有限度的马可夫动力系统(见定义1.1)可以看成是迭代函数系统的一种特殊情形。过去人们对压缩迭代函数系统有着较深刻和广泛的研究。近年来关于非一致扩张的有限度马可夫动力系统的研究非常活跃，这类动力系统可以看成是非一致压缩的迭代函数系统。关于非一致压缩迭代函数系统的深入研究甚少，但非一致扩张的动力系统，更进一步地，非一致压缩的迭代函数系统在动力系统和分形几何中又是常见和重要的。本研究项目将在我们以前工作的基础上设法填补这个空白。我们将对非一致压缩迭代函数系统以及与其相关的动力系统进行深入、系统的研究。本研究工作将涉及实和复动力系统，分形几何，以及转移算子理论。

Finite Markovian dynamical systems (that is, dynamical systems with finite Markov partitions) are a special case of iterated function systems. In the past, there was a deep and systematic study of contractive iterated function systems. Recently, non-uniformly expanding finite Markovian dynamical systems became an active research area. This kind of dynamical systems can be also treated as non-uniformly contracting iterated function systems. However, there is still no systematic study of these non-uniformly contracting iterated function systems. And, non-uniformly expanding dynamical systems, furthermore, non-uniformly contracting iterated function systems are popular and important both in dynamical systems and fractal geometry. Based on our previous work, this research project tries to fill the gap. We will work on a systematic study of these non-uniformly contracting iterated function systems and related dynamical systems. Our study will involve real and complex dynamical systems, fractal geometry, and transfer operator theory.