Exact solutions and mixing in an algebraic dynamical system. (English) Zbl 1178.82028
Theor. Math. Phys. 143, No. 1, 599-614 (2005); translation from Teor. Mat. Fiz. 143, No. 1, 131-149 (2005).
Summary: 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 \(I: {\mathcal A}\mapsto{\mathcal A}^{-1}\) and the entry-wise or Hadamard inversion \(J: a_{ij}\mapsto a_{ij}^{-1}\). We study an algebraic dynamical system generated by iterations of the product \(J\circ I\). We construct the complete solution of this system for \(n\leq 4\). For \(n = 4\), it is obtained using an ansatz in theta functions. For \(n\geq 5\), 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.
MSC:
82B23 | Exactly solvable models; Bethe ansatz |
81R12 | Groups and algebras in quantum theory and relations with integrable systems |
33E05 | Elliptic functions and integrals |
References:
[1] | I. G. Korepanov, ?Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics,? solv-int/9506003 (1995). |
[2] | M. P. Bellon, J.-M. Maillard, and C.-M. Viallet, Phys. Lett. A, 159, 221-232 (1991). · doi:10.1016/0375-9601(91)90516-B |
[3] | R. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982). · Zbl 0538.60093 |
[4] | M. Bellon, J.-M. Maillard, and C. Viallet, Phys. Rev. Lett., 67, 1373-1376 (1991); hep-th/9112067 (1991). · Zbl 0990.37517 · doi:10.1103/PhysRevLett.67.1373 |
[5] | J.-C. Anglès d?Auriac, J.-M. Maillard, and C.-M. Viallet, ?A classification of four-state spin edge Potts models,? cond-mat/0209557 (2002). · Zbl 1097.82511 |
[6] | I. G. Korepanov, ?Exact solution for a matrix dynamical system with usual and Hadamard inverses,? nlin.SI/0303007 (2003). |
[7] | J. M. Landsberg, ?On an unusual conjecture of Kontsevich and variants of Castelnuovo?s lemma,? alg-geom/9604023 (1996). |
[8] | N. Kitanine, J. M. Maillet, and V. Terras, Nucl. Phys. B, 567, 554-582 (2000); math-ph/9907019 (1999). · Zbl 0955.82010 · doi:10.1016/S0550-3213(99)00619-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.