关于临界有限有理函数组合不变量的存在性研究

The two types of important combination invariants——Yoccoz puzzles and Hubbard trees for complex polynomials play a crucial role in proving the local connectivity of Julia sets and in characterizing the dynamics of polynomials in the Julia sets. People are concerned about the existence of similar combinatorial invariants for general rational maps, which is also an open question posted by McMullen in 1995. The applicant and the collaborators have proved the existence of combinatorial invariants for carpet rational maps. Based on the existing technologies and research results, we believe that this conclusion can be generalized to all postcritically finite rational maps. In this project, we aim to construct combinatorial invariants for arbitrary postcritically finite rational maps, by using the technology of “from initial graphs to homotopic invariants and to combinatorial invariants”. The main purpose of this project is to provide new ideas for combinatorial characterization of postcritically finite rational maps from the way quite different from that of Thurston Theorem, and to deepen our understanding of the dynamics of postcritically finite rational maps, which is widely distributed in the parameter space.

复多项式的两类重要组合不变量——Yoccoz拼图与Hubbard树分别在证明Julia集的局部连通性以及刻画多项式在整个Julia集上的动力系统过程中发挥着至关重要的作用。人们普遍关注非多项式有理函数是否存在类似的组合不变量。这也是McMullen于1995年所提出的公开问题。申报人与合作者已经证明了地毯有理函数组合不变量的存在性。基于现有技术与研究成果，我们认为可以将这个结论推广到任意临界有限有理函数。本项目拟采用“从初始图到同伦不变量再到组合不变量”技术，构造出临界有限有理函数的组合不变量。本项目旨在于，从不同于Thurston定理的角度，为组合刻画临界有限有理函数提供新的研究思路；加深人们对参数空间中普遍存在的临界有限有理函数动力系统的认识。

组合不变量在研究有理函数动力系统过程中发挥着至关重要的作用。本项目将扩张有理函数与Sierpinski有理函数存在不变曲线的结论推广到所有临界有限的有理函数，证明了：任意临界有限有理函数存在包含临界值轨道的不变图。本项目完成了对高次Newton映射游荡连续统问题及其Julia集局部连通性问题的研究。另外本项目刻画了Julia集局部连通的多项式的动力系统：给出了两条外射线着陆到同一点的充要条件。以此为应用，证明了W.Thurston引进的核拓扑熵在著名的Mandelbrot集上任意vein上的单调性。