Dynamics on algebraic varieties and Painleve equations

在Painleve系统的代数几何和模理论基础上，我们发展了对第六Painleve方程的动力学研究。当参数位于仿射Weyl群的壁上时，我们建立了系统的混沌性质，并证明了孤立周期解数的指数增长。为了得到这些结果，我们发展了射影曲面上保面积双调映射的周期点的一般理论。还讨论了构造正熵的有理曲面自同构的问题。

We developed a dynamical study of the sixth Painleve equation on the algebro-geometrical and moduli theoretical foundations of the Painleve system. When the parameter lies on the walls of an affine Weyl group, we established the chaotic nature of the system and proved the exponential growth of the number of isolated periodic solutions. To obtain these results, we developed a general theory of periodic points for area-preserving birational maps on a projective surface. Constructing rational surface automorphisms of positive entropy has also been discussed.