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 $K=I\circ J$\mathrm{.}Theset${\cal SC}_q$ofsymmetric,cyclicmatricesisinvariantunder$K$. In this paper, we determine the degrees of the iterates $K^n=K\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ K$ restricted to ${\mathcal{S}\mathcal{C}}_q$.