一维覆盖映射的拟正则延拓及其在复动力系统中的应用

The main topic of this project is to study the quasi-regular extension of the covering maps from the real line to the unit circle. The importance of the research lies in the following two facts. In the aspect of the theory of quasiconformal mappings, it is a natural generalization of Beurling-Ahlfors quasiconformal extension. In the aspect of the application in complex dynamics, it allows us to construct a quasi-regular model with an invariant line field, by which we may construct an entire function with certain desired properties through quasi-conformal surgery. In particular, we have constructed the models for Stallard’s Cauchy integrals in the study of the Hausdorff dimension of the Julia sets of entire functions. We may expect that the further study in this direction may produce more and deeper applications in complex dynamics.

该项目的主要内容是研究实轴到单位圆周的一维覆盖映射的拟正则延拓。这项研究有两个重要意义。在拟共形映射的基础理论方面，这种拟正则延拓是Beurling-Ahlfors拟共形延拓的一个自然推广。在复动力系统中的应用方面，这样的延拓可以使得我们获得一个具有不变线域的拟正则模型，从而通过拟共形手术得到一个满足特定性质的整函数。特别地，在我们已经完成的工作中，我们对Stallard关于实现Julia集的Hausdorff维数的Cauchy积分给出了相应的几何模型。我们有理由相信对该课题的进一步深入研究将会在复动力系统的研究中产生更广泛且更深刻的应用。