β-展式和 β-动力系统的研究起源于著名数学家Rényi的工作。从数论的角度看，β-展式极大地丰富了实数的表示理论。从动力系统的角度看，β-动力系统是一类典型的非 Markov 系统，并且其动力学行为完全由1的展式决定。这些与整数进制展式间的本质区别，使得 β-动力系统的研究一直是度量数论、动力系统研究的重要课题之一。.本项目拟展开对 β-动力系统的动力丢番图逼近及相关问题的研究：研究 β-动力系统的常返性问题和一致丢番图逼近问题；研究轨道的分布状况对 β的依赖关系及相应的度量性质和分形结构。建立 β-动力系统中动力丢番图逼近的度量理论，发展 β-动力系统的分形维数理论，并将研究上述问题时发现的新方法和技巧应用到分形理论的研究中。

The research on β-expansions and β-dynamical systems derives from the work of the famous mathematician Rényi. From the perspective of number theory, β-expansions greatly enriched the representation theory of real numbers. β-dynamical systems are typical non-Markov systems from the viewpoint of dynamical systems, and the dynamic behavior of the β-dynamical systems is already contained in the orbit of 1. It is the essential difference between β-dynamical systems and the dynamical systems corresponding to the expansions on integer bases, making the investigation of β-dynamical systems has been always one of the important topics in the metric number theory and dynamical system..This project intends to initiate a study on the dynamic Diophantine approximation of β-dynamical systems and the related issues. At first, we will study the recurrence and the uniform Diophantine approximation problems in β-dynamical systems; then, the dependence of the distribution of orbits on the parameter β will be studied, the measure theory and fractal dimension corresponding to this issue will be further explored. In this project, the metric theory of the dynamic Diophantine approximation problems of β-dynamical systems will be built up, and the fractal dimension theory of β-dynamical systems will be developed. Then, the new methods and techniques derived from our research will be applied to the fractal theory.