Measurable dynamics: theory and applications

这是一份未来五年数学研究计划的提案，研究方向是可测量动力系统或遍历理论。在这个领域，我们使用测量理论和现代概率论的技术来研究随时间进化的数学系统（或物理系统的数学模型）的渐近或远程行为。遍历理论为动力系统的分析提供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或循环结构，人们构造了一个庞加莱截面。对于哈密顿系统，Liouville测度推导出截面上离散时间映射的不变测度，可以应用遍历理论的工具。实际上，不变测度编码了系统横截面的平衡行为，从中可以计算长期统计数据。另一个例子是，世界海洋的流动可以被建模为一个离散时间动力系统，在这个系统中，海洋的小盒子的演变在一些固定的正时间步长（根据问题的不同，从几小时到几个月不等）上被跟踪。虽然理想的洋流应该保持体积，但在实践中，观测数据仅与体积或面积保持大致一致，这对建模提出了挑战。一个更基本的数学挑战是，洋流的参数随时间而变化；系统是非自治的。这个系统的平衡现在采用等变测度族的形式，随着基础参数值的变化而演变。尽管如此，人们仍然可以使用遍历理论的理论结果（在这种情况下，乘法遍历理论）来提取海洋流动的长期统计数据。虽然上面的例子是确定性系统，但众所周知，它们可以模拟随机或概率行为。例如，遍历理论有大数定律（趋于均值）、中心极限定理（观测值向正态分布收敛）、大偏差估计等。这是目前遍历理论研究中最令人兴奋的领域之一。本提案中描述的工作旨在进一步梳理包括自主和非自主动力学在内的一系列模型中确定性和随机行为之间的重要联系。为了实现这些目标，该计划将依赖于PI与该领域现有国际专家之间的协调研究，并将在PI的监督下整合维多利亚大学HQP的培训活动。

This is a proposal for a mathematical 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 study asymptotic or long-range behaviour of mathematical systems (or mathematical models of physical systems) 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. In effect, the invariant measure encodes the equilibrium behaviour of the system's cross section, from which one can compute long term statistics. As another 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, presenting a modelling challenge.  A more fundamental mathematical challenge is that the parameters in ocean flow change over time; the system is non-autonomous. Equilibria for this system now take the form of equivariant families of measures, evolving along with the changing values of the underlaying parameters. Still, one can use theoretical results from ergodic theory (in this case, the multiplicative ergodic theory) to extract long term statistics of ocean flow. Although the examples above are deterministic systems, it is well-known that they can mimic stochastic or probabilistic behaviour. For example, ergodic theory has a version of the law of large numbers (tendency to the mean), the central limit theorem (distributional convergence of observations to normal), large deviations estimates, and so-on. This is one of the most exciting areas of current research in ergodic theory.   The work described in this proposal aims to further tease out this important connection between deterministic and random-like behaviour in a range of models that include both autonomous and non-autonomous dynamics. 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 supervision of the PI.