Mathematical Sciences: Measures, Dynamics and Dimensions

Mauldin 9502952 我们的项目包括三个部分。在第一部分中，我们建议 处理无穷系统的共形映射，调查的拓扑和 对应极限集的几何结构。我们打算进行研究， 比较共形，不变，调和，Hausdorff和包装措施在一个 一般设置以及在一些特殊的情况下，系统主要给出 通过不规则系统，具有限制项的连续分数， 阿波罗填料我们还计划探讨这样的共轭问题， 系统，研究随机系统和吸引子的不变连续统， 驱虫剂在第二部分中，我们计划处理的相变 单连通流形上Julia集的压力与Riemann映射的积分平均 周期稳定点或理性中立点的吸引域 周期点我们打算提供适当的压力定义， 建立压力和积分平均值之间的关系，并研究 相变现象的动力学系统和热力学方法 形式主义我们还想探索平衡的几何学和动力学 保持器连续函数的状态。我们特别感兴趣的是 Perron-Frobenius算子迭代的渐近性态和 由保持器连续函数生成的随机变量序列。 在最后一节中，我们提出处理的几何和动力学的 Julia集上无常返临界点的有理函数 和Collet-Eckmann地图，探索豪斯多夫和盒子的问题 的有限性的影响 σ有限不变测度我们还计划研究调和测度， 从雅可比行列式的保持器连续性开始。 我们提出的研究涉及各种措施和维度， 类型，例如Hausdorff、packing、conformal和harmonic。这些 测量是空间和物体中“体积”的自然概念， study.这样的空间和对象可以由某种算法给出， 使我们能够逐步逼近最终的物体和体积。这些 空间也有一些自然的“运动”或动力作用于它们。我们 研究涉及开发技术，使我们能够研究 我们系统的几何和动态特征。的具体问题 这一建议产生于许多背景，其解决方案将涉及 来自不同领域的技术的混合：复杂分析，连续 理论，动力学，功能分析-理论的积极运营商的 Perron-Frobenius型，测度论与概率，统计 物理学-平衡态，相变和热力学形式主义。 ***

Mauldin 9502952 Our project consists of three sections. In the first section we propose to deal with infinite systems of conformal maps, investigating the topological and geometric structure of the corresponding limit set. We intend to study and to compare conformal, invariant, harmonic, Hausdorff and packing measures in a general setting as well as in the case of some special systems primarily given by irregular systems, continued fractions with restricted entries and apollonian packings. We also plan to explore the problem of conjugacies of such systems, to study random systems and invariant continua for attractors and repellors. In the second section we plan to deal with the phase transitions of pressure on Julia sets and integral means of Riemann maps of a simply connected basin of attraction to a periodic stable point or rationally indifferent periodic point. We intend to provide appropriate definitions of pressure, to establish relations between pressure and integral means and to investigate the phenomenon of phase transitions by means of dynamical systems and thermodynamic formalism. We also want to explore the geometry and dynamics of equilibrium states of Holder continuous functions. In particular we are interested in the asymptotic behavior of of iterates of the Perron-Frobenius operator and the sequence of random variables generated by Holder continuous functions. In the last section we propose to deal with geometry and dynamics of rational functions with no recurrent critical points in the Julia sets and Collet-Eckmann maps, exploring the problems of Hausdorff and box dimension and the influence of critical points on finiteness of sigma-finite invariant measures. We also plan to study harmonic measures, beginning with Holder continuity of their Jacobians. We propose research involving measures and dimensions of various kinds, for example Hausdorff, packing, conformal, and ha rmonic. These measures are natural notions of "volume" in the spaces and objects we study. Such spaces and objects may be given by some algorithm which allows us to approximate the final object and volume step by step. These spaces also have some natural "motion" or dynamics acting on them. Our research involves developing techniques which allow us to study both geometric and dynamic features of our system. The specific problems of this proposal arise from many contexts and their solutions will involve a mixture of techniques from various areas: complex analysis, continua theory, dynamics, functional analysis-theory of positive operators of the Perron-Frobenius type, measure theory and probability, statistical physics-equilibrium states, phase transitions and thermodynamic formalism. ***