2003年, 著名数学家Schweiger定义了一类新的连分数（GCF）变换，见[Sc1]。2008年，本项目申请人获得了GCF的一些运算和度量性质。2010年，Shen和Zhou应用这些性质导出了GCF的一系列随机性质。最近我们又确定了一些GCF例外集的Hausdorff维数，发现其与正规连分数的类似集之间存在值得进一步探讨的关系。 本项目拟在此基础上进一步研究GCF动力系统：（1）研究GCF的代数意义和几何意义，推导GCF的更多运算性质和度量性质，以完善GCF的理论体系；（2）广泛研究GCF的各种例外集的分形维数及量纲函数，包括类似于正规连分数的Jarnik集、Good集、Hirst集以及GCF动力系统的常返集等具有无穷多迭代分枝的动力系统生成的吸引子；（3）揭示GCF与正规连分数之间的内在关系，为一般连分数动力系统的研究与应用提供新的思想、方法和技巧。

In 2003, famous mathematician Schweiger defined a class of new continued fractions(GCF) transforms (see[Sc1]). In 2008, the applicant obtained some arithmetic and metric properties of GCF. In 2010,Shen and Zhou deduced a series of random properties for GCF using the results of the applicant. Recently, we determined the Hausdorff dimensions of some GCF exceptional sets, and found that they have some relations with similar sets of regular continued fractions(RCF), whch are worthy of further study. The project intends to make more study on GCF dynamical systems: (1) Study the algebra and geometric meanings of GCF, deduce more arithmetic and metric properties of GCF, to complete the theoretical system of GCF. (2) Widely research on fractal dimensions and dimension functions of exceptional sets in GCF, including the attractors with infinite iteration branchs generated by GCF dynamical systems, such as Jarnik sets, Good sets, Hirst sets and the recurrence sets of GCF dynamical systems. etc, which are silimar with RCF's, (3) Explore the internal relations between GCF and RCF, to provide some new ideas, methods and techniques for research and application on general continued fractions dynamical systems.