Measurable Dynamics, Theory and Applications

可测量的动力学、理论和应用

• 批准号: RGPIN-2014-04958
• 负责人: Bose, Chris
• 金额: $ 1.02万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566849
• 关键词: Measurable; Dynamics; Theory; Applications

This is a proposal for a research program over the next five years in an area called measurable dynamical systems or ergodic theory. In this field we use techniques from measure theory and modern probability to understand asymptotic or long range behaviour of mathematical objects (or mathematical models of physical objects) that evolve with time. Ergodic theory brings a range of powerful tools to the analysis of dynamical systems with applications both in and outside the field of mathematics. For example, in ordinary differential equations one constructs a Poincare section in order to study periodic or recurrent structures in the continuous time flow. In the case of Hamiltonian systems, Liouville measure induces an invariant measure for the discrete-time map on the section, and the tools from ergodic theory can be applied. As another example, the sequence of digits in the continued fraction expansion of a real number can be generated by iteration of the Gauss map on the unit interval. There is a finite, invariant, Borel measure (Gauss measure) on the interval preserved by the Gauss map. The orbit of a point under iteration of the Gauss map determines the continued fraction digits and their statistical properties via this measure preserving system. As a third example, the flow of the world's oceans can be modelled as a discrete-time dynamical system, where the evolution of little boxes of ocean are tracked over some fixed positive time step (ranging from hours to months, depending on the problem). While an idealized ocean flow should preserve volume, in practice, observational data only roughly coincides with volume or area preservation. A very basic problem is to find an invariant measure for this observed discrete-time transformation and to use that measure along with the tools from ergodic theory to study this important dynamical system. Our proposal considers both theoretical and computational aspects of the field. We will treat both closed (traditional) and open dynamical systems as well as non-autonomous or time-dependent systems arising as random maps. Our objectives are (1) To develop efficient, rigorous numerical schemes to approximate dynamical structures such as invariant measures, conditionally invariant measures, almost invariant structures and other such barriers to mixing in dynamical systems. The same methods will be used to estimate relaxation parameters such as escape rate and correlation decay. (2) Ergodic theory has an analogue of the transition matrix from Markov chains theory called the Perron-Frobenius operator. A second objective of this program is to extend powerful theoretical techniques for spectral analysis of the Perron-Frobenius operator to the multidimensional setting, particularly for non-uniformly hyperbolic examples. This is an important, modern area of focus in the field of ergodic theory. (3) To investigate stability and bifurcation issues that arise from random perturbation of a single transformation and/or more generally to understand the effects of small-scale parameter variation in non-autonomous systems. This is highly relevant for real-life applications such as the ocean flow model outlined above. In order to achieve these objectives the program will rely on coordinated research between the PI and existing international experts in the field and will integrate training activities for HQP located at the University of Victoria and under direct supervision of the PI.

这是一个未来五年的研究计划的建议，在一个称为可测动力系统或遍历理论的领域。在这一领域，我们使用测量理论和现代概率的技术来理解随时间演化的数学对象（或物理对象的数学模型）的渐近或长期行为。遍历理论为动力系统的分析带来了一系列强大的工具，在数学领域内外都有应用。例如，在常微分方程中，为了研究连续时间流中的周期性或递归结构，人们构造了一个庞加莱截面。在Hamilton系统的情形下，Liouville测度导出了截面上离散映射的不变测度，并且可以应用遍历理论的工具。作为另一示例，真实的数的连分式展开中的数字序列可以通过高斯映射在单位区间上的迭代来生成。有一个有限的，不变的，波莱尔测度（高斯测度）的区间保持高斯映射。高斯映射迭代下的点的轨道通过这个保测度系统确定连分数位数及其统计性质。作为第三个例子，世界海洋的流动可以被建模为离散时间动力系统，其中海洋的小盒子的演变在一些固定的正时间步长（从几小时到几个月不等，取决于问题）上被跟踪。虽然理想化的海洋流动应该保持体积，但在实践中，观测数据只能大致符合体积或面积的保持。一个非常基本的问题是找到一个不变的措施，这个观察到的离散时间变换，并使用该措施沿着的工具，从遍历理论来研究这个重要的动力系统。我们的建议考虑了该领域的理论和计算方面。我们将同时处理封闭（传统）和开放的动力系统，以及作为随机映射产生的非自治或时间依赖系统。我们的目标是：（1）发展有效的，严格的数值格式来近似动力学结构，如不变测度，条件不变测度，几乎不变结构和其他这样的障碍混合动力系统。同样的方法将被用来估计弛豫参数，如逃逸率和相关衰减。 (2)遍历理论有一个类似于马尔可夫链理论的转移矩阵，称为Perron-Frobenius算子。该计划的第二个目标是扩展强大的理论技术谱分析的Perron Frobenius运营商的多维设置，特别是非均匀双曲的例子。这是一个重要的，现代领域的焦点在遍历理论领域。(3)研究由单个变换的随机扰动引起的稳定性和分叉问题，和/或更一般地理解非自治系统中小尺度参数变化的影响。这与现实生活中的应用高度相关，例如上面概述的海洋流动模型。为了实现这些目标，该计划将依靠PI和该领域现有国际专家之间的协调研究，并将整合位于维多利亚大学的HQP的培训活动，并在PI的直接监督下进行。