Tropical geometry and integrable systems

利用热带椭圆曲线上的点的加法构造了一个8参数二维双分段线性映射族，并得到了每个映射族的通解.通过超离散化过程中，我们将家庭与一个18参数家庭的二维双有理映射称为QRT映射，包括其一般解决方案。同样地，将这种方法应用于热带椭圆曲线上点的重复所产生的可解混沌映射，得到了它们的通解，并阐明了它们与椭圆曲线上点的重复所产生的有理映射的对应关系.此外，我们还证明了存在一族元胞自动机，每一族元胞自动机都具有一个称为线性化的性质。然后，我们得到一个公式的基本周期相对于时间的发展施加周期边界条件的家庭。

We construct an 8-parameter family of two-dimensional bi-piecewise linear maps in terms of the addition of the points on tropical elliptic curves, and obtain the general solution for each member of the family. Through the ultradiscretization procedure, we associate the family with an 18-parameter family of two-dimensional birational maps called the QRT map including its general solutions. Similarly, applying such technique to solvable chaotic maps induced from the duplication of points on tropical elliptic curves, we obtain their general solution and clarify the correspondence to the rational maps induced from the duplication of points on elliptic curves. Moreover, we show that there exists a family of cellular automata each of which has a property called the linearizability. Then we obtain a formula concerning the fundamental period with respect to the time evolution of the family imposing periodic boundary conditions.