TY - JOUR

T1 - Ultradiscrete QRT maps and tropical elliptic curves

AU - Nobe, Atsushi

PY - 2008/3/28

Y1 - 2008/3/28

N2 - It is shown that the polygonal invariant curve of the ultradiscrete QRT (uQRT) map, which is a two-dimensional piecewise linear integrable map, is the complement of the tentacles of a tropical elliptic curve on which the curve has a group structure in analogy to classical elliptic curves. Through the addition formula of a tropical elliptic curve, a tropical geometric description of the uQRT map is then presented. This is a natural tropicalization of the geometry of the QRT map found by Tsuda. Moreover, the uQRT map is linearized on the tropical Jacobian of the corresponding tropical elliptic curve in terms of the Abel-Jacobi map. Finally, a formula concerning the period of a point in the uQRT map is given, and an exact solution to its initial-value problem is constructed by using the ultradiscrete elliptic theta function.

AB - It is shown that the polygonal invariant curve of the ultradiscrete QRT (uQRT) map, which is a two-dimensional piecewise linear integrable map, is the complement of the tentacles of a tropical elliptic curve on which the curve has a group structure in analogy to classical elliptic curves. Through the addition formula of a tropical elliptic curve, a tropical geometric description of the uQRT map is then presented. This is a natural tropicalization of the geometry of the QRT map found by Tsuda. Moreover, the uQRT map is linearized on the tropical Jacobian of the corresponding tropical elliptic curve in terms of the Abel-Jacobi map. Finally, a formula concerning the period of a point in the uQRT map is given, and an exact solution to its initial-value problem is constructed by using the ultradiscrete elliptic theta function.

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U2 - 10.1088/1751-8113/41/12/125205

DO - 10.1088/1751-8113/41/12/125205

M3 - Article

AN - SCOPUS:41149092560

VL - 41

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 12

M1 - 125205

ER -