关于超越整函数动力系统若干问题的研究

复解析动力系统是现代数学中一个非常值得研究的重要领域。复解析动力系统研究主要分为有理函数动力系统的研究和超越整函数动力系统的研究。由于超越整函数迭代的复杂性，人们对超越整函数动力系统的认识还不是很深刻，仍有大量的问题有待解决。本项目主要研究超越整函数动力系统中超越整函数 Fatou分支的有界性和超越整函数Julia 集的维数。所要探究的问题是：1.Baker问题即级小于1/2的超越整函数的Fatou分支的有界性；2.由缺项幂级数表示的超越整函数Fatou分支的有界性和超越整函数的复合函数的Fatou分支的有界性；3.特殊超越整函数和函数族Julia集的Hausdorff维数；4.特殊超越整函数Julia集的各种维数和Julia 集及其Hausdorff 维数的连续性。对上述问题的研究，我们预期能取得一些实质性结果，这将具有重要的学术研究价值，并对整函数动力系统的发展和完善有较大理论价值。

Complex dynamics is one of important research areas in modern mathematics. There are two branches in the research of complex dynamics, one is the research of iteration of rational functions, and the another is iteration of transcendental entire functions. Since the iteration of transcendental entire functions is more complex than iteration of rational functions, the development of iteration theory about transcendent entire functions is slowly. There are many basic problems still need to be solved. In this project we will study the bounded of Fatou components of transcendental entire functions and the dimensions of the Julia sets of entire functions. It includes following problems: 1.Baker's problem, i.e.the bounded of Fatou components of transcendental entire functions with small growth; 2.The bounded of Fatou components of transcendental entire functions with gaps and the bounded of Fatou components of composite function of transcendental entire functions; 3.The Hausdorff dimensions of some transcendental entire functions or families of transcendental entire functions; 4.The continuity of Julia sets and Hausdorff dimensions of Julia sets of some transcendental functions.