Studies on the geometry of discrete integrable systems and solvable chaotic systems

利用周期离散Toda格的谱曲线与其他两条曲线的交集，建立了任意系统大小的周期离散Toda格的几何实现。利用周期离散Toda格的守恒量，将这种实现归结为在谱曲线的对称乘积上加点，并利用周期离散Toda格的守恒量具体给出了实现中的曲线，并通过热带平面曲线建立了任意系统大小的超离散周期Toda格的几何实现。利用超离散周期Toda格子的守恒量，给出了实现中的热带曲线，其中之一是超离散周期Toda格子的谱曲线(热带超椭圆曲线)。

We establish a geometric realization of the periodic discrete Toda lattice of an arbitrary system size by using the intersection of its spectral curve and other two curves. This realization is reduced from the addition of points on the symmetric product of the spectral curve, and the curves in the realization are concretely given by using the conserved quantities of the periodic discrete Toda lattice.We also establish a geometric realization of the ultradiscrete periodic Toda lattice of an arbitrary system size, which is an ultradiscrete analogue of the periodic discrete Toda lattice, via tropical plane curves. The tropical curves in the realization, one of which is the spectral curve of the ultradiscrete periodic Toda lattice (a tropical hyperelliptic curve), are also given by using the conserved quantities of the ultradiscrete periodic Toda lattice.