含抛物不动点的有理函数动力系统与抛物不动点的扰动

This project contains the following contents:.1. We give a new constructive proof of the result proved by Kozlovski and van Strien before that each finitely renormalizable polynomial with only hyperbolic periodic points can be approximated by hyperbolic polynomials, and obtain a better result than before. .2. We study the measurable dynamic systems of the rational maps with totally disconnected Julia set. We want to prove the number of the ergodic components of the conformal measures supported on the Julia sets is finite and the hyperbolic dimension equals the Hausdorff dimension of the Julia sets of such maps. .3. We study the parabolic and near-parabolic renormalization theory introduced by Inou and Shishikura. We will improve this theory by the technique of quasi-conformal surgery, we hope to obtain some local properties of the perturbation of the parabolic fixed points, and obtain several results about the topological and geometric property of the Julia sets. Moreover, we can get some consequences about the parameter space. 4. We will prove a result about a conjecture raised by J. Milnor that a rational map with totally disconnected Julia set which has a parabolic fixed point can be perturbed to a rational map also with totally disconnected Julia set which has an attracting fixed point, moreover the two rational maps are topologically conjugate in their Julia sets.

本项目的主要研究内容包括：.1.对Kozlovski-van Strien之前的结果："只含双曲周期点的有限次可重整的多项式可被双曲多项式逼近"，给出新的构造性的证明，得到更细致的结果。.2.研究具有完全不连通Julia集的有理函数的可测动力系统，证明这类函数Julia集上共形测度的遍历分支个数有限，并且Julia集的双曲维数与Hausdorff维数相等。.3．用拟共形手术的方法研究Inou-Shishikura抛物不动点的重整化理论与抛物点的扰动，希望对抛物不动点扰动以后的局部性质，Julia集的拓扑与几何性质得到一些结果，并进一步得到参数空间的性质。.4.给出一个与Fields奖得主J. Milnor提出的猜想有关的结果，证明有抛物不动点的具有完全不连通Julia集的有理函数可以扰动为有吸性不动点的具有完全不连通Julia集的有理函数并使它们在Julia集上拓扑共轭。

本项目的研究成果主要包括以下四方面内容:.1. 研究了有理函数的可测动力系统, 证明了对于Julia集为Cantor集, Fatou域为吸性域, Julia集上不含持续回归临界点的有理函数, 其Julia集的双曲维数与Hausdorff维数相等..2. 研究了与统计力学有关的重整化变换T_\lambda的参数空间与动力学性质, 证明了Misiurewicz点在类Mandelbrot集边界上的稠密性, 并且类Mandelbrot集与实轴只相交于一列Misiurewicz点. 同时还证明了T_\lambda的Julia集的Hausdorff维数关于实参数的连续性..3. O. Kozlovski and S. van Strien曾经证明了只含双曲周期点的至多有限次可重整的复多项式可由双曲多项式逼近. 我们利用拟共形手术的方法对这一定理给出新的构造性的证明, 事实上, 利用我们的证明方法可以得到比之前更强的结论, 就是只含双曲周期点的至多有限次可重整的复多项式可由双曲多项式连续逼近..4. 证明了一个复分析领域中与几何函数论有关的一个结果, 我们给出了一个构造单位圆内保持边界任意多个指定点不动的共形映射的方法. 这个结论在复动力系统中也有着非常重要的应用.