Measures, Dynamics and Dimensions

[摘要]Mauldin/Urbanski研究者将继续进行各种测量方法的研究——hausdorff、packing、保形、Gibbs状态等；相关功能-容量、压力和相关尺寸。这些概念将从两个相互交织的观点来研究。一个是各种动力系统的问题，另一个是应用于递归生成对象的几何测度理论问题。这些方法自然地相互融合，产生了一些有趣的想法。在过去的几年里，我们发展了一个相当广泛的关于无限多个一致收缩共形映射（双曲系统）迭代的理论，并将其应用于一些众所周知的问题。我们希望扩展这一理论，以涵盖更结构化的映射迭代——它由有向图或替换和随机迭代控制。我们还希望将我们的理论应用于作为Kleinian群的极限集的几何对象，或者更一般地说是非一致收缩的映射系统（抛物系统）。我们打算开发必要的工具来应用我们的理论继续获得关于连分式集合的几何结构以及与经典算术密度结果的关系的新结果。我们还计划为我们的系统开发一个合适的多重分形形式。我们想用这些工具来描述Collett-Eckmann有理函数的Julia集合的一些几何方面。最后，我们想要得到一些近似定理，这些近似定理对于得到已知的关于Hausdorff测度和维数的截面定理和投影定理对填充测度和维数及其他混合几何测度的类似物是有用的。本提案的具体问题出现在许多背景下，其解决方案将涉及不同领域的技术混合：复杂分析，动力学，泛函分析- reule - perron - frobeni - us型正算子理论（转移算子），测量理论和概率论，统计物理-平衡态和热力学形式主义。这项工作源于几个不同领域发生的问题-统计物理，热力学，湍流理论，几何，算法或生成各种几何物体的配方，这些物体受到随机误差的影响，最后是动力系统，其中系统的长期行为由一些看似奇怪的物体控制。我们的工作集中在所有这些系统的一些共同特征上。首先，我们打算展示如何以一种自然的方式生成与每个系统相关的度量。对于某些系统，这些度量为我们提供了一种度量系统长期行为的方法。对于某些系统，它给了我们一种量化所产生物体的精细几何结构的方法，或者说明几何结构的平均样子。每一个度量都与一个维度相关联，一个可能不是整数的数字——分形维数。这些量纲数可以用我们开发的一些基本公式来计算或估计。这对应用程序很有用。这部分可以在计算机研究的帮助下进行，因此出于实际原因是直接感兴趣的。我们打算继续发展适用的公式来确定这些方面，并从中产生产生关于该系统的大量信息的相关措施。

Abstract Mauldin/Urbanski The investigators will continue ongoing research involving measures of various kinds-Hausdorff, packing, conformal, Gibbs state, etc.; associated functions-capacities, pressure and associated dimensions. These notions will be studied from two interlaced viewpoints. One is that of various dynamical systems and the other is that of geometric measure theory as applied to recursively generated objects. These approaches naturally meld with one another and lead to some interesting mixtures of ideas. During the last few years we have developed a fairly extensive theory of the iteration of infinitely many uniformly contracting conformal maps(hyperbolic systems) and have made some applications of it to some well known problems. We would like to extend this theory to cover a more structured iteration of maps-whose governed by directed graphs or substitutions and random iterations. We also wish to apply our theory to geometric object obtained as limit sets of Kleinian groups or more generally a system of maps which are not uniformly contracting(parabolic systems). We intend to develop the tools necessary to apply our theory to continue obtaining new results about the geometric structure of sets of continued fractions and relationships to classical arithmetic density results. We also plan to develop an appropriate multifractal formalism for our systems. We want to use these tools to describe some aspects of the geometry of the Julia sets of Collett-Eckmann rational functions. Finally, we want to obtain some approximation theorems which should be useful for obtaining analogues of known section and projection theorems concerning Hausdorff measures and dimension to packing measures and dimension and other mixed geometric measures. The specific problems of this proposal arise from many contexts and their solutions will involve a mixture of techniques from various areas: complex analysis, dynamics, functional analysis-theory of positive operators of the Reulle-Perron-Frobeni us type(transfer operators), measure theory and probability, statistical physics-equilibrium states, and thermodynamic formalism. This work arises from problems occurring in several diverse areas-statistical physics, thermodynamics, the theory of turbulence, geometry, algorithms or recipes for generating various geometric objects which are subject to random errors and finally, to dynamical systems where the long term behavior of the system is governed by some seemingly strange object. Our work focuses on some common features of all of these systems. First, we intend to show how, in a natural way, to produce measures associated with each system. For some systems, these measures give us a means of measuring the long term behavior of the system. For some systems, it gives us a means of quantifying the fine scale geometric structure of the object produced or of stating what the geometric structure will be like on average. To each of these measures is associated a dimension, a number which might not be an integer-a fractal dimension. These dimension numbers can be computed or estimated by some basic formulas which we have developed. This is useful for applications. This part can be carried out with the aid of computer studies and thus is of direct interest for pratical reasons. We intend to continue to develop applicable formulas for determining these dimensions and from that how to produce the associated measures which yield so much information about the system.