Exact solutions and mixing in an algebraic dynamical system
Teoretičeskaâ i matematičeskaâ fizika, Tome 143 (2005) no. 1, pp. 131-149
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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
Mots-clés : exact solutions
@article{TMF_2005_143_1_a8,
author = {I. G. Korepanov},
title = {Exact solutions and mixing in an algebraic dynamical system},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {131--149},
publisher = {mathdoc},
volume = {143},
number = {1},
year = {2005},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2005_143_1_a8/}
}
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/