Geodesic
Exact solutions and mixing in an algebraic dynamical system
Teoretičeskaâ i matematičeskaâ fizika, Tome 143 (2005) no. 1, pp. 131-149 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: algebraic dynamical systems, mixing, star-triangle relation symmetries.
Mots-clés : exact solutions 
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I. G. Korepanov. Exact solutions and mixing in an algebraic dynamical system. Teoretičeskaâ i matematičeskaâ fizika, Tome 143 (2005) no. 1, pp. 131-149. http://geodesic.mathdoc.fr/item/TMF_2005_143_1_a8/

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