Degree Complexity of Birational Maps Related to Matrix Inversion

Abstract

For a q × q matrix x = (x _{i, j}) we let ${J(x)=(x_{i,j}^{-1})}$ be the Hadamard inverse, which takes the reciprocal of the elements of x. We let ${I(x)=(x_{i,j})^{-1}}$ denote the matrix inverse, and we define ${K=I\circ J}$ to be the birational map obtained from the composition of these two involutions. We consider the iterates ${K^n=K\circ\cdots\circ K}$ and determine the degree complexity of K, which is the exponential rate of degree growth ${\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}}$ of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.