Exact solutions and mixing in an algebraic dynamical system

Let $\mathcal A$ be an $n\times n$ matrix with entries $a_{ij}$ in the field $\mathbb C$. We consider two involutive operations on these matrices: the matrix inverse $I\colon\mathcal A\mapsto\mathcal A^{-1}$ and the entry-wise or Hadamard inverse $J\colon a_{ij}\mapsto a_{ij}^{-1}$. We study the algebraic dynamical system generated by iterations of the product $J\circ I$. We construct the complete solution of this system for $n\le4$. For $n=4$, it is obtained using an ansatz in theta functions. For $n\ge 5$, 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.