多项式的有理扰动的动力系统

Complex dynamical system is one of the main research directions of mathematical field . There are strong connections between complex dynamic system and many other mathematical fields, such as Klein group theory, real one-dimensional dynamical systems, ergodic theory, geometry group theory, computational complexity and so on. Moreover, complex dynamical system have a wide application in satistical physics, thermodynamics and other disciplines. In this project, we plan to study some important problems in the field of complex dynamic system. For example,the local connectivity of Julia sets of McMullen maps, the continuity problem of Julia sets of generalized McMullen maps，the quasisymmetric classification problem of Julia sets,the dynamics of rational perturbations of complex polynomials.

复动力系统是现代数学研究的主流方向之一。与Klein群理论，一维实动力系统、遍历理论、几何群论、计算复杂性等数学领域有密切联系和相互影响，并在统计物理、热力学等其他学科有广泛的应用。本申请项目将研究复动力系统领域中受到广泛关注的重要问题。如McMullen函数族的局部连通性问题, 广义McMullen函数族的Julia集的连续性问题, Julia集的拟对称分类问题，多项式的有理扰动的动力系统。

复动力系统研究中的一个有趣而又重要的问题是描叙有理函数的Julia集的拓扑，如连通性，局部连通性等。本项目主要研究了单项式z^n的有理扰动所得到的两类有理函数的动力系统, 根据其临界轨道的动力学行为，对其Julia集和Fatou集的拓扑进行了较完整的刻画。