ) = X 0 . Let ϕ : X → Y is a birational map which conjugates (f, X) to an automorphism (g, Y ). Then there is a birational mapφ : X →Ŷ such that the strict transformŶ 0 :=φ(X 0 ) is a nonsingular hypersurface, and the induced mapĝ :=φ • f •φ −1 gives an automorphsm ofŶ .
Proof. We may assume that X 0 is irreducible. Since X 0 is a hypersurface, we may take its strict transform ϕ(X 0 ). If ϕ(X 0 ) is a point in Y , then it is fixed by g. If π 1 : Y 1 → Y be the blowup of the point ϕ(X 0 ), then g lifts to an automorphism of Y 1 . Let φ 1 := π −1 1 • ϕ. If ϕ 1 (X 0 ) is again a point, we can repeat this blowing-up process until ϕ 1 (X 0 ) has dimension > 0, which we may assume to be 1. If the singular locus of ϕ 1 (X 0 ) is nonempty, it is finite and invariant under f 1 . Now we can blow up the singular set of ϕ 1 (X 0 ) finitely many times and have a new blow up space π 2 : Y 2 → Y 1 . Since we were blowing up invariant sets, the induced birational map g 2 of Y 2 is again an automorphism. Now the image ϕ 2 (X 0 ) must be a nonsingular curve, which must be invariant. We can blow up this curve, and repeat the process finitely many times so that ϕ 3 (X 0 ) has dimension > 1. We continue this process until ϕ N (X 0 ) is a nonsingular hypersurface in Y N , and now we setŶ = Y N . Corollary 1.6. Let f be a birational map of X. Let X 0 ⊂ X be a hypersurface for which the strict transform is f (X 0 ) = X 0 . Let ϕ : X → Y is a birational map which conjugates (f, X) to an automorphism (g, Y ). Then the restriction (f X 0 , X 0 ) is birationally conjugate to an automorphism.
Proof of Theorem 4. Let f be as in Theorem 4. In §C we study the restriction of f 8 to the plane Σ 3 = {[x 0 : x 1 : x 2 : x 3 ] ∈ P 3 : x 3 = 0}. There we show that this restricted mapping is not birationally equivalent to an automorphism of Σ 3 . Thus Theorem 4 is a consequence of Corollary 1.6. §2. Linear fractional recurrences. The maps (2.2) are among the Cremona transformations of 3-space which are discussed in Chapter 10 of [H]. We discuss general properties of these transformations, and for the generic parameters (2.10) we construct a new space π : X → P 3 , such that passing to the induced map f X effectively eliminates one of the exceptional components.
Note that if one of the first two conditions does not hold, then f is linear, and if the third condition does not hold, then f is independent of x 1 and thus f is actually a 2-step recurrence. If we set γ = β 1 α − α 1 β, then we have and e 1 = [0 : 1 : 0 : 0] = Σ 023 .
The Jacobian determinant of f is given by 2x 0 (γ · x)(β · x) 2 . Thus we see that the exceptional hypersurfaces are E = {Σ 0 , Σ β , Σ γ }. The action of f on the exceptional varieties is given as follows: for λ 2 , λ 3 ∈ C, (λ 2 , λ 3 ) = (0, 0), where we setα = (α 0 , α 2 , α 3 , 0),β = (β 0 , β 2 , β 3 , 0), and B = (−α 1 , 0, 0, β 1 ), Thus Σ β is blown down to a point. The pencil of lines in Σ γ passing through e 1 ∈ Σ 0 ∩ Σ γ are all mapped to points in Σ BC . The pencil of lines in Σ 0 passing through e 1 are all mapped to points on the line Σ 03 , which is again one of the exceptional lines. We have strict transforms: The inverse is given by 6) and the indeterminacy locus is (2.7) Now let us construct the space π 1 : X 1 → P 3 by blowing up a point e 1 , and then the space π 2 : X → X 1 obtained by blowing up a line Σ 03 . We set π = π 1 • π 2 : X → P 3 . (2.8) Let S 03 denote the blowup fiber over the strict transformation of Σ 03 in X 1 and E 1 for the strict transformation of π −1 1 e 1 in X 1 . For the induced map on X, the orbit of Σ 0 becomes If X and Y are irreducible, we will say that a rational map f : X Y is dominant if the rank of df is equal to the dimension of Y on a dense open set. Let us define a generic condition: β 1 = 0, β 1 α 2 = α 1 β 2 , and β 1 α 3 = α 1 β 3 . (2.10) For simplicity we use the same notation for both a variety and its strict transform, if there is no possibility of confusion.
Thus in passing to f X , we have removed one exceptional hypersurface and one point of indeterminacy. There is a group of linear conjugacies acting on the family (0.2). For (λ, c, µ) ∈ C * × C * × C, we set (α, β) → (λα, λβ) (2.11a) (2.11c) The first action corresponds to the homogeneity of f . The action (2.11b) corresponds to a scaling of (x 1 , x 2 , x 3 ) in affine coordinates, and (2.11c) comes from translation by the vector (µ, µ, µ). Note that these actions preserve the form of the recurrence relation. §3. Non-critical maps. A map f of the form (2.2) is critical if (3.1) holds: β 2 = β 3 = 0, and β 1 α 2 α 3 = 0. (3.1) In this section we establish the following: Theorem 3.1. If f is not critical, then f is not periodic.
We will use the following criterion: Proof. Since E is exceptional, then codim(f (E)) ≥ 2. Let us consider the sequence of varieties V j := f j X (Σ). If V j ⊂ I(f X ) for all j, then applying the strict transform of f repeatedly, we have f j+1 X (E) = f X (V j ) for all j, so codim(f (V j )) ≥ 2 for all j. On the other hand, we must have f p X (E) = E = V p . The proof of Theorem 3.1 will involve several cases, so we start with some Lemmas. Lemma 3.3. Let π : X → P 3 be the complex manifold defined in (2.8). If β 1 = 0, then there is a hypersurface V ⊂ X such that f n X V is a point of X − I either for all n ≥ 1 or for all n ≤ −1.
Proof. We use the local coordinates for a neighborhood of S 03 and a neighborhood of E 1 defined in §2. Since β = (β 0 , 0, 0, 0), we first assume that β 3 = 0 (and thus we may suppose β 3 = 1) and to consider various cases.
(i) Case β 2 = 0 : In this case, the orbit of Σ β is: Thus the orbit is periodic and remains a regular point of f X .
Proof. Let us first assume that β 3 = 0. Then Now suppose β 3 = 0 and β 2 = 0. In this case we have and this last point is fixed under f −1 .
(i) Case α 3 = 0 and α 2 = 0 : (ii) Case α 2 = 0 and α 3 = 0 : Theorem 3.6. If f is not critical, then there exists a complex manifold X such that either there is an exceptional hypersurface E ⊂ X for an induced birational map f X such that f n X E ⊂ I(f ) for n = 1, 2, . . . , or the analogous statement holds for f −1 X . Proof. Let X denote either the space X or Z in the Lemmas above. This Theorem follows from the Lemmas 3.3-5.
Proof of Theorem 3.1. If f is not critical, then Theorem 3.6 says that in each case that there is an exceptional hypersurface that does not map into I(f X ). By Proposition 3.2, then, f is not periodic. §4. Critical Maps. Here we study critical maps in general. Let us recall the condition for f being critical : β 2 = β 3 = 0 and β 1 α 2 α 3 = 0. Using the action (2.11a-c) we may assume that a critical map satisfies: (4.1) In this section, we show (Lemma 4.2) that for every critical map there is a blowup space π : Y → X such that the induced map f Y has only one exceptional hypersurface, which is Σ γ . We determine the indeterminacy locus of f Y (Corollary 4.6) and the dynamical degree for the generic case (Theorem 4.8).
Proposition 4.1. If f is critical then f −1 is conjugate to a critical map.
If (4.1) holds, it follows that e 3 = Σ 0 ∩ Σ β ∩ {x 2 = 0} ∈ I, and We define a new complex manifold π Y : Y → P 3 by blowing up e 1 and e 3 , then the strict transform of Σ 01 , followed by the strict transform of Σ 03 . (Equivalently, we start with X and blow up the strict transform of e 1 and Σ 03 .) For j = 1, 3, we denote the exceptional divisor over e j by E j and the exceptional divisor over Σ 0j by S 0j for j = 1, 3. The induced birational Lemma 4.2. The maps in (4.3) are dominant; Σ γ is the unique exceptional hypersurface for f Y , and Σ C is the unique exceptional hypersurface for f −1 Y . Proof. Using the local coordinates defined in §2, we have In Proposition 2.1 we showed that the maps Σ 0 → S 03 → E 1 → Σ B are dominant. It follows that Σ γ is the only exceptional hypersurface for f Y , and Σ C is the only one for f −1 Y . For p ∈ P 3 , we will say that a point of π −1 Y p is at level 1 if it could have been obtained by a blowup a point or curve in P 3 . Thus the points of all fibers are of level 1, unless they lie over e 1 , e 3 , or e 2 = Σ 01 ∩ Σ 03 . The fibers E 1 ∩ S 03 and E 3 ∩ S 01 represent the points of E 1 and E 3 which are at level 2. Over e 2 , we define F 1 e 2 := S 01 ∩ π −1 Y e 2 and F 2 e 2 := S 03 ∩ π −1 Y e 2 . We see that F i e 2 , for i = 1, 2 is at level i. We see that the three curves on level 2 are not indeterminate: Proof. Let us first consider the blowup fiber over E 3 ∩ Σ 01 . For this fiber let us use a local coordinate gives a local coordinates near S 01 . It follows that the second blowup fiber E 3 ∩ S 01 is not indeterminate for f Y . The computations for f −1 Y and for E 1 ∩ S 03 are essentially the same, and we see that E 1 ∩ S 03 and E 3 ∩ S 01 are not indeterminate for f Y or to f −1 Y . To consider the second blowup fiber F 2 e 2 , we use local coordinates (ξ, s, x 3 ) 01 → [ξs : s : 1 : x 3 ]. In this coordinates we see that S 01 = {s = 0} and the strict transform of Σ 03 = {ξ = 0, x 3 = 0}. Thus the local coordinates near the blowup of Σ 03 is given by (η, s, t) 03 → (ηt, s, t) 01 → [ηts : s : 1 : t] and we have F 2 Recall from §2 that in P 3 each point on Σ βγ blows up to a line in Σ C . Note that [0 : 0 : 1 : −α 2 ] = Σ βγ ∩Σ 01 , and let F 0βγ := π −1 Y (Σ βγ ∩Σ 01 ). Note that the base point is the intersection of Σ 01 and Σ βγ , two indeterminate lines. Similarly, we write Proof. Let us consider a local coordinates in a neighborhood of the fiber F 0βγ and a local coordinates in a neighborhood of F 0BC : we see that for each η 1 we have Lemma 4.5. If f is critical, then Σ 02 is indeterminate for f Y . Each point of Σ 02 blows up to F 1 e 2 , and F 1 e 2 is mapped smoothly to Σ 02 . The set Σ 02 ∪ F 1 e 2 is totally invariant. Proof. Recall that f Σ 02 = e 2 , and the point e 2 was blown up. We consider points [s : 1 : sξ : x] which are close to Σ 02 when s is small. We see that Using the same local coordinates we also see that For the second statement, we first notice that from (4. The behavior of f Y at Σ 02 is, in suitable coordinates, given by the third model (1.5). The behavior of f Y at F 0βγ , as seen in Lemma 4.4, is different from the model (1.5). Further, we note that by Proposition 4.1 and the remark following it, the analogues of Lemmas 4.2-5 all hold for Proof. By Lemma 4.5, it suffices to consider the case j = 0. By (4.1), e 2 / ∈ Σ γ in P 3 , so the fiber over e 2 remains disjoint from Σ γ inside Y . Now e 1 = Σ 02 ∩ Σ γ in P 3 and we see that Σ 02 and Σ γ are separated when we blow up e 1 to make Y .
Recall that the degree complexity is δ(f ) = lim n→∞ (deg(f n )) 1/n . If δ(f ) > 1, then the degrees of the iterates f n grow exponentially in n. In particular, f cannot be periodic if δ(f ) > 1.
Thus δ(f ) is the spectral radius of f * Y . Inside the Picard group P ic(Y ), we let H Y be the class of a generic hyperplane in Y , and we have The characteristic polynomial of this transformation is ( Now we give the existence of Green currents, which are invariant currents with the equidistribution properties given in the following: The Theorem will then be a consequence of Theorem 1.3 of [Ba]. Up to a scalar multiple, we may write In this case, we have that σ · S 01 and σ · S 03 , are ±1, with opposite signs, depending on the order of blow-up of Σ 01 and Σ 03 . Thus σ · α + Y = ±c 01 ∓ c 03 = 0 The other possibility is that σ ⊂ Σ C . In this case, we have σ · H = deg(σ) = σ · S 01 . Further, if we let m 3 denote the multiplicity of σ at e 3 , then m 3 is bounded above by deg(σ). If σ is a curve in Σ C , then it is represented by L + m 1 F 1 01 + m 3 ǫ 3 , where F 1 01 represents a fiber of S 01 , and ǫ 3 = E 3 ∩ Σ C . The multiplicities m 1 and m 3 are bounded below by −deg(σ).
Proof. Let us describe the indeterminate behavior of f Y at Σ βγ and F 0βγ . Up to coordinate changes in domain and range, we may assume that the indeterminate curve is {ξ 2 = ξ 3 = 0}, and the maps behave like The behavior near Σ βγ is given in (2.7), and F 0βγ is given in Lemma 4.4. We will track the forward orbit f i Y Σ γ . Without loss of generality, we may assume that . By making coordinate changes in the range, we may represent the iterated map near Σ γ as (u 1 , u 2 v, v) as long as f i Y Σ γ is not an exceptional curve in Σ γ , and not a component , which we will write as {ξ 2 = ξ 3 = 0}, as above. Thus we may assume that (s, ξ 2 , ξ 3 ) = (u 1 , u 2 v, v), so f i+1 has the form (u 1 , u 2 , v) → (u 1 u 2 , u 2 v, v), which is a map of rank 1.
As we continue to iterate f Y , the other possibility is that f i Y Σ γ ⊂ Σ γ is an exceptional curve. The coordinate of the map which varies when v = 0 must be inside {u 1 = const} ⊂ Σ γ , which means that the map must be like Σ γ ∋ (u 2 v, u 1 u 2 , v) → (u 2 v, u 1 u 2 v, v), which belongs to Σ BC . Now we continue to iterate this point forward. It cannot re-enter Σ γ , because otherwise the orbit would re-enter Σ BC and become pre-periodic. Thus the only possibility is to enter the indeterminacy locus. Let N denote the first positive integer for which f N Y Σ γ ⊂ Σ γ was an exceptional curve.
Thus the forward orbit of the point can intersect the indeterminacy locus only finitely many times. If the point enters which has rank 1 again, and we continue as before. §5. Pseudo-automorphisms. In this section we assume that f is critical and we consider the condition We give conditions for f Y to be birationally equivalent to a pseudo-automorphism (Theorem 5.1). Suppose that (5.1) holds. For 1 ≤ j ≤ N − 1, we consider four possibilities: Theorem 5.1. Suppose that a critical map f satisfies (5.1), and that whenever case (iii) occurs above, then f j Y Σ γ = F 0βγ . Then there is a blowup space π : Z → Y such that f Z is a pseudo-automorphism.
It follows that In case the second possibility (ii), i.e. f j Y Σ γ is a point in Σ βγ occurs is essentially identical to the first possibility (i). For the third possibility (iii), due to Lemma 6.3 we only need to check the induced map on the blowup fiber over f j Y Σ γ = F 0βγ . We use local coordinates near F j , the blowup fiber over F 0βγ , and F j+1 , the blowup fiber over F 0BC : Using these local coordinates we see that We have The last part we have to check is f Z on F N , the blowup fiber of the line of indeterminacy Σ βγ . The local coordinates we use near F N is given by and we get From (5.2-7) we see that the induced mapping f Z is dominant on the orbit of Σ γ and therefore f Z has no exceptional hypersurface. By Lemma 4.1, it follows that f −1 Z also has no exceptional hypersurface, so f Z is a pseudo-automorphism.
Lemma 5.2. Suppose that a critical map f satisfies (5.1), and that whenever case (iii) occurs above, then the possibility (iii) can occur at most once. Let N be the smallest positive integer such that f N Y Σ γ = Σ βγ . Let m d positive integers d 1 < d 2 < · · · < d m d denote the number of iteration to have the possibility (i), that is for each We also set m s be a positive integer such that f m s +2 Y Σ γ ⊂ F 0βγ if such possibility occurs. If there is no such case we set m s = ∞. To illustrate this numbering scheme, a hypothetical orbit of Σ γ is given in Figure 5.1. Here we have assumed that we are in the simpler case m s = ∞, which means that the orbit never enters F 0βγ , so the case (iii) does not occur. Thus in Figure 5.1 the dimension can increase from 0 to 1 only via case (ii).
Let us use the numbers m s , m u , m d , u j , d j and N to define four Laurent polynomials: Theorem 5.3. If f is pseudo-automorphism and if the possibility (iii) can occur at most once, then the dynamical degree of f is given by the largest root of the polynomial Proof. By Corollary 6.2 and Lemma 6.3, we see that f Y satisfies the hypotheses of Theorem 5.1. Let f Z be the corresponding pseudo-automorphism. The dynamical degree will be the modulus of the largest root of the characteristic polynomial of f * Z . In the Appendix we show that the characteristic polynomialis given by χ f . §6. Periodic maps. In this section, we determine all possible periodic 3-step recurrences. By §3, we may assume (4.1). The question of periodicities for maps (4.1) with β 0 = 0 has been answered by Csörnyei and Laczkovic [CL]: they have shown that the only periodicities in this case are the two period 8 maps given in the Theorem stated in the Introduction. We will consider the general case where β 0 is possibly nonzero. We start by giving a necessary condition for a map to be periodic.
Proposition 6.1. If f is pseudo-automorphism and if E is an exceptional hypersurface then there is an exceptional hypersurface E ′ for f −1 such that f n E = E ′ for some n > 0 and the co-dimension of f j E is ≥ 2 for all j = 1, . . . , n − 1.
Proof. Suppose f has period p. Since f p E = E and codim f E ≥ 2, it follows that there exists 0 < n ≤ p such that codim f n−1 E ≥ 2 and codim f n E = 1. Thus f n E is an exceptional for f −1 .
Since f is critical, dimf j Σ β < 2 for j = 1, 2, and f 3 Σ β = Σ 0 ; further, dimf j Σ 0 < 2 for j = 1, 2, and f 3 Σ 0 = Σ B . By Lemma 4.2 the only exceptional hypersurface for f Y is Σ γ , and the only exceptional hypersurface for f −1 Y is Σ C . This gives us the following necessary condition for f to be periodic.
Corollary 6.2. If f is periodic, then f is critical and there is some Proof. If f is pseudo-automorphism then so is f Y . Since both f Y and f −1 Y have the unique exceptional hypersurface, there exists n ≥ 0 such that f n Y Σ γ = Σ βγ which blows up to a hypersurface Σ C . If f is periodic then so is f −1 and thus f −n Y Σ C = Σ BC .
Proof. Suppose f is a periodic map with period p. For each i = 1, . . . , p, let us set It follows that f satisfies every condition in Theorem 5.1 and therefore f Z is a pseudo-automorphism.
is a product of cyclotomic factors and thus χ(t) is self-reciprocal. Furthermore by Lemma 6.3 f Z is a pseudo-automorphism and therefore (f * Z ) −1 = (f −1 Z ) * . It follows that χ f and χ f −1 are integer polynomials with the same roots.
Proof. From (5.8) we see that the characteristic polynomial χ = χ f (t) is given by (6.1) By Lemma 6.4, χ(t) should be self-reciprocal. Since the first part of χ and the first line of (6.1) are self-reciprocal, it suffices to consider the case m s = ∞ and m u m d = 0. In this case Thus it is clear that we have m = m u = m d < N and 1 < d 1 < u 1 < · · · < d m < u m < N for some positive integer m. Thus we have By interchanging the roles of Σ β , Σ γ and Σ B , Σ C , we see that the characteristic polynomial for f −1 is given byχ Since both f and f −1 have the same characteristic polynomial, we obtain the second statement of the Lemma by comparing χ f and χ f −1 .
Lemma 6.6. Suppose f is periodic.
Proof. Suppose j * is the smallest positive integer such that u j * − d j * > d 1 . Then we have It follows that the exceptional hypersurface Σ γ is pre-periodic which contradicts to the hypothesis f is periodic.
Direct computation shows the following properties: ). Lemma 6.9. Suppose that m ≥ 2 and that f is critical satisfying (5.1). Then if m is even Thusφ ′ (1) < 0, soφ has a root greater than 1.
Theorem 6.10. If f is periodic with m = 0 and m s = ∞ then f is one of the following: • α = (−1, 0, −1, 1), β = (0, 1, 0, 0) : f αβ has period 8 and there is a conic Q such that • α = (−1/2, 0, −1, 1), β = (1, 1, 0, 0): f αβ has period 12, and Proof. The polynomial defined in (5.8) is also given by It follows that χ(t) has a root bigger than 1 if and only if N ≥ 8 and in case N = 7 the matrix representation of f * Z has 3 × 3 Jordan block with eigenvalue 1. Thus we need to check the situation f n+1 Σ γ = Σ βγ only for n ≤ 5. For this, let us parametrize Σ BC Since equations in (6.3) are polynomials in t whose coefficients are integer polynomials in the variables β 0 , α 0 , and α 2 , we may use the computer show that for 0 ≤ n ≤ 5, the only two possibilities are those listed above.
Proof. From (5.8) the characteristic polynomial of f * Z is given by It follows that χ(t) has a root bigger than 1 if and only if u 1 ≥ 8. If u 1 = 7, the f * Z has a 3 × 3 Jordan block. Thus If f * Z is periodic then d 1 ≤ 5 < u 1 . By direct computation of f n Σ γ = f n−1 Σ BC for n = 1, . . . , 5, we can easily check the two conditions (i) f n−1 Σ BC ⊂ Σ γ , (ii) f n−1 Σ BC ⊂ {x 3 = λx 2 } for some λ ∈ C and thus we see that there are only two possibilities listed in this Theorem.
Theorem 6.12. If m ≥ 2, m s = ∞ then f has exponential degree growth (and is not periodic).
Proof. By Lemmas 6.8 and 6.9 we see that χ N (1) = 0 and χ ′ N (1) = 2φ ′ (1) < 0. Since the leading coefficient of χ N is 1, there exist a real root which is strictly bigger than 1. It follows that the dynamical degree of f is strictly bigger than 1.
Theorem 6.13. If 1 ≤ m s < ∞, then f is not periodic.

Proof of Theorem 5:
The statement of the Theorem 5 in the Introduction follows from Theorems 6.10-13.
We remark that in the proof of Theorem 6.13, we see that if m s = 3 and m = 0, then the degree of f n is quadratic in n. This case occurs for α = (a, 0, 1, 1) and β = (0, 1, 0, 0), which is the so-called Lyness process and will be discussed in §8. §7. Pseudo-automorphisms with positive entropy In this section we consider the case β = (0, 1, 0, 0) and α = (a, 0, ω, 1) (7.1) where ω 2 + ω + 1 = 0 and a ∈ C \ {0}. With this choice of parameters, we see that f is critical and that Σ B = Σ 3 and Σ β = Σ 1 . Since the maps f : Σ 3 → Σ 2 → Σ 1 are dominant, (4.3) gives an 8-cycle of dominant maps Since this 8-cycle is fundamental to our understanding of f in this case, we will refer to the union of these 8 hypersurfaces as the rotor and denote it as R. Clearly, f 8 Y fixes each component of the rotor; in addition, it has a relatively simple expression. On Σ 3 , or example, we have: The restriction of f 8 Y to the rotor is studied in §C. Note that by (7.1), Σ BC = Σ 3 ∩ Σ C and Σ βγ = Σ 1 ∩ Σ γ . Using (7.2) we may verify that f Y satisfies condition (5.1), which in this case is f j Y Σ γ ⊂ F 0βγ for all 1 ≤ j ≤ 10, and f 11 Y = Σ βγ (7.4) We define the space π Z : Z → Y by successively blowing up the 11 curves γ j := f j Y Σ γ , 1 ≤ j ≤ 11. The dynamical degree, being a birational invariant, is independent of the order in which the γ j 's are blown up.
Theorem 7.1. The induced map f Z is a pseudo-automorphism, and the dynamical degree of f is greater than 1.
Proof. From (7.4) we see that f Y satisfies the condition in Theorem 5.1, so f Z is a pseudoautomorphism. By Theorem 5.3, the characteristic polynomial of f * Thus δ(f ) is the largest root of this polynomial, which is approximately 1.28064.
The space Z has been defined earlier, but now let us be more precise: we define Z as the space obtained by blowing up first γ 11 ⊂ Y , then we blow up the strict transform of γ 10 in the resulting space, followed by blowing up the strict transform of γ 9 , and continuing this way until we blow up the strict transform of γ 1 . We will use the notation Γ j to denote the exceptional divisor of the blowup of γ j . There are no points where three distinct γ j 's intersect. If p = γ j ∩ γ k , with j > k, then we blow up γ j first, and we refer to the fiber in Γ j over p as the first fiber over p, and write it as F 1 p . We then blow up the strict transform of γ k , and the blowup fiber over the point γ k ∩ Γ j is equal to Γ j ∩ Γ k .
Lemma 7.4. For 1 ≤ j ≤ 10, I(f Z j ) = F 1 0βγ ∪ Σ 02 ∪ γ 11−j . Proof. Suppose p is a point of γ j ∩ γ k 1 ≤ j < k ≤ 10. Because of the order of blowup, γ k is blown up before γ j and γ k+1 is blown up before γ j+1 . Since f Y is regular at p and the order of blowups at p is consistent with the order of blowups at f Y (p), the induced map f Z i is a local biholomorphism in a neighborhood of the exceptional divisor over p for 12 − j ≤ i ≤ 11.
Notice that for all 1 ≤ j ≤ 11 the strict transformation of γ j does not intersect Σ 02 in Y . Suppose γ j intersects F 1 0βγ at a point q. Using the local coordinates in the neighborhood (s, ζ, x 3 ) S 01 , we may assume that q = (0, ζ * , −ω) S 01 and γ j (s) = (Q 1 (s), Q 2 (s) + ζ * , Q 3 (s) − ω) S 01 , where γ j = {γ j (s), s ∈ C}, and γ j (0) = q. Consider two local coordinate charts covering the exceptional divisor over the point q : With a computation similar to Lemma 7.2, We see that the induced map is regular everywhere on the exceptional divisor over q, F(q), except the point of intersection F(q) ∩ F 1 0βγ . Now since the curve γ 11−j is the pre-image of γ 12−j , we have From the previous Lemma we have I(f Z 10 ) = F 1 0βγ ∪ Σ 02 ∪ γ 1 . Since Σ γ is the pre-image of γ 1 , we have I(f Z ) ⊂ F 1 0βγ ∪ Σ 02 ∪ Σ γ . From (5.2) we see that for all most every line ℓ ⊂ Σ γ , through e 1 in Σ γ , f maps ℓ regularly to a point q ∈ γ 1 . In our construction of Z, we blew up γ 11 , . . . , γ 2 before γ 1 . Thus the map f Z will map ℓ regularly to the fiber of Γ 1 over q unless q is an intersection point of γ 1 ∩ γ j for some 2 ≤ j ≤ 11.
Lemma 7.5. Suppose q ∈ γ 1 ∩ γ j for some j = 2, . . . , 11 and ℓ j ⊂ Σ γ be the line which mapped to q by f Y . The line ℓ j ⊂ I(f Z ) and every point in ℓ j blows up to the first blowup fiber F 1 q .
Using the induced map f Z , we see that that is, each point in ℓ j blows up to a whole first blowup fiber over q.
Before Lemma 7.2, we enumerated the possibilities for lines ℓ and points q as in the hypotheses of Lemma 7.5. Thus we may combine Lemmas 7.2-5 to have the following Theorem: Theorem 7.6. The indeterminacy locus I(f Z ) = Σ 02 ∪ F 1 0βγ ∪ ℓ 2 ∪ ℓ 3 ∪ ℓ 7 ∪ ℓ ′ 9 ∪ ℓ ′′ 9 . If ζ is a point of one of the lines ℓ, then f Z blows up ζ to the first fiber F 1 f (ℓ) .
Now we give the existence of Green currents for the invariant class α = α + Z ∈ H 1,1 (Z).
Theorem 7.7. There is a positive closed current T + Z in the class of α + Z with the property: if Ξ + is a smooth form which represents α + Z , then lim n→∞ δ 1 (f ) −n f n * Z Ξ + Z = T + Z in the weak sense of currents on Z.
Proof. The map f * Z is given in Appendix A, where we are in Case (II). Working directly with the matrix (A.1), we see that the invariant class is given by: where c 1 , c 3 > 0, c 1 + c 3 = 1, c ′ 11 > c ′ 10 > · · · > c ′ 1 > 0, and c 01 = c 03 > c ′ 8 . As in Theorem 4.9, we will show that α + Z · σ for each curve σ inside the forward image of I(f Z ). The result will the follow from Theorem 1.3 of [Ba].
Let us start with F 0βγ ⊂ I(f Z ). Points of this curve are blown up to F 0BC . The curve σ = F 0BC is the exceptional fiber inside S 03 over the point Σ BC ∩Σ 03 ∈ P 3 . Thus σ ·S 03 = −1.
Points of the indeterminate curve Σ 02 blow up to σ = F 1 e 2 . In this case, we have that σ · S 01 and σ · S 03 , are ±1, with opposite signs, so σ · α + Z = ±c 01 ∓ c 03 = 0 as was seen in the proof of Theorem 4.9.
The other possibility is ℓ ⊂ I(f Z ), for one of the indeterminate lines in Σ γ . This blows up to one of the first fibers σ = F 1 ζ . In this case, σ crosses Γ 1 transversally, so σ · Γ 1 = 1. On the other hand, σ ⊂ Γ j for some j > 1, so we have σ · Γ j = −1.
Thus we can apply a similar argument to α − Z to obtain the Green current for f −1 Z . Corollary 7.8. There is a positive closed current T − Z in the class of α − Z with the property: if Ξ − is a smooth form which represents α − Z , then lim n→∞ δ 1 (f ) −n f −n * Z Ξ − Z = T − Z in the weak sense of currents on Z.
Next we show what happens to the invariant fibration when we lift it to Z. Let us set P 0 = x 0 x 1 x 2 x 3 , and let P 1 be a homogeneous quartic polynomial defined in Appendix B. For c ∈ C, let us set S c = {cP 0 + P 1 = 0}, so the rotor R corresponds to c = ∞. Since we have f (S c ) = S ωc , the surface S 0 is invariant.
Proposition 7.9. The variety S 0 := {P 1 = 0} ⊂ P 3 has singular points at e 1 , e 3 and the fixed points p ± . If p ± are blown up (in additional to the e 1 and e 3 which were blown up to construct Y ), then the strict transform of S 0 is a nonsingular K3 surface.
Proof. Using the computer, we find that the critical points of P 1 occur exactly at e 1 , e 3 and p ± = (x ± , x ± , x ± ) ∈ C 3 where x ± are the roots of x 2 = a + (1 + ω)x. (Mathematica, for instance, can do this.) Further, p ± are singular points of type A 1 . The singular points e 1 and e 3 are type A 1 unless a = (1 + 2ω)/(1 − ω), in which case they are type A 2 . In either case, it follows (see, for instance, [EJ,Lemma 3.1 and Remark 3.2]) that S 0 is K3.
Corollary 7.10. For all but finitely many values of c ∈ C, the strict transform of S c in Z is a nonsingular K3 surface.
Let P ⊂ Y denote the (finite) set of all intersection points of distinct curves γ j ∩ γ k . Since the γ j lie in the rotor, we have P ⊂ R. The rotor is the union of 8 smooth hypersurfaces which intersect transversally, so the singular locus of R is the set where two (or more) of these surfaces intersect. We will write P s (resp. P r ) for the points of P which are contained in the singular (resp. regular) locus of R.
While Z itself depends on the order in which the curves γ j are blown up, the following Propositions are valid for any ordering of the blowups.
Proposition 7.11. For p ∈ P r , there is a unique c p ∈ C such that S c p ⊂ Y is singular at p. This is a conical singularity, and the strict transform S c p ⊂ Z contains the first fiber F 1 c p . Proof. Without loss of generality, we may choose coordinates (x, y, z) so that p = 0, L = z near p, and R = {z = 0}. Let us suppose that Since the curves f j Y Σ BC are contained in R and intersect transversally, we may suppose that near p the curves f j Y Σ BC and f k Y Σ BC coincide with the x-and y-axes. Thus the tangent to {M = 0} at p is given by z = 0, so we may suppose that M = λz + xy + · · ·. The surfaces are then S c = {M + cL = 0} = {λz + xy + cz + · · · = 0}. The surface S c is singular if c = −λ. We blow up the x-axis by the coordinate change (x, s, η) → (x, s, sη). The first fiber is F 1 p = {x = s = 0}. The strict transforms of the surfaces are S c = {(λ + c)η + x = 0}. The strict transform of the y-axis is now the s-axis, which is contained in each S c . Otherwise, the S c 's are disjoint. The strict transform of S −λ contains F 1 p . After we blow up the s-axis, the surfaces are all disjoint and smooth.
Proposition 7.12. For p ∈ P s , S c is smooth at p for all c ∈ C. The first fiber is contained in the rotor: F 1 p ⊂ R ⊂ Z. Proof. We may assume that p is a normal crossing of two of the hypersurfaces of R. Thus we may choose coordinates (x, y, z) such that p = 0, and L = xy near p. We may assume that f j Y Σ BC is the x-axis, and f k Y Σ BC is the y-axis. Since M contains both axes, we may assume that M = z + ϕ, where ϕ is divisible by xy. Thus S c = {M + cL = z + ϕ + cxy = 0} is smooth for all c ∈ C. When we blow up the x-axis, we use coordinates (x, s, η) → (x, s, sη). The strict transforms are then S c = {sη +φ + csx = 0}, whereφ is divisible by xs. Dividing this equation by s, we have S c = {η + ψ(x, s, η) + cx = 0}, where ψ(0, s, 0) = 0, since S c contains the s-axis (the strict transform of the y-axis). We have F 1 p = {x = s = 0}. Now we blow up the s-axis via the coordinates (ξ, s, t) → (ξt, s, t) = (x, s, η). This gives the new strict transforms S c = {1 +ψ(ξ, s, t) + cξ = 0}, whereψ(ξ, s, t) = t −1 ψ(ξt, s, t) is regular. The strict transform of F 1 p is now {ξ = s = 0}, which is disjoint from the S c s. If p ′ ∈ γ 1 ∩ γ 9 , then there is a unique c ′ ∈ C be such that S c ′ is singular at p ′ . Let ℓ ′ 9 denote the line for which f (ℓ ′ 9 ) = p ′ . By Theorem 7.6 and Proposition 7.11, it follows that f Z maps ℓ ′ 9 to the strict transform of S c ′ inside Z. Thus the total transform of ℓ ′ 9 under f n Z is contained in S ω n c ′ . Let p ′′ denote the other point of γ 1 ∩ γ 9 , and let c ′′ ∈ C denote the corresponding parameter. LetŜ = S c ′ ∪ S ωc ′ ∪ S ω 2 c ′ ∪ S c ′′ ∪ S ωc ′′ ∪ S ω 2 c ′′ we see thatŜ is a f Z -invariant set which contains ℓ ′ 9 ∪ ℓ ′′ 9 . Let R denote the strict transform of the rotor in Z. The sets R,Ŝ, and Σ 02 ∪ F 1 e 2 are totally invariant, and we break the indeterminacy locus into three sets: By Propositions 7.11 and 7.12, R is disjoint from the strict transform of each S c . Thus f Z is regular on Ω, and Ω is invariant under f Z .
Proposition 7.13. For every S c in Ω the dynamical degree of the restriction is δ(f 3 c ) = δ 1 (f ) 3 . Proof. Let us denote Γ a hypersurface in Z whose cohomology class in H 1,1 (Z) is H Z . It follows that the degree of f −3n Z Γ grows like δ 1 (f ) 3n . On the other hand S c ⊂ Ω does not contain an irreducible component of the indeterminacy locus for f Z . It follows that we have Using the fact that f Z is regular on the large invariant set Ω, we avoid the difficulties that can occur in defining the entropy of a map (see [G1]).
Theorem 7.14. The entropy of f is log δ 1 (f ).
Proof. Since f is equivalent to a pseudo automorphism, and f * Z is conjugate to (f * Z ) −1 , both the first and the second dynamical degrees are equal. Combining the result in [DS] and the fact that h top (f ) ≥ h top (f 0 ), we have the inequality which gives the result.
Since Ξ ± and f Z are regular on Ω, the potential of g ± is continuous on Ω. Thus may define the wedge product T 2 := T + ∧ T − as a positive, closed (2,2)-current on Ω, and we have: We have seen that the restrictions f 3 | S c are automorphisms, and there are invariant currents µ ± c on S c , as well as invariant measures µ c := µ + c ∧ µ − c (see [C]). The following property leads us to consider T ± and T 2 as the "bifurcation currents" for the family {f 3 | S c } (see [DuF]).
Theorem 7.16. For S c ⊂ Ω the slices by S c are well-defined and give the corresponding dynamical objects: T ± | S c = µ ± c , and T 2 | S c = µ c . Proof. If we set h = f 3 , then the class [S c ] is invariant under h * . Thus α + · [S c ] ∈ H 1,1 (S c ) is a class that is expanded by a factor of δ 1 (f ). It follows that the restriction Ξ + | S c gives the expanded class, and this converges to µ + c . Similarly, the normalized pullbacks/push-forwards of Ξ + ∧ Ξ − on S c will converge to µ c . at e. The induced map has only two exceptional lines which are mapped to fixed points and therefore the induced map is algebraically stable. The action on P ic is given by the matrix 2 1 −1 0 which has an eigenvalue 1 with 2 × 2 jordan block. It follows that the degree of restriction map grows linearly. The analysis in the case (8.1a) is essentially the same. The induced rotor map is now: This map has three exceptional lines. Two of them are mapped to fixed points [1 : 0 : −1] and [0 : 1 : −a]. The third exceptional line is mapped to [1 : −1 : 0], which is indeterminate. After we blow up the point [1 : −1 : 0], the induced map is algebraically stable and the action on P ic has an eigenvalue 1 with 2 × 2 jordan block. Finally, since the restriction of f W to the rotor has linear degree growth. It follows from [DiF] that this restriction is not an automorphism.
We consider first the Lyness map,i.e.,case (8.1a). This is known to be integrable, and the invariant polynomials are given in [CGM1] and [KoL]. These invariant polynomials, which satisfy (B.1) with t = 1, are: The set {Q 0 = 0} gives an invariant 8-cycle of rational surfaces, which is the rotor R ⊂ Y . (Although Q 0 = 0 consists of 4 irreducible components in P 3 , it yields an 8-cycle inside Y because these components map through the indeterminacy locus, which is blown up to yield an additional 4 divisors.) The set {Q 1 = 0} gives an invariant 4-cycle, and {Q 2 = 0} gives an invariant 3-cycle; the components of the 8-, 4-, and 3-cycles are rational surfaces. As we observed in §4, f Y induces dominant maps on each of these cycles. And as in Proposition 8.2, we may show that the restriction of f 4 to the 4-cycle, and the restriction of f 3 to the 3-cycle both have linear degree growth.
Let us define the surfaces S c = {Q c = 0} with Q c := c 0 Q 0 + c 1 Q 1 + c 2 Q 2 . If we also write S c for its strict transform inside Z, we have f S c = S c Theorem 8.3. For generic c, the surface S c is an irreducible K3 surface.
Proof. For generic c, we find that S c has 16 singular points: two of them are e 1 , e 3 , which are type A 2 , and there are 14 more which are of type A 1 . In the construction of Z, we blew up e 1 and e 3 . Then we blew up f j Σ BC , 0 ≤ j ≤ 10, and the other 14 singular points are contained in these curves. It follows that the strict transform of S c inside Z is smooth and thus K3.
Theorem 8.4. For generic c and c ′ , the intersection S c ∩S c ′ is an elliptic curve. The restriction of f 3 to S c has quadratic degree growth.
Proof. Since S c is a K3 surface, it has trivial canonical bundle. Thus the birational map f 3 of S c must be an automorphism. For generic c and c ′ = c, the intersections S c ∩ S c ′ give an invariant fibration of S c . Since f 3 |S c is an automorphism, then by [DiF] the intersection S c ∩ S c ′ is an elliptic curve and the restriction of f to the family of K3 surfaces has quadratic degree growth.
The map (8.1b) is similar. In this case the solutions to (B.1) take the form: where t R 0 = 1, t R 1 = ω 2 , and t R 2 = ω 2 . As before, we see that f Z will have an invariant 8-cycle given by the rotor R ⊂ Z. And {R 1 = 0} will give a 4-cycle of rational surfaces. For generic c, the singularities of the surface S c = { c j R j = 0} are e 1 , e 3 (type A 2 ) and e 2 (type A 1 ). As in Theorems 8.3 and 8.4, we have: Theorem 8.5. In case (8.1b): for generic c, S c is a K3 surface, f 3 is an automorphism of S c with quadratic growth, and the intersections S c ∩ S c ′ are elliptic curves. §A. Appendix: Computing the Characteristic Polynomial for f * Z . Let us consider a critical map f satisfying condition (5.1), and let m u , m d , m s , d j , u j and N be the numbers defined in §5. We define the (N + 5) × (N + 5) matrix where the * 's indicate that the 7 th through the N + 5 th rows remain to be specified. We will define the j th row r j in terms of the elements e k , which are vectors of length N + 5 in which the k th entry is 1, and all other entries are 0: (a) if j = N − d i for some i = 1, . . . , m d , then r j+6 = e j+5 − e N+5 (b) if j = N − u i for some i = 1, . . . , m u , then r j+6 = −e 1 − e 5 + e j+5 − e N+5 (c) if j = N − m s , r j+2 = −e 1 − e 3 + e j+1 − e N+5 , r j+3 = −e 3 + e j+2 r j+4 = −e 1 − e 3 − e 5 + e j+3 − e N+5 , r j+5 = −e 1 − e 3 − e 5 + e j+4 r j+6 = −e 5 + e j+5 (d) Otherwise, r j+6 = e j+5 . Let π : Z → P 3 be the space constructed in Theorems 5.1 and 5.3.
Proposition A.1. The matrix (A.1) represents f * Z . Proof. There are three cases to consider. Although we define a different basis in each case, the matrix (A.1) representing f * Z is the same.

Case (I) :
There is no 1 ≤ j ≤ N such that f j Y Σ γ ⊂ F 0βγ ∪ Σ βγ . In this case for all j = d i , u k , m s , m s+1 , . . . , m s+4 , 1 ≤ i ≤ m d , 1 ≤ k ≤ m u we have πF j ⊂ Σ 0 ∪ Σ β ∪ Σ γ and therefore . . , N , and F 1 → {Σ γ } using the ordered basis {H Z , E 1 , S 03 , S 01 , E 3 , F N , F N−1 , . . . , F 2 , F 1 } for Pic(Z) we see that (A.1) is the matrix representation for f * Z . Case (II) : There are κ positive integers 1 < s 1 < · · · < s κ < N such that for j = 1, . . . , κ f Using this new ordered basis we can see that In a similar way we may compute f * Z of H Z ,S 01 ,Ẽ 3 and F N and see that the matrix representation withB is given by (A.1).
Case (III) : There are τ positive integers 1 < q 1 < · · · < q τ < N such that f It follows that we have For the other basis elements, computations are essentially identical and thus we see that (A.1) represents f * Z with respect to the ordered basisB. According to the previous Proposition, we see that the characteristic polynomial of f * Z only depends on m u , m d , m s , d j , u j and N .
Lemma A.2. The characteristic polynomial of f * Z is given by Proof. We subtract tI from the matrix (A.1) and perform a sequence of row operations on it.
Proof. The exceptional curves C 2 and C 4 mapped to a three cycle : g : ]. For C 3 we see that g j C 3 = [1 : −ω 2 (ω/a) j−1 : 0] for all j ≥ 1. It follows that these three curves have orbits that do not encounter the indeterminacy locus of g. The remaining exceptional curve C 1 mapped to e 1 = [0 : 1 : 0], which is indeterminate. We let W be the space obtained by blowing up Σ 3 at e 1 , and we let E 1 be the corresponding exceptional divisor. Under the induced map g W we have g W (E 1 ) = E 1 and the orbit of the strict transform of C 1 remains in E 1 and does not encounter the indeterminacy locus of g W . Now if H denote the class of a generic line in W , then H, E 1 is an ordered basis for P ic(W ). The action on P ic is given by the matrix g * W = 3 1 −1 0 . The largest eigenvalue is λ = (3 + √ 5)/2 and invariant class is given by θ = λH − E 1 . Since θ 2 = λ 2 − 1 = 0, it follows from [DF, Theorem 5.4] that g is not birationally conjugate to an automorphism. Lemma C.2. If a j = ω j−2 for some j ≥ 2 then g is not birationally conjugate to an automorphism.
Proof. In case a j = ω j−2 for some j ≥ 2, the orbits of three exceptional curves C 2 , C 3 , and C 4 are the same as the previous Lemma. After we blowup e 1 on Σ 3 , the strict transformation of C 1 mapped to a point of indeterminacy after j-th iteration of g W . We let W 2 be the space obtained by blowing up W at g k W C 1 for k = 1, . . . , j and we let F k , 1 ≤ k ≤ j be the corresponding exceptional divisors. Under the induced map g W 2 , the exceptional line C 1 is removed and the orbits of remaining three exceptional curves do not encounter the indeterminacy locus of g W 2 .
Let H, F j , F j−1 , . . . , F 1 , E 1 be the ordered basis for P ic(W 2 ). The characteristic polynomial of the action on P ic is given by t j+2 − 4t j+1 + 3t j + t 2 − 2t + 1. It follows that the dynamical degree is not a Salem number. Thus by [DiF], g is not birationally conjugate to an automorphism. Lemma C.3. If a j = ω j+2 for some j ≥ 2 then g is not birationally conjugate to an automorphism.

Proof.
When a j = ω j+2 , the orbit of C 3 is different from Lemma C.1, that is g j+1 C 3 = [1 : −1 : 0], which is indeterminate. We let W 3 be the space obtained by blowing up Σ 3 at e 1 and g k C 3 , 1 ≤ k ≤ j + 1, and we let E 1 and F k , 1 ≤ k ≤ j + 1 be the corresponding exceptional divisors. Using the ordered basis H, F j+1 , F j , . . . , F 1 , E 1 for P ic(W 3 ), we see that the characteristic polynomial of the action on P ic is given by t j+3 − 3t j+2 + t j+1 + t. Similarly as in Lemma C.2, the dynamical degree is not a Salem number and therefore g is not birationally conjugate to an automorphism. Lemma C.4. If a = ω then g is not birationally conjugate to an automorphism.
Proof. In this case we see that C 2 is mapped to a point of indeterminacy under 2 iterations and C 4 is also mapped to a point of indeterminacy under 3 iterations. After we blowup e 1 , we can check that the orbits of other two remaining exceptional lines does not encounter the indeterminacy locus. After we blow up the orbit of C 2 and the orbit of C 4 , we see that the dynamical degree of g is given by the largest root of the polynomial t 3 − t 2 − 2t − 1. Again since this number is not a Salem number we have our result. Lemma C.5. If a = ω 2 then g is not birationally conjugate to an automorphism.
Proof. If a = ω 2 the each component of g has the same factor x 0 + x 1 + ω 2 x 2 . It follows that the restriction of f 8 Y to Σ 3 is a degree 2 birational map. There are two exceptional lines and both exceptional lines are mapped to points of indeterminacy. After we blowup the points on the orbits of three exceptional lines, we see that the induced map has one exceptional line which is mapped to a point of indeterminacy. Once we blow up this point of indeterminacy, we see that the induced map has no exceptional lines and therefore the induced map is algebraically stable. Furthermore the characteristic polynomial of the action on P ic is t(1+t)(t−1) 3 and the action on P ic has 2 × 2 Jordan block. It follows that the degree of g grows linearly. According to [DiF], we have that g is not birationally conjugate to an automorphism. Lemma C.6. If a = 1 then the degree g grows quadratically.
Proof. For this case all four exceptional curves are mapped to points of indeterminacy: g : C 1 → e 1 , C 2 → [0 : 1 : −ω], C 3 → [1 : −ω 2 : 0] → [1 : −1 : 0] and g : C 4 → [1 : 0 : −ω 2 ]. We let Z be the space obtained by blowing up Σ 3 at all five points in the orbit of exceptional curves and we let E 1 , Q 2 , Q 3 , Q 4 , and Q 5 be the corresponding exceptional divisors. Under the induced map g Z , there is a unique exceptional line which is the strict transformation of C 1 . We see that g Z C 1 is a point of indeterminacy of g Z . By blowing up one more point on E 1 we make the induced map an algebraically stable. Let us denote Q 1 the exceptional divisors corresponding to the point blow ups on E 1 . Let us use H, F 1 , F 2 , F 3 , F 4 , F 5 , E 1 as the ordered basis of P ic. The characteristic polynomial of the action on P ic is given by (t − 1) 4 (t + 1)(t 2 + t + 1) and the matrix representation of the action on P ic has 3 × 3 Jordan block. It follows that the degree of g grows quadratically.