自仿系统中的变尺度和收缩靶问题

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.

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。

在过去30多年间，分形几何理论取得了许多重要进展，其中一个原因就是得益于结合动力系统和数论发展起来的维数理论。分形几何可以将一些在不同学科中出现的对象用统一的思想和方法来处理。许多自然出现于动力系统和数论中的集合与测度为分形几何提供了研究对象。由经典Poincaré常返定理诱发了一系列关于动力系统中量化常返性质的研究，如收缩靶问题，轨道分布，击中深度等。这些问题同样出现在度量数论中，如丢番图逼近中的许多经典问题亦可归为某些特定动力系统中的量化常返性问题。本项目结合分形几何和动力系统中的方法和技巧，研究了一些特定动力系统中量化常返性的度量理论。建立了连分数部分商乘积的度量理论；丰富了实数的表示理论中run-length函数的维数理论；确定了Besicovitch集与Erdös-Rényi集交集的Hausdorff维数；解决了三分Cantor集上的动力覆盖性问题；刻画了二进制pattern序列的关联维数和关联测度；给出了β-动力系统中Birkhoff平均发散点集的重分形维数谱。上述问题的解决推动了分形几何、动力系统和丢番图逼近领域间的交叉发展。