Dimension theory of dynamically defined sets

我提出的研究计划是维度理论和动力系统的交叉，这是一个令人兴奋的领域，涉及研究在动力系统背景下自然产生的高度不规则的几何物体。动态系统的目的是提供关于状态集合的长期行为的定性和定量信息，这些状态集合根据一些固定的规则随着时间的推移而演变。通常，满足某些规定的统计规律的状态的集合形成一个分形集；在各种放大倍数下都能显示复杂细节的几何物体。系统的动力学行为和某些子集的分形几何之间的这种联系，已经导致它被用来模拟自然现象，如云边界和流体湍流。我研究计划的第一个目标是研究具有重叠的自相似集，这是维数理论的核心研究课题，与遍历理论、动力系统和几何测量理论有很强的联系。这样的集合，尽管以一种简单的方式定义，却显示出复杂的细节，而且通常没有面积或体积，因此经典几何的工具被证明不是那么有用。分形维数量化了物体的不规则程度，并提供了分析这些集合的重要工具。在这些集合上强加一个技术条件，使丰富的理论得以发展，并显示出与遍历理论和动力系统等领域的深刻关系。在没有施加这种条件的情况下，只知道部分结果。我的意图是研究没有分离条件的情况，以关注豪斯多夫维度和符号维度重合的必要条件。即使是这个方向的部分结果也会对我们对自相似性的理解产生深远的影响，并将在这个领域产生一个全新的研究方向。我的第二个目标是研究开放动力系统的统计性质。动力系统往往对初始条件表现出敏感的依赖性，这意味着单个轨道的长期行为极难预测。在统计物理中类似问题的激励下，数学家发展了热力学形式主义，它为研究动力系统中典型轨迹的长期行为提供了一条途径。我在这个领域的首要目标是在开放动力系统的背景下更好地了解这些现象的统计特性。这项研究的一个新应用将是计算机科学领域。Lempel-Ziv-Welsch是一种用于压缩数据的算法，它通过关联数据的某些子块来操作。与此目标相关的技术将直接应用于此设置，并将使我们更好地了解该算法的效率。这样的结果会引起计算机科学领域工作人员的极大兴趣。该方案的最终目标是在分形设置丢番图近似的研究。丢番图近似关注的是用有理数近似实数的效果。这项工作的经典背景是在欧几里得空间的勒贝格测度中，其中许多亮点之一是著名的钦钦定理。近年来，人们对分形集和测度的这些问题产生了浓厚的兴趣，以期在这种情况下证明Khinchin定理的类比。在这个方向上已经有了部分结果，我打算在自相似测量的背景下研究这些问题。

My proposed programme of research lies at the intersection of dimension theory and dynamical systems, an exciting area which is concerned with the study of highly irregular geometric objects which arise naturally in the context of dynamical systems.Dynamical systems aims to give qualitative and quantitative information about the long term behaviour of collections of states which evolve over time according to some fixed rule. Often the collection of states which satisfy some prescribed statistical law form a fractal set; a geometric object which displays intricate detail at all magnifications. This link between the dynamical behaviour of a system and the fractal geometry of certain subsets has led to it being used to model natural phenomena such as cloud boundaries and fluid turbulence.The first objective in my programme of research is the study of self-similar sets with overlap, a research topic which lies at the heart of dimension theory, and has strong connections with ergodic theory, dynamical systems, and geometric measure theory. Such sets, despite being defined in a simple fashion, display intricate detail and often have zero area or volume, and so the tools of classical geometry prove to be not so useful. Fractal dimension quantifies how irregular an object is and provides an important tool with which to analyse these sets.Imposing a technical condition on these sets has enabled a rich theory to be developed that has shown deep relations with fields such as ergodic theory and dynamical systems. In the case that no such condition is imposed only partial results are known. It is my intention to investigate the case where no separation conditions are assumed with a view of focussing on necessary conditions for the coincidence of the Hausdorff and symbolic dimensions. Even partial results in this direction would have a profound effect on our understanding of self-similarity, and would spawn an entirely new line of research in this field. My second objective is to investigate the statistical properties of open dynamical systems. Often a dynamical system will display sensitive dependence on initial conditions, which means that the long term behaviour of an individual orbit is extremely difficult to predict. Motivated by a similar problem in statistical physics mathematicians developed the thermodynamic formalism, which provides an avenue to study the long term behaviour of typical trajectories in the dynamical system. My overarching goal within this field is to gain a better picture of the statistical properties of these this phenomena in the context of open dynamical systems. A novel application of this research would be to the field of computer science. Lempel-Ziv-Welsch is an algorithm used for compressing data, which operates by relating certain sub-blocks of data. The techniques involved with this objective would directly apply in this setting and would afford us a better picture of the efficiency of this algorithm. Such a result would be of great interest to those working in the field of computer science.The final objective for the programme is the study of diophantine approximation in a fractal setting. Diophantine approximation is concerned with how well one may approximate real numbers by rationals. The classical setting for this work has been in Euclidean space with the Lebesgue measure, with one of the many highlights being the celebrated theorem of Khinchin. Recently, there has been great interest in studying these problems for fractal sets and measures, with a view of proving an analogue of Khinchin's theorem in this setting. There have been partial results in this direction, it is my intention to investigate these problems in the setting of self-similar measures.