Ultradiscrete QRT maps and tropical elliptic curves

Abstract

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.

Tropical geometric interpretation of ultradiscrete singularity confinement

“…which is in the class of ultradiscrete QRT equations studied by Nobe [21]. The invariants are We now calculate the difference between the left and right derivatives are (left minus the right) where a non-zero discrepancy indicates non-differentiability.…”

Ultradiscretization of solvable one-dimensional chaotic maps

“…The dynamics of the map (2.8) is described as follows: if the initial value X 0 is in [0, L], the map is the tent map and 10) and otherwise the dynamics is trivial. Now let us consider the limit of the solution by using the ultradiscretization of the elliptic theta functions [32](see also [14,24,25]). Jacobi's elliptic functions are expressed in terms of the elliptic theta functions…”

“…The key of the method is that one can obtain not only the equations but also their solutions simultaneously. It also allows us to understand the underlying mathematical structures of the ultradiscrete systems [2,3,5,8,11,13,14,15,16,25,33].…”

From Polygons to Ultradiscrete Painlevé Equations

“…They also admit symmetry groups of affine Weyl type [18,19] and special solutions of rational and hypergeometric type [26,34,50]. The ultradiscrete QRT maps may also be obtained as autonomous limits of the ultradiscrete Painlevé equations [29,39].…”