Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices

We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to ${\cal SC}_q$.


§0. Introduction
Let M q denote the space of q × q matrices, and let P(M q ) denote its projectivization. For a matrix x = (x ij ) we consider two maps. One is J(x) = (x −1 ij ) which takes the reciprocal of each entry of the matrix, and the other is the matrix inverse I(x) = (x ij ) −1 . The involutions I and J, and thus the mapping K = I • J, arise as basic symmetries in Lattice Statistical Mechanics (see [BM], [BMV]). This leads to the problem of determining the iterated behavior of K on P(M q ) (see [AABHM], [AABM], [AMV2], [BV]). A basic question is to know the degree complexity δ(K) := lim n→∞ (deg(K n )) 1/n = lim n→∞ (deg(K • · · · • K)) 1/n of the iterates of this map. The quantity log δ is also called the algebraic entropy (see [BV]). We note that PM q has dimension q 2 − 1, I has degree q − 1, and J has degree q 2 − 1. Thus a computer cannot directly evaluate the composition K 2 = K • K (or even K = I • J) unless q is small.
The q ×q matrices correspond to the coupling constants of a system in which each location has q possible states. In more specific models, there may be additional symmetries, and such symmetries define a K-invariant subspace S ⊂ P(M q ) (see [AMV1]). In general, the degree of the restriction K|S will be lower than the degree of K, and the corresponding question in this case is to know δ(K|S) = lim n→∞ (deg(K n |S)) 1/n . An example of this, related to Potts models, is the subspace C q of cyclic matrices, i.e., matrices (x ij ) for which x ij depends only on j − i (mod q). A cyclic matrix is thus determined by numbers x 0 , . . . , x q−1 according to the formula M (x 0 , . . . , x q−1 ) =       x 0 x 1 x q−1 x q−1 . . . . . .
The degree growth of K|C q was determined in [BV]. Another case of evident importance is SC q , the symmetric, cyclic matrices. The degree growth of K|SC q was determined in [AMV2] for prime q. In this paper we determine δ(K|SC q ) for all q. In doing this, we expose a general method of determining δ, which we believe will also be applicable to the study of δ(K|S) for more general spaces S.
Main Theorem. The dynamical degree δ(K|SC q ) = ρ 2 , where ρ is the spectral radius of an integer matrix M . When q is odd, M is defined by (4.3-7); when q = 2×odd, M is defined by (5.6-12); and when q is divisible by 4, M is defined by (6.5-12).
The mappings K|C q and K|SC q lead to maps of the form f = L • J on P N , where L is linear, and J = [x −1 0 : · · · : x −1 N ]. In the case of K|C q , we have L = F , the matrix representation of the finite Fourier transform, and the entries are qth roots of unity. By the internal symmetry of the map, the exceptional hypersurfaces Σ i = {x i = 0} all behave in the same way, and δ for these maps is found easily by the method of regularization described below. The family of "Noetherian maps" was introduced in [BHM] and generalized to "elementary maps" in [BK1]. These maps have the feature that all exceptional hypersurfaces behave like Σ i → * → · · · → e i V i , (0.2) which means that Σ i blows down to a point * , which then maps forward for finite time until it reaches a point of indeterminacy e i , which blows up to a hypersurface V i . The reason for deg(f n ) < (deg(f )) n comes from the existence of exceptional hypersurfaces like Σ i , called "degree lowering" in [FS], which are mapped into the indeterminacy locus. As we pass from K|C q to K|SC q , a number of symmetries are added. Because of these additional symmetries, the dimension of the representation f = L•J on P N changes from N = q − 1 to N = ⌊q/2⌋. The new matrix L, however, is more difficult to work with explicitly; its entries have changed from roots of unity to more general cyclotomic numbers. The exceptional hypersurfaces all blow down to points, but their subsequent behaviors are richly varied, showing phenomena connected to properties of the cyclotomic numbers.
If f : P N P N is a rational map, then there is a well-defined pullback map on cohomology f * : H 1,1 (P N ) → H 1,1 (P N ). The cohomology of projective space is generated by the class of a hypersurface H, and the connection between cohomology and degree is given by the formula In our approach, we construct a new complex manifold π : X → P N , which will be obtained by performing certain (depending on f ) blow-ups over P N . This construction induces a rational map f X : X X which has the additional property that (f n X ) * = (f * X ) n on H 1,1 (X). (0.4) Once we have our good model X, we find δ(f ) = δ(f X ) by computing the spectral radius of the mapping f * X . Diller and Favre [DF] showed that such a construction of X with (0.4) is always possible for birational maps in dimension 2. This method for determining δ then gives a tool for deciding whether f is integrable (which happens when δ = 1) or has positive entropy (in which case δ(f ) > 1). This was used in the integrable case in [BTR], [T1,2] and in both cases in [BK2].
We note that the space X which is constructed by this procedure is useful for understanding further properties of f . For instance, it has proved useful in analyzing the pointwise dynamics of f on real points (see [BD]).
An important difference between the cases of dimension 2 and dimension > 2, as well as a reason why the maps K|SC q do not fall within the scope of earlier approaches, is that exceptional hypersurfaces cannot always be removed from the dynamical system by blowups. In fact, the new map f X can have more indeterminate components and exceptional hypersurfaces than the original map.
Our method proceeds as follows. After choosing subspaces λ 0 , . . . , λ j as centers of blowup, we construct X. The blowup fibers Λ i over λ i , i = 0, . . . , j, together with H, provide a convenient basis for P ic(X). A careful examination of f −1 lets us determine f −1 X H and f −1 X Λ i , and thus we can determine the action of the linear map f * X on P ic(X). In order to see whether (0.4) holds, we need to track the forward orbits f n E for each exceptional hypersurface E. By Theorem 1.1, the condition that f n E ⊂ I X for each n ≥ 0 and each E is sufficient for (0.4) to hold. We develop two techniques to verify this last condition for our maps K|SC q . One of them, called a "hook," is a subvariety α E ⊂ I X such that f X α E = α E , and f j E ⊃ α E . The simplest case of this is a fixed point. The other technique uses the fact that f = L • J is defined over the cyclotomic numbers, and we cannot have f n X E ⊂ I X for number theoretic reasons.
Let us describe the contents of this paper. In §1 we discuss blowups and the map J. We show how to write blowups in local coordinates, how to describe J X , and how to determine J * X . We also give sufficient conditions for (0.4).
In §2, we show how this approach may be applied to K|C q . In this case, the exceptional orbits are of the form (0.2). We construct the space X by blowing up the points of the exceptional orbits. After these blowups, the induced map f X has no exceptional hypersurfaces, which implies that (0.4) holds. A calculation of f * X and its spectral radius leads to the same number δ(K|C q ) that was found in [BV].
In §3, we give the setup of the symmetric, cyclic case. When q is prime, the map K|SC q exhibits the same general phenomenon: the orbits of all exceptional hypersurfaces are of the form (0.2). As before, we construct X by blowing up the point orbits, and we find that the new map f X has no exceptional hypersurfaces. Thus we recapture the δ(K|SC q ) from [AMV2].
When q is not prime, however, the map K|C q develops a new kind of symmetry as we pass to SC q . Now there are exceptional orbits where p i blows up to a variety W i of positive dimension but too small to be a hypersurface, yet W i blows up further and becomes a hypersurface V i . In §4, we work with the case where q is a general odd number. We construct our a blowup space π : X → SC q , and we obtain an induced map f X . If i is relatively prime to q, then the orbit of Σ i has the form (0.2), and after blowing up the singular orbit, Σ i will no longer be exceptional. On the other hand, if i is not relatively prime to q, then the exceptional orbit has the form (0.5). Let r divide q, and letr = q/r, and define the sets S r = {1 ≤ j ≤ (q − 1)/2 : gcd(j, q) = r}. We will see below that if i ∈ S r and j ∈ Sr, then there is an interaction between the (exceptional) orbits of Σ i and Σ j (see Figure 4.1). After blowing up along certain linear subspaces, we find a 2-cycle hook α r ↔ αr for all hypersurfaces Σ i , i ∈ S r ∪ Sr.
In §5, we consider the case q = 2 × odd. We construct a new space by blowing up along various subspaces. We find that for each odd divisor r > 1 of q, the exceptional varieties Σ i , i ∈ S r ∪ S 2r act like the case where q is odd. As before, we construct a hook α r ↔ αr for all i ∈ S r ∪ S 2r ∪ Sr ∪ S 2r . However, there is also a new phenomenon, which we call the "wringer" (see Figure 5.1), which consists of an f -invariant 4-cycle of blowup fibers. All of the exceptional hypersurfaces Σ i , i ∈ S 1 ∪ S 2 enter the wringer. We find hooks for all of these hypersurfaces, which shows that (0.4) holds for f X .
In §6, we consider the case where q is divisible by 4. Again, we construct X and obtain a new map f X . In this case, f X has some exceptional hypersurfaces with hooks. Yet a number of exceptional hypersurfaces remain to be analyzed. These hypersurfaces are of the form Σ i → c i → · · · : they blow down to points, and we must show that no point of this orbit blows up, i.e., f n X c i / ∈ I X for all n ≥ 0. The complication of one such orbit is shown in Figure 6.1.
We approach this problem now by taking advantage of cyclotomic properties of the coefficients of f . We show that we can work over the integers modulo µ, for certain primes µ, and the orbit {f n X c i : n ≥ 0} is pre-periodic to an orbit which is disjoint from I X and periodic in this reduced number ring.
In each of these cases, we regularize f by constructing an X such that (0.4) holds, and we write down f * X explicitly. Thus δ(K|SC q ) is the spectral radius of this linear transformation, which is given as modulus of the largest zero of the characteristic polynomial of f * X . We write down general formulas for the characteristic polynomials in the cases q =odd and q = 2×odd.
We give some Appendices to show how our Theorems may be used to calculate δ(f ) in an efficient manner.
The structures of the sets of exceptional hypersurfaces are both complicated and different for the various cases of q. So at the beginning of each section, we give a visual summary of the exceptional hypersurfaces and their orbits. §1. Complex Manifolds and their Blow-ups Recall that complex projective space P N consists of complex N + 1-tuples [x 0 : · · · : x N ] subject to the equivalence condition [x 0 : · · · : x N ] ≡ [λx 0 : · · · : λx N ] for any nonzero λ ∈ C.
A rational map f = [F 0 : · · · : F N ] : P N → P N is given by an N + 1-tuple of homogeneous polynomials of the same degree d. Without loss of generality we may assume that these polynomials have no common factor. The indeterminacy locus I = {x ∈ P N : F 0 (x) = · · · = F N (x) = 0} is the set of points where f does not define a mapping to P N . Since the F j have no common factor, I has codimension at least 2. Clearly f is holomorphic on P N − I, but if x 0 ∈ I, then f cannot be extended to be continuous at x 0 . If S ⊂ P N is an irreducible algebraic subvariety with S ⊂ I, then we define the strict image, written f (S), as the closure of f (S − I). Thus f (S) is an algebraic subvariety of P N . We say that S is exceptional if the dimension of f (S) is strictly less than the dimension of S.
In a similar fashion, we may construct the blow-up space π ′′ := π 2 • π 21 : M 21 → C 3 by blowing up X 2 first and then X 1 . We say that a map h : X 1 → X 2 is a pseudo-isomorphism if it is biholomorphic outside a subvariety of codimension ≥ 2. Thus (π ′ , M 12 ) and (π ′′ , M 21 ) are pseudo-isomorphic, since (π ′′ ) −1 • π ′ extends to a biholomorphism between M 12 − (π ′ ) −1 0 and M 21 − (π ′′ ) −1 0. In our discussion of degree growth, we will be concerned only with divisors, and in this context pseudo-isomorphic spaces are equivalent. Thus when we perform multiple blowups, we will not be concerned about the order in which they are performed since the spaces obtained will be pseudo-isomorphic.
Next we discuss the map J : P N P N given by J[x 0 : · · · : For a subset T ⊂ {0, . . . , N } we use the notation A point x is indeterminate for J exactly when two or more coordinates are zero. That is to say The total image of an indeterminate point is given by The exceptional hypersurfaces for J are exactly the hypersurfaces Σ i for 0 ≤ i ≤ N , and we have f (Σ i ) = e i := [0 : · · · : 0 : 1 : 0 : · · ·]. Let π : X → P N denote the blowup of the point e i , and let E i := π −1 e i ∼ = P N−1 . We introduce the notation x ′ = [x 0 : · · · : x i−1 : 0 : x i+1 : · · · : x N ] and J ′ x ′ = [x −1 0 : · · · : x −1 i−1 : 0 : x −1 i+1 : · · · : x −1 N ]. Thus near Σ i we have Letting t → 0, we find that the induced map J X : X X is given by The effect of passing to the blowup X is that Σ i is no longer exceptional. Since J is an involution, we also have . . , N } be a subset with i / ∈ T and #T ≥ 2, and let Σ T denote its strict transform inside X. We see that Σ T ∩ E i is nonempty and indeterminate for J X , and the union of such sets gives E i ∩ I(J X ). Now let us discuss the relationship between blowups and the indeterminate strata of J. For T ⊂ {0, . . . , N }, #T ≥ 2, we have Σ T ⊂ I, and f * : Σ * T ∋ p → Π T . Let π : X → P N be the blowup of P N along the subspaces Σ T and Π T . Let S T = π −1 Σ T and P T = π −1 Π T denote the exceptional fibers. The induced map J X : X X acts to interchange base and fiber coordinates: where J ′′ (ξ) = ξ −1 on Π T , and J ′ (x) = x −1 on Σ T . In particular, J X is a birational map which interchanges the two exceptional hypersurfaces, and acts again like J, separately on the fiber and base, and interchanges fiber and base. Now let π : X → P N be a complex manfold obtained by blowing up a sequence of smooth subspaces. If r = p/q is a rational function (quotient of two homogeneous polynomials of the same degree), we will say that π * r := r • π is a rational function on X. We consider the group Div(X) of integral divisors on X, i.e. the finite sums D = n j V j , where n j ∈ Z, and V j is an irreducible hypersurface in X. We say that divisors D, D ′ are linearly equivalent if there is a rational function on X such that D − D ′ is the divisor of r. We define P ic(X) to be the set of divisors on X modulo linear equivalence.
For a rational map f : X Y , there is an induced map f * : P ic(Y ) → P ic(X): if D ∈ P ic(Y ), its pullback is well defined as a divisor on X − I because f is holomorphic there. Taking its closure inside X, we obtain f * D. Let H = {ℓ = 0} denote a linear hypersurface in P N . The group P ic(P N ) is generated by H. If f : P N P N is a rational map, then f * H = deg(f )H. Let H X = π * H be the divisor of π * ℓ = ℓ•π in X. A basis for P ic(X) is given by H X , together with the (finitely many) irreducible components of exceptional hypersurfaces for π. We may choose an ordered basis H X , E 1 , . . . , E s for P ic(X) and write f * with respect to this basis as an integer matrix M f . It follows that deg(f ) is the (1,1) entry of M f .
Let us consider the blowup π : Y → P N of Σ 0,...,M = {x 0 = · · · = x M = 0}, with M < N . We write F(x) := π −1 x for the fiber over x ∈ Σ 0,...,M , and we let Λ := π −1 Σ 0,...,M denote the exceptional divisor of the blowup. It follows that H Y and Λ give a basis of P ic(Y ). Let J Y : Y Y denote the map induced by J. For j > M , the induced map J Y |Σ j : Σ j F(e j ) may be written in coordinates in a fashion similar to (1.5) and is seen to be a dominant map. Since F(e j ) ∼ = P N−M −1 , we see that Σ j is exceptional.
We have noted that Σ 0,...,M ⊂ I and that Σ 0,...,M ∋ p → f * p = Π 0,...,M . The indeterminacy locus I Y of J Y has codimension 2 and thus does not contain Λ. In fact, can be written in coordinates similar to (1.6) and is thus seen to be birational. Observe that there is a subspace Γ ⊂ P N of codimension M + 1 such that JΓ = Σ 0,...,M . It follows that J Y blows up Γ to Λ, and thus Γ ⊂ I Y . T ⊃ {0, . . . , M } holds if and only if Σ T ⊂ Σ 0,...,M , or Σ T has a strict transform in Y . We see, then, that We see that we have multiplicity 1 for the divisors Σ j because the linear factor t in (1.4), we means that the pullback of the defining function will vanish to first order. Now let us write the class of Σ j ∈ P ic(Y ) in terms of the basis {H Y , Λ}. First, we see that Σ j = {x j = 0} = H is the class of a general hypersurface in P ic(P N ), so π * Σ j = H Y . Since we have Σ 0,...,M ⊂ Σ j if and only if j ≤ M , we have We have seen that J Y maps Λ − I to the strict transform of Π 0,...,M which is not contained in a general hyperplane. Thus f −1 Y {ℓ = 0} will not contain Λ. Pulling back by π * , we have If (M f ) n = M f n , then the matrix M f allows us to determine the degrees of the iterates of f , since the degree of f n is given by the (1,1)-entry of M f n . The following result gives a sufficient condition for this to hold. Fornaess and Sibony [FS] showed that when X = P N , this condition is actually equivalent to (1.14). Theorem 1.1 is a special case of Propositions 1.1 and 1.2 of [BK1].
Theorem 1.1. Let f : X X be a rational map. We suppose that for all exceptional hypersurfaces E there is a point p ∈ E such that f n p / ∈ I for all n ≥ 0. Then it follows that (1.14) Proof. Condition (1.14) is clearly equivalent to condition (0.4). Thus we need to show that We note that if there is a point p ∈ E such that f n p / ∈ I for all n ≥ 0, then the set E − n≥0 I(f n ) has full measure in E. Thus the forward pointwise dynamics of f is defined on almost every point of E. The following three results are direct consequences of Theorem 1.1.
Corollary 1.2. If for each irreducible exceptional hypersurface E, we have f n E ⊂ I for all n ≥ 1, then condition (1.14), or equivalently (0.4), holds . Proposition 1.3. Let f : X X be a rational map. Suppose that there is a subvariety S ⊂ X such that S, f S . . . , f j−1 S ⊂ I, and f j S = S. If E is an exceptional hypersurface such that E, f 2 E, . . . , f ℓ−1 E ⊂ I, and f ℓ E ⊃ S, then there is a point p ∈ E such that f n p / ∈ I for all n ≥ 0.
In this situation, we will say that S is a hook for E. Sometimes, instead of specifying f S = S, we will say that f : S → S is a dominant map, which means that the generic rank of f |S is the same as the dimension of the target space S.
Theorem 1.4. Let f : X X be a rational map. If there is a hook for every exceptional hypersurface, then (0.4) and (1.14) hold. §2. Cyclic (Circulant) Matrices Let ω denote a primitive qth root of unity, and let us write F = (ω jk ) 0≤j,k≤q−1 , i.e., Given numbers x 0 , . . . , x q−1 , we have the diagonal matrix A basic property (cf. [D,Chapter 3]) is that F conjugates diagonal matrices to cyclic matrices. Specifically, gives an isomorphism between C q and P q−1 . The map I : C q → C q may now be represented as where F : P q−1 → P q−1 denotes the matrix multiplication map x → F x. A computation (see [D, p. 31]) shows that F 2 is q times the permutation matrix corresponding to the permutation x j ↔ x q−j for 1 ≤ j ≤ q − 1, so F 4 is a multiple of the identity matrix. On projective space, F 2 simply permutes the coordinates, so we have F 2 • J = J • F 2 . From this and the identity Following the discussion in §1, we know that the exceptional divisors of f : We let π : X → P q−1 denote the complex manifold obtained by blowing up the orbits {f j , e j }, 0 ≤ j ≤ q − 1. Let F j and E j denote the blow-up fibers in X over f j and e j . It follows that Further, by §1 or [BK1] we have that f X is 1-regular, and We take {H X , E 0 , F 0 , . . . , E q−1 , F q−1 } as an ordered basis for H 1,1 (X). Thus the linear transformation f * X is completely defined by (2.1) and (2.2), and we may write it in matrix form as: It follows that deg(f n ) is the upper left hand entry of the nth power of the matrix (2.3). Further, the characteristic polynomial of (2.3) is Summarizing our discussion, we obtain the degree complexity numbers which were found earlier in [BV]: To work with symmetric, cyclic matrices, we consider separately the cases of q even and odd. In §3 and §4 we will assume that q is odd, and we define p := (q − 1)/2.
With this isomorphism, we transfer the map F • J : SC q SC q to a map where A is a (p + 1) × (p + 1) matrix which will we now determine. It is easily seen that the 0th column a 0 is the same as the 0th column f 0 = (1, . . . , 1). For 1 ≤ j ≤ p, the symmetry of ιx means that the jth column of A is the sum of the jth and (q − j)th columns of F . Thus we have Immediate properties are Summing over roots of unity, we find Thus, by (3.3), A 2 = qI, so A acts as an involution on projective space.
As in the general cyclic case, we see that we have the orbit Now we consider the orbit of Σ i for i = 0. Let us define v 1 = [1 : t 1 : · · · : t p ] ∈ P p to be the point whose entries are ±1 and which is given by (3.4) Lemma 3.1. Ja 1 = Av 1 .
For ω 3 to cancel, we must have t 4 = −t 2 , etc. We continue in this fashion and determine t j = −t j−2 for all even j. Using (3.2), we see that ω p−1 = ω p , so this equation ends like Thus we have t p = −t p−1 . Now we can come back down the indices and determine t j−2 = −t j for all odd j. We see that these values of t j are consistent with (3.4), which shows that the right hand equation holds for k = 1. Now for general k, we have and we can repeat the argument that was used for k = 1.
Let us define v k = [1 : t ′ 1 : · · · : t ′ p ], t ′ π(j) = t j with t j as in (3.4), so v k is obtained from v 1 by permuting the coordinates.
Proof. As in Lemma 3.1, we will show that ω ik (1 + J ω Ji t ′ J ) = 2t ′ k for all 1 ≤ i ≤ p. By Lemma 3.1, we have ω I (1 + ω Ij t j ) = 2t 1 for all 1 ≤ I ≤ p. First observe that π(1) = k, so t ′ k = t 1 . Now set I = π(i) and J = π(j). It follows that the second equation is obtained from the first one by substitution of the subscripts, which amounts to permuting various coefficients.
Theorem 3.3. If k ∈ S 1 , then f maps: The second equality follows from Lemma 3.2, and the third equality follows because A is an involution.
To conclude this Section, we suppose that q is prime. This means that S 1 = {1, . . . , p}. Let X be the complex manifold obtained by blowing up the points a j and e j for 0 ≤ j ≤ p as well as v j and Av j for 1 ≤ j ≤ p. Let f X : X X be the induced birational map. It follows from §1 that f X has no exceptional divisors and is thus 1-regular. By Theorem 3.3, then, we have: (3.6) The linear map f * X is determined by (3.6). Thus we may use (3.6) to write f * X as a matrix and compute its characteristic polynomial. We could do this directly, as we did in §2. In this case, simply observe that Theorem 3.3 implies that f = AJ is an elementary map. A formula for the degree growth of any elementary map was given in [BK1,Theorem A.1]. By that formula we recapture the numbers obtained in [AMV2]: We observe that in the odd case, we have {i/r : i ∈ S r } = {j : gcd(j, q/r) = 1}. (4.1) We will use this observation to bring ourselves back to certain aspects of the "relatively prime" case. Let 1 < r < q be a divisor of q, and setq = q/r,p = (q − 1)/2. Let us fix an element k ∈ S r and setk = k/r. It follows from (4.1) that gcd(k,q) = 1. The numberω := ω r is a primitiveqth root of unity. LetÃ denote thep ×p matrix constructed like A but using the numbersω j =ω j +ωq −j . Letṽ 1 = [1 :t 1 : · · · :tp] denote the vector (3.4). Let η r = [1 : 0 : · · · : 0 :t 1 : 0 : · · ·] ∈ Π 0 mod r ⊂ P p be obtained fromṽ 1 by inserting r − 1 zeros between every pair of coordinates.
Lemma 4.1. Let 1 < r < q be a divisor of q. Then Ja r = Aη r , and f a r = v r .
Lemma 4.2. If k ∈ S r , then η k := f a k is obtained from v r by permuting the nonzero entries.
Proof. This Lemma follows from Lemma 4.1 exactly the same way that Lemma 3.2 follows from Lemma 3.1.
Let us construct the complex manifold π X : X → P p by a series of blow-ups. First we blow up e 0 and all the a j . We also blow up the points v j , Av j and e j for all j ∈ S 1 . Next we blow up the subspaces Π 0 mod r for all divisors r of q. If r 1 and r 2 both divide q, and r 2 divides r 1 , then we blow up Π 0 mod r 1 before Π 0 mod r 2 . As we observed in §1, we get different manifolds X, depending on the order of the blowups of linear subspaces that intersect, but the results in any case will be pseudo-isomorphic, and thus equivalent for our purposes. We will denote the exceptional blowup fibers over a j , v j , Av j , and e j by A j , V j , AV j and E j . We use the notation P r for the exceptional fiber over Π 0 mod r . Now let us discuss the exceptional locus of the induced map f X : X X. As in §3, we have Since A is invertible, f X is locally equivalent to J X , so by (1.5) and (1.6) we see that none of these hypersurfaces is exceptional for f X . P ic(X) is generated by H = H X , the point blow-up fibers, and the P r 's. By (4.2) we have where we use the notationÊ = i∈S 1 E i andP = r P r The left hand part of the first line follows from (4.2). Now to explain the right hand side of the same line, we note that {Σ 0 } X denotes the class generated by the strict transform of Σ 0 in P ic(X). To write this in terms of our basis, we observe that of all the blowup points, the only ones contained in Σ 0 are e i for i ∈ S 1 . On the other hand, none of the blowup subspaces Π 0 mod r is contained in Σ 0 . Thus H X is equal to {Σ 0 } X plus E j for j ∈ S 1 , which gives the first line of (4.3). For the second line, we have H X = {Σ i } X + · · ·, where the dots represent all the blowup fibers lying over subsets of Σ i . The the sums of the E's correspond to all the blowup points contained in Σ i , and for theP term recall that if i ∈ S 1 and r divides q, then i ≡ 0 mod r, and thus Π 0 mod r ⊂ Σ i . If j / ∈ S 1 , then j ∈ S r for r = gcd(j, q). For η ∈ Π 0 mod r we let F(η) denote the P r fiber over η. For the special points η j , we write simply F j := F(η j ). For each η, the induced map is birational by (1.8). Since all the fibers map to the same space Λ r , it follows that P r is exceptional. In particular, we have Thus by (1.5) Σ j is not exceptional. A similar calculation shows that A j F j is dominant, and in particular, the A j are exceptional for j ∈ S r .
Since each F j is contained in P r when j ∈ S r , we have Also, for j ∈ S r , we have Hooks. In the sequel we will repeatedly use the notationr := q/r, where 1 < r < q divides q. Thuŝ r = r. Let us define the point τ r := [r − 1 : 0 : · · · : 0 : −1 : 0 : · · · : 0 : −1 : 0 : · · ·] ∈ Π 0 modr , and let us define ξ r := [0 : 1 : · · · : 1 : 0 : 1 : · · · : 1 : 0 : 1 : · · ·] ∈ Σ 0 modr . We define α r ∈ Pr to be the point whose base coordinates are τ r and whose fiber coordinates are ξ r . Now to show that (f n X ) * = (f * X ) n we will follow the procedure which is sketched in Figure  4.1. That is, we suppose that i 1 , i 2 ∈ S r and j 1 , j 2 ∈ Sr, so the orbits are as in (4.4). We will show that there is a 2-cycle α r ↔ αr with α r ∈ Λ r − I and αr ∈ Λr − I. This 2-cycle will serve as a hook for P r and for all A j with j ∈ S r (see Proposition 1.3).
Theorem 4.4. The action on cohomology f * X is given by: Proof. Everything except the last line is a consequence of (4.4), (4.6) and (4.7). It remains to determine f * X H, which is the same as J * X H. We recall from §1 that J * X H is equal to N · H minus a linear combination of the exceptional blowup fibers over the indeterminate subspaces that got blown up. Here N = p, the dimension of the space X. The multiples of the exceptional blowup fibers are, according to (1.12) and (1.13), given by −M , where M is one less than the codimension of the blowup base. This gives the numbers in the last line of the formula above.
Let us consider the prime factorization q = p m 1 1 p m 2 2 · · · p m k k . For each divisor r > 1 of q, we set µ r := ⌊ q−1 2r ⌋ + 1, κ r = #S r , and κ = q−1 2 − r κ r . We define (4.8) Theorem 4.5. The map f X satisfies (0.4), and the dynamical degree δ(K|SC q ) is ρ 2 , where ρ is the largest root of (4.9) Proof. We have found hooks for all the exceptional hypersurfaces of f X , so (0.4) holds by Theorem 1.4. The proof that formula (4.9) gives characteristic polynomial of f * X is given in Appendix E. §5. Symmetric, Cyclic Matrices: q = 2×odd For the rest of this paper we consider the case of even q. Let us set p = q/2 and ι(x 0 , . . . , x p ) = (x 0 , . . . , x p−1 , x p , x p−1 , . . . , x 1 ). For even q, the matrix in (0.1) is symmetric if and only if it has the form M (ι(x 0 , . . . , x p )). As in §3, we have an isomorphism With this isomorphism we transfer the map F • J to the map f := A • J : P p P p .
Lemma 5.2. Let r be an odd divisor of q. For j ∈ S r , we have η j := f a j ∈ Π r mod 2r , and η 2j := f a 2j ∈ Π 0 mod 2r .
Proof. Let us first consider the case AΠ odd . A linear subspace AΠ odd is spanned by column vectors {a 1 , a 3 , . . . , a p }. When j is odd, a j = [2 : ω j : ω 2j : · · · : ω (p−1)j : −2]. By (5.1) With the fact that ω (p−k)j = ω kj for even j, the proof for AΠ even is similar. With this formula for AΠ odd , we see that it is invariant under J. Now since A is an involution, we have f AΠ odd = Π odd .
Let us construct the complex manifold π : X → P p by a series of blow-ups. First we blow up the points e 0 , e p and a j for all j. Next we blow up the subspaces Π even , Π odd , AΠ even , and AΠ odd . Then we blow up the subspaces Π 0 mod 2r , Π r mod 2r and Π 0 mod r for all r / ∈ S 1 ∪ S 2 . We continue with our convention that if r 2 divides r 1 then we first blow up Π 0 mod 2r 1 , Π r 1 mod 2r 1 , then Π 0 mod r 1 , and then the corresponding spaces for r 2 . We will use the following notation for (π-exceptional) divisors of the blowup: and for every proper divisor r of p we will write: π : P e,r → Π 0 mod 2r , P o,r → Π r mod 2r , P r → Π 0 mod r .
For 1 ≤ i ≤ p − 1, we let F i = F(η i ) denote the fiber over η i . We define Λ r as the strict transform of AΣ 0 mod r in X, and Λ e/o,r as the strict transforms of AΣ 0/r mod 2r .
We will do two things in the rest of this Section: we will compute f * X on P ic(X), and we will show that f X : X X is 1-regular. It is frequently a straightforward calculation to determine f * X and more difficult to show that the map is 1-regular. Let us start by computing f * X . We will take H = H X , E 0/p , A i , i = 0, . . . , p, P e/o , AP e/o , P e/o,r , P r as a basis for P ic(X).
We see that Σ 0 contains e p as well as Π odd , as well as Π r mod 2r ⊂ Π odd ; and Σ 0 contains no other centers of blow-up. Thus we have whereP e = P e + r P e,r . Next, consider a divisor r of p = q/2, so r is odd. If i ∈ S r , then i is odd, and the set Σ i contains the following centers of blowup: e 0 , e p , Π even , Π s mod 2s and Π 0 mod s for all s which divide p but not r. Thus we have where I r is the set of numbers 1 ≤ k ≤ p − 1 which divide r, andP = r P r . Thus we have By a similar argument, we have If i ∈ S 1 , then f a i ∈ Π odd . Further f AΠ odd = Π odd and f X Π o = AΠ e . We observe that for every divisor r, we have P r → Λ r , P e/o,r → Λ e/o,r , so AP o and A i , i ∈ S 1 are the only exceptional hypersurfaces which is mapped by f X to π −1 (Π odd ). Thus we have For a divisor r of p we have Theorem 5.4. Equations (5.6-12) define f * X as a linear map of P ic(X). Next we discuss the exceptional locus of the induced map f X : X → X. As in §3, we have Using (1.5), (1,6) and (1.8), we see that Σ 0/p , A 0/p , and E 0/p are not exceptional.
Lemma 5.5. For i ∈ S 1 ∪ S 2 , Σ i is not exceptional for f X , and f X |A i : A i F i ⊂ P e/o is a dominant map; thus A i is exceptional.
Lemma 5.6. The maps f X : P e AP o P o AP e P e are dominant. In particular, P e , AP o , P o , and AP e are not exceptional.
Proof. Since AΠ odd and AΠ even are not indeterminate, it is sufficient to show that only for P e and P o . We will show the mapping f X : P e AP o is dominant. The proof for P o is similar. The generic point of P e is written as x; ξ where x = [x 0 : 0 : x 2 : 0 : · · · : x p−1 : 0] and ξ = [0 : ξ 1 : 0 : ξ 3 : · · · : 0 : ξ p ]. It follows that f X (x; ξ) = i: odd (1/ξ i )a i ; j: even (1/x j )a j . It is evident that the mapping is dominant and thus P e is not exceptional. By Lemma 5.6, there is a 4-cycle {P e , AP o , P o , AP e } of hypersurfaces, which we call "the wringer"; this is pictured in Figure 5.1. For i ∈ S 1 , the orbit f X : Σ i A i F i enters this 4-cycle, which illustrates Lemma 5.5. The fibers ε ⊂ P e are the fibers F(e j ) for even j, 1 < j ≤ p − 1, and the fibers ε = F(e i ) ⊂ P o correspond to i odd. If, for some n ≥ 0, we have f n X F i ⊂ ε ⊂ I X , then the next iteration will blow up to a hypersurface.
Let us identify Π e , and Π o , with Pp,p = (p − 1)/2 as follows: where for each v = [v 0 : · · · : vp] ∈ Pp we set φ v : [w 0 : · · · : wp] → [w 0 v −2 0 : · · · : wpv −2 p ]. If we set h := h 2 • h 1 , then since i 2 reverses the coordinates, we have In other words, ι e and ι o conjugate the action of f 2 X on the wringer to the map h on Pp × Pp. If i ∈ S 2 , thenĩ = i/2 is relatively prime toq, and we writeṽĩ ∈ Pp for the vector in Lemma 3.2. Thus we have ι e (η i ) =ṽĩ, and we have ι e F i = {ṽĩ} × Pp. Similarly, if i ∈ S 1 , ı = (p − i)/2 is relatively prime toq, and we have ι o (η i ) =ṽĩ, and we may identify F i with the vertical fiber overṽĩ.
From §1 we have the following: Lemma 5.11. When 1 < r < p divides p, f X induces dominant maps P e,r Λ e,r , P o,r Λ o,r , and P r Λ r . In particular, the hypersurfaces P e,r , P o,r , and P r are exceptional.
We also have ω Similarly for all odd k = p, ω k is a pth root of −1 and i odd ω ki − 1 = 0.

Subtracting
i odd a i from i odd, i≡0 mod r a i , it follows that τ ′ = i odd, i ≡0 mod r a i ∈ AΣ 0 mod r . The proof for τ ′′ is similar.
Let us define u ′ e,r = (u ′ i ) ∈ P p to be the vector such that u ′ i = 1 if i ≡ p/r mod 2p/r and u ′ i = 0 otherwise. We set u ′′ e,r = (u ′′ i ) where u ′′ i = 0 if i ≡ 0 mod p/r and u ′′ i = 1 otherwise. Let us define u ′ o,r = (u ′ i ) ∈ P p to be the vector such that u ′ i = 1 if i ≡ 0 mod 2p/r and u ′ i = 0 otherwise. We set u ′′ o,r = (u ′′ i ) where u ′′ i = 0 if i ≡ 0 mod p/r and u ′′ i = (−1) i otherwise. We let ℓ e,r to be the line containing u ′ e,r and u ′′ e,r , and let α e,r be the line in P e,r lying over the basepoint τ e,r and having fiber coordinate in ℓ e,r . We define α o,r similarly.
Note that both α e,r f 2 X α e,r are 1-dimensional linear subspaces in fiber over τ e,r . Using the computation in Lemma 5.12 we have f 2 X α e,r = α e,r . We use a similar argument for α o,r . Corollary 5.14. Let r > 1 be an odd divisor of q. Then for j ∈ S r , α o,r is a hook for A j , and P o,r and P r ; and α e,r is a hook for A 2j , P e,r , and P r .
Proof. We have determined all the exceptional hypersurfaces for f X and have found a hook for each of them. Thus by Theorem 1.4, condition (0.4) holds for f X . Thus δ(f ) is the spectral radius of f * X . Consider f * X as in Theorem 5.4 and let χ(x) denote its characteristic polynomial.
Lemma 6.1. Suppose that r = 2 m−1 r ′ , and r ′ divides q odd . If i ∈ S r , then f a i ∈ Π r mod 2r , and if j ∈ S 2r , then f a j ∈ Π 0 mod 2r .
Proof. Sinceω = ω 2r is a primitive p/rth root of unity and p/r is odd, the proof is the same as Lemma 5.2.
Now we construct the space π : X → P p by a series of blowups. We blow up a 0 , e 0 , a p , e p , and a p/2 . For each divisor of the form r in (6.4), we blow up a i for all i ∈ S r ∪ S 2r . As before, A i denotes the blowup fiber of a i . We also blow up Π 0 mod 2r and Π r mod 2r ; we denote the blowup fibers as P e,r and P o,r , respectively. For each divisor of the form ρ in (6.4) (or equivalentlyρ), we blow up Σ ρ mod 2ρ ; we denote the blowup fiber by Γ ρ . Let f X : X → X denote the induced birational map.
Let us take H = H X , E 0/p , A 0/p , A p 2 , A i , i ∈ S r ∪ S 2r , P e/o,r , and Γ ρ as a basis for P ic(X). As in §5, we have (6.5) whereP e/o = r P e/o,r . And for a divisor r of q in (6.4), we have We see that Σ p/2 contains e 0/p , Π 0 mod 2r and Π r mod 2r as well as Γ ρ . Let us suppose q = 2 m · odd. We setΓ = ρ:2 m−2 ·odd Γ ρ . Since a j ∈ Σ p/2 for all odd j, if p/2 is odd we have Thus we have Let us consider a divisor ρ of q in (6.4). We have We observe that Σ ρ mod 2ρ ⊂ Σ odd·ρ and a p/2 = [1 : 0 : −1 : 0 : · · · : ±1] ∈ Σ j for all odd j. Thus for i ∈ S ρ we have ρ even This accounts for all of the basis elements of P ic(X), so we have: Theorem 6.4. Equations (6.5-12) define f * X as a linear map of P ic(X).
From §1 we have the following: Lemma 6.6. Let r be a divisor of the form (6.4). If i ∈ S r , we let F i denote the fiber of P o,r over f a i . In this notation, we have dominant maps: f X : Σ i A i F i . In fact, for every fiber F of P o,r , f X : F AΣ r mod 2r is a dominant map. Similarly, suppose j ∈ S 2r . With corresponding notation, we have dominant maps f X : Σ i A i F i ⊂ P e,r and f X : F AΣ 0 mod 2r .
Proposition 6.7. AΠ odd ⊂ AΣ 0 mod 2r ∩ AΣ r mod 2r is a hook for the spaces: A p/2 , and P e,r , P o,r , A i , i ∈ S r ∪ S 2r , for every divisor r in (6.4).
Lemma 6.10. For i ∈ S 1 , f n X a i / ∈ I X for all n ≥ 0.
Proof. By Proposition 6.7, i 1 f X a 1 = [p − 1 : 3 − p : p − 5 : · · · : ±1]. Let us set u 1 = (p − 1, 3 − p, p − 5, · · · , ±1). It suffices to show that ϕ n (u 1 ) / ∈ i 1 I X for all n ≥ 0. For this we need to know that for each n, at most one coordinate of ϕ n u 1 can vanish. Let us choose a prime number p/2 < µ ≤ p − 1. One of the coordinates of u 1 is equal to ±µ. Suppose it is the jth coordinate. Then 2j − 1 must be relatively prime to q, so we can apply Lemma 6.9. Working modulo µ, we see that ϕu 1 = b j u j , where u j is obtained from u 1 by permuting the coordinates, and b j = 2(p/2) 2 ((u 1 ) j ) p/2 . For each k = j, the kth coordinate of u 1 is nonzero modulo µ. Thus b j is a unit modulo µ, and so ϕu 1 is a unit times a permutation of u 1 . The permutation preserves the set S 1 , so if j 2 denotes the coordinate of i −1 1 ϕu 1 which vanishes modulo µ, then j 2 ∈ S 1 . Thus we may repeat this argument to conclude that, modulo µ, ϕ n u 1 is equal to a unit times a permutation of u 1 . Thus at most one entry of ϕ n u 1 can vanish, even modulo µ.
Lemma 6.15. For j ∈ S ρ , f n X λ j ⊂ I X for all n ≥ 0. Proof. We apply Lemma 6.14 modulo µ following the line of argument of Lemma 6.10.
. . . U (a n ) D(a n ) where the empty spaces are filled by zeros.
Lemma E.1. det(M n (a 1 , . . . , a n )) = n j=1 (x 2 −a j ). Any of the blocks U (a j ) may be replaced by 2 × 2 blocks of zeros without changing the determinant.
Proof. We first expand in minors along the next to last row which contains a 1 in the second slot and then expand in minors along the second row which has only one entry 1 in the first slot. It follows that det(M ′ n (a 1 , . . . , a n )) = a 1 · det(M ′′ n−1 (a 2 , . . . , a n )) where B ′ = ( 1 −x ) and M ′′ n−1 (a 2 , . . . , a n ) =       H(a 2 ) H(a 3 ) . . . H(a n ) D(a 2 ) U (a 3 ) . . .
. . . U (a n ) Now we use the first row to compute minors. It is not hard to see that each minor can be computed from the matrix of the form M k−3 (a 2 , . . . , a k−2 ) * 0 M ′′ n−k+1 (a K , . . . , a n ) .
The result follows using Lemma E.1 and its proof.
Proof of Theorem 4.5. We use the symmetry of M f noted in Appendix A and work with a symmetrized basis for P ic(X): H = H X , P r , A (r) , E 0 , A 0 , E (1) , AV (1) , V (1) , A (1) . We order the basis so that if r 1 |r 2 then P r 1 , A (r 1 ) appears before P r 2 , A (r 2 ) ; thus we we start with the prime factors of q. To compute the characteristic polynomial, we consider a matrix M f − xI.
For a simpler format, we add first row to the row corresponding to P r , E 0 and E (1) . After the series of row operations, we have the determinant of (M f − xI) is equal to the deteminant of  p − x H(a 1 ) H(a 2 ) · · · H(a κ ) H(1) H(0) H(1) V (b 1 − x) D(a 1 ) U (a 2 ) · · · U (a κ ) U (1) D(a 2 ) . . . U (a κ ) U (1) . . . . . . U (a κ ) U (1) where the empty spaces are filled by zeros and each a j b j is determined by a proper divisor of q and κ is the number of proper divisors, and V (a) = a 0 . Now we expand in the minors along the first column. For the (j, 1)-minor we move the first row to the jth row and then expand in minors along the jth row. The rest of the computation follows using Lemmas E.1 and E.2.