Fractal and multifractal structure of non-conformal repellers

在过去的30年里，在量纲和动力学领域的研究有了很大的发展。本课题将研究具有高度复杂分形的非共形动力系统的吸引子和排斥子的几何结构和尺寸。一个动力系统本质上是由一个集合上的映射组成的，比如平面上的一个区域到它自身的映射。特别有趣的是，将映射重复应用到某个初始点之后的轨迹。排斥者是一个映射到自身的集合，但在映射的迭代下，该集合外的邻近点会远离该集合。驱虫剂通常具有分形结构，在继续放大的情况下，越来越复杂的细节变得明显。Dimension提供了一种度量这些集合的大小和复杂性的方法。光滑的几何形状具有整数的维度：曲线是一维的，表面是二维的，等等，反映了在物体中定位点所需的坐标数量。为了应对分形排斥的复杂性，需要不需要整数的维度概念，例如Hausdorff，包装和盒计数维度，这些分数维度将在该项目中发挥关键作用。许多研究都集中在保角映射上，其中的驱避器是经常被描绘的复杂动力学的Julia集。它们通常是局部自相似的，即由许多更小的、几乎相似的自身副本组成，它们的尺寸通常可以通过求解一个被称为鲍文-鲁尔公式的方程来确定。非保角映射导致局部自仿射驱避器，其可能具有非常不同的特征，在放大时，驱避器的一小部分变得越来越长或扭曲。考虑到自仿射集分析的最新进展，它可以被认为是非共形驱避器的分段线性类似物，将维数公式扩展到非共形非线性设置是及时的。然而，求自仿射集的维数比求自相似集的维数要尴尬得多，尤其是因为维数不需要随定义变换连续变化。通过类比，可能很难获得在每种情况下都有效的维度公式，因此项目的主要部分将是寻找在适当意义上“几乎总是”有效的通用公式。在共形环境中，热力学形式是寻找维度的重要工具，但对于非共形系统，则需要热力学形式的“次加性”版本。当然，确切地知道特定驱蚊剂的尺寸是可取的，而不仅仅是寻找一般有效的公式，因此该项目还将确定可以保证通用公式给出实际尺寸值的函数类。进一步的研究将考虑非保形驱蚊剂的质量分布，目的是使用分数维的类似物来量化分布的不规则性，这一领域被称为多重分形分析。

Over the past 30 years research in the area of dimension and dynamics has developed enormously. This project will study the geometric structure and dimension of attractors and repellers of certain non-conformal dynamical systems, which often have a highly complicated fractal form.A dynamical system essentially consists of a mapping on a set, such as a region of the plane, into itself. Of particular interest are the trajectories followed by repeatedly applying the mapping to some initial point. A repeller is a set that is mapped into itself, but such that nearby points outside the set move away from the set under iteration of the mapping. Repellers often have a fractal structure, with ever more intricate detail becoming apparent under continued enlargement. Dimension provides a way of measuring the size and complexity of such sets. Smooth geometrical shapes have dimensions that are whole numbers: curves are 1-dimensional, surfaces are 2-dimensional, etc., reflecting the number of coordinates needed to locate points in the object. To cope with the intricacy of fractal repellers, notions of dimension that need not be whole numbers are required, such as Hausdorff, packing and box-counting dimensions, and such fractional dimensions will play a key role in this project.Much research has focused on conformal mappings, where the repellers are the often-pictured Julia sets of complex dynamics. These are generally locally self-similar, that is made up of many smaller, nearly similar copies of themselves, and their dimensions often may be found by solving an equation known as the Bowen-Ruelle formula.Non-conformal mappings lead to locally self-affine repellers which may be of a very different character, with small parts of the repellers appearing increasingly elongated or distorted under enlargement. Given recent progress in the analysis of self-affine sets, which may be thought of as piecewise linear analogues of non-conformal repellers, it is timely to extend the dimension formulae to a non-conformal, nonlinear setting. However, finding dimensions of self-affine sets is far more awkward than for self-similar sets, not least because the dimensions need not vary continuously with the defining transformations. By analogy, it may be difficult to obtain dimension formulae that are valid in every situation, so a major part of the project will be to seek generic formulae that are 'almost always' valid in an appropriate sense. In the conformal setting, the thermodynamic formalism is an important tool for finding the dimensions, but for non-conformal systems a 'subadditive' version of the thermodynamic formalism will be needed.Of course, it is desirable to know with certainty the dimension of specific repellers, rather than just finding formulae that are valid generically, so the project will also identify classes of functions for which the generic formulae can be guaranteed to give the actual value of dimension. A further line of investigation will consider distributions of mass across non-conformal repellers, with the aim of using analogues of fractional dimensions to quantify the irregularities of distribution, an area known as multifractal analysis.