Exact solutions and mixing in an algebraic dynamical system

Let be an n×n matrix with entries a_ij in the field . We consider two involutive operations on these matrices: the matrix inverse I: ^–1 and the entry-wise or Hadamard inverse J: a_ij a_ij^–1. We study the algebraic dynamical system generated by iterations of the product J. I. We construct the complete solution of this system for n 4. For n = 4, it is obtained using an ansatz in theta functions. For n 5, the same ansatz gives partial solutions. They are described by integer linear transformations of the product of two identical complex tori. As a result, we obtain a dynamical system with mixing described by explicit formulas.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 131–149, April, 2005.