ar X iv : 0 90 7 . 33 19 v 1 [ m at h . D S ] 1 9 Ju l 2 00 9 Degree Complexity of Matrix Inversion Eric Bedford and Tuyen

For a q × q matrix x = (xi,j) we let J(x) = (x −1 i,j ) be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x) = (xi,j) −1 denote the matrix inverse, and we define K = I ◦J to be the birational map obtained from the composition of these two involutions. We consider the iteratesK = K◦· · ·◦K and determine degree complexity of K, which is the exponential rate of degree growth δ(K) = limn→∞ (deg(K )) 1/n of the degrees of the iterates. §0. Introduction Let Mq denote the space of q×q matrices with coefficients in C, and let P(Mq) denote its projectivization. We consider two involutions on the space of matrices: J(x) = (x i,j ) takes the reciprocal of each entry of the matrix x = (xi,j), and I(x) = (xi,j) −1 denotes the matrix inverse. The composition K = I ◦ J defines a birational map of P(Mq). For a rational self-map f of projective space, we may define its nth iterate f = f ◦ · · · ◦ f , as well as the degree deg(f). The degree complexity or dynamical degree is defined as δ(f) := lim n→∞ (deg(f)). In general it is not easy to determine δ(f), or even to make a good numerical estimate. Birational maps in dimension 2 were studied in [DF], where a technique was given that, in principle, can be used to determine δ(f). This method, however, does not carry over to higher dimension. In the case of the map Kq, the dimension of the space and the degree of the map both grow quadratically in q, so it is difficult to write even a small composition Kq ◦ · · · ◦Kq explicitly. This paper is devoted to determining δ(Kq). Theorem. For q ≥ 3, δ(Kq) is the largest root of the polynomial λ 2 − (q − 4q+2)λ+ 1. The map K and the question of determining its dynamical degree have received attention because K may be interpreted as acting on the space of matrices of Boltzmann weights and as such represents a basic symmetry in certain problems of lattice statistical mechanics (see [BHM], [BM]). In fact there are many K-invariant subspaces T ⊂ P(Mq) (see, for instance, [AMV1] and [PAM]), and it is of interest to know the values of the restrictions δ(K|T ). The first invariant subspaces that were considered are Sq , the space of symmetric matrices, and Cq , the cyclic (also called circulant) matrices. The value δ(K|Cq) was found in [BV], and another proof of this was given in [BK1]. Anglès d’Auriac, Maillard and Viallet [AMV2] developed numerical approaches to finding δ and found approximate values of δ(Kq) and δ(K|Sq) for q ≤ 14. A comparison of these values with the (known) values of δ(K|Cq) led them to conjecture that δ(K|Cq) = δ(Kq) = δ(K|Sq) for all q. The Theorem above proves the first of these conjectured equalities. We note that the second equality, δ(K|Sq) = δ(Kq), involves additional symmetry, which adds another layer of subtlety to the problem. An example where additional symmetry leads to additional complication has been seen already with the K-invariant space Cq ∩ Sq: the value 1 of δ(KCq∩Sq) has been determined in [AMV2] (for prime q) and [BK2] (for general q), and in the general case it depends on q in a rather involved way. The reason why the cyclic matrices were handled first was that K|Cq (see [BV]) and K|Cq∩Sq (see [AMV2]) can be converted to maps of the form L ◦ J for certain linear L. In the case of K|Cq , the associated map is “elementary” in the terminology of [BK1], whereas K|Cq∩Sq exhibits more complicated singularities, i.e., blow-down/blow-up behavior. In contrast, the present paper treats matrices in their general form, so our methods should be applicable to much wider classes of K-invariant subspaces. Our approach is to replace P(Mq) by a birationally equivalent manifold π : X → P(Mq) and consider the induced birational map KX := π −1 ◦ K ◦ π. A rational map KX induces a well-defined linear map K X on the cohomology group H (X ), and the exponential growth rate of degree is equal to the exponential growth rate of the induced maps on cohomology: δ(K) = lim n→∞ ( ||(K X ) ||H1,1(X ) 1/n . Our approach is to choose X so that we can determine (K X ) ∗ sufficiently well. A difficulty is that frequently (K) 6= (K) on H. In the cases we consider, H, the cohomology group in (complex) codimension 1, is generated by the cohomology classes corresponding to complex hypersurfaces. So in order to find a suitable regularization X , we need to analyze the singularity of the blow-down behavior of K, which means that we analyze K at the hypersurfaces E with the property that K(E) has codimension ≥ 2. Let us give the plan for this paper. In general, deg(K ◦ K) ≤ deg(K), so δ(K) ≤ deg(K). On the other hand, δ decreases when we restrict to a linear subspace, so δ(K) ≥ δ(K|Cq). The paper [BV] shows that δ(K|Cq) is the largest root of the polynomial λ 2 − (q − 4q + 2)λ + 1, so it will suffice to show that this number is also an upper bound for δ(K). In order to find the right upper bound on δ(Kq), we construct a blowup space π : Z → P(Mq). Such a blowup induces a birational map KZ of Z. Each birational map induces a linear mapping K Z on the Picard group Pic(Z) ∼= H(Z). A basic property is that δ(KZ) ≤ sp(K ∗ Z), where sp(K ∗ Z) indicates the spectral radius, or modulus of the largest eigenvalue of K Z . Thus the goal of this paper is to construct a space Z such that the spectral radius of K Z is the number given in the Theorem. §1. Basic properties of I, J , and K For 1 ≤ j ≤ q − 1, define Rj as the set of matrices in Mq of rank less than or equal to j. In P(Mq), R1 consists of matrices of rank exactly 1 since the zero matrix is not in P(Mq). For λ, ν ∈ P , let λ ⊗ ν = (λiνj) ∈ P(Mq) denote the outer vector product. The map P × P ∋ (λ, ν) 7→ λ⊗ ν ∈ R1 ⊂ P(Mq) is biholomorphic, and thus R1 is a smooth submanifold. We let I : P(Mq) → P(Mq) denote the birational involution given by matrix inversion I(A) = A. We let x[k,m] denote the (q − 1) × (q − 1) sub-matrix of (xi,j) which is obtained by deleting the k-th row and the m-th column. We recall the classic formula