分形几何中最基本的分形集是自相似集。对满足开集条件的自相似集的研究已经比较完善。自仿集作为另一类重要的分形集，在过去30多年中，一直是大家关注和重点研究的对象。然而，即使对没有重叠结构的自仿集而言，都存在许多悬而未决的问题。著名的Bedford-McMullen集是帮助我们从自相似框架过渡到自仿框架的一类重要集合。本项目拟研究存在于自仿系统中的，以及和动力系统密切相关的问题。包括：Bedford-McMullen集上的收缩靶问题，自仿集上的遍历测度的下局部维数，以及变尺度自仿集中的相关问题。项目将综合利用分形几何、动力系统、遍历理论及组合学中的方法和技巧，建立变尺度自仿集中普适的维数公式和确定具体的Bedford-McMullen集上的回归集的维数。项目的研究涉及到分形几何与动力系统的交叉领域，旨在发现和探索不同领域间的新联系和新现象。

The most basic fractal objects are self-similar sets. Self-similar sets that satisfy the open set condition are well understood. Self-affine sets are also an important class of fractals, which have received a great deal of attention over the last 30 years. However self-affine sets even without overlap still retain many mysteries. The famous Bedford-McMullen sets which provide a bridge between the relatively well understood world of self-similar sets and the far from understood world of general self-affine sets. This project will make an attack on several problems arise in affine framework and interrelated problems also in dynamical systems. It involves the investigation of shrinking target problems on Bedford-McMullen sets, lower local dimensions of ergodic measures on self-affine sets and non-homogeneous affine fractals that is the limit sets of non-homogeneous affine iterated function systems in which the affine contractions applied at each step in time are allowed to vary. The goal of this project is to find generic formulae that hold almost always for non-homogeneous affine fractals and to seek the dimensions of recurrent sets on specific Bedford-McMullen sets. The methods and techniques used will not be limited to those from fractal geometry but will also come from other fields, including dynamical systems, ergodic theory and combinatorics. This project is in the area of interaction between fractal geometry and dynamical systems, and aims at discovering new phenomena and new connections between different fields.