Invariant measures for non-autonomous dynamical systems.

动力系统是数学中一个令人兴奋的分支。他们使用许多其他分支的结果和方法，并在数学和其他科学的许多分支中找到应用。他们也被认为是重要的：2014年国际数学大会上的四个菲尔兹奖中有两个和2018年国际数学大会上的一个被授予了动力学研究人员。提出的研究的主要目的是研究非自治动力系统的行为和预测未来。标准（自主）动力系统是空间X上的映射T，它保留了一些度量（平稳分布）。我们研究了T的迭代以及它们如何改变某个初始分布。例如，在种群动力学中，给定种群的初始分布和一年内控制其运动的规则（对应于地图T），我们可以研究多年后的变化（T的迭代），如果我们能确定平稳分布，我们就能理解未来种群的行为。非自治动力系统是通过n步作用于空间的一组映射n。点x在x中的轨迹为x， T1(x), T2(T1(x)), T3(T2(T1(x)))，…非自治模型对于自然科学和社会科学的过程更为现实，因为它们的参数一直在轻微变化。由于我们绘制的图是不同的，平稳分布的标准概念不再适合描述系统的长时间行为。我们所能期望的最好结果是某种“渐近几乎不变”的度量，在这种度量中，这个概念以不同的方式定义。我要探讨的另一个问题是土星环内粒子的混沌运动。这些环吸引了数百年的研究。1992年，J. Froyland提出了动力学的动力学系统模型，但迄今为止还没有对其进行理论研究。计算机实验表明，该系统至少有11个不变区域。对应11个环。该系统是理论上尚未解决的三体问题的一种特殊情况，在这方面的任何进展都会引起人们的兴趣。我要研究的第三个问题是一个二维动力系统，它有大量周期性移动的“岛屿”。在每个岛屿内，点在地图的迭代下混乱地移动。这是一个被称为“弱混沌”的普遍现象的例子，在这种现象中，看似周期性的轨道实际上只是近似周期性的，它的每一个点都不是一个点，而是一小簇点。人们强烈怀疑行星绕太阳运行的轨道具有这种特征。在我的研究中，我将与我们在康考迪亚大学的动力学研讨会小组密切合作，该小组由A. Boyarsky博士、H. Proppe博士、我自己和我们的研究生组成。我所指导的研究生将会全身心地投入到所建议的研究中，以帮助实现所建议的目标。

Dynamical Systems are an exciting branch of Mathematics. They use results and methods from many other branches and find application in many branches of mathematics and other sciences. They are also recognized as important: two out of four Fields Medals at the 2014 and one at the 2018 International Mathematical Congresses were awarded to researchers in dynamics. The main objective of the proposed research is to study the behaviour and predict the future of a non-autonomous dynamical system. A standard (autonomous) dynamical system is a map T on a space X which preserves some measure (stationary distribution). We study the iterations of T and how they change some initial distribution. For example, in population dynamics, given an initial distribution of population and the rules which govern its movements within a year (corresponding to map T), we can study the changes after many years (iterations of T) and if we can identify the stationary distribution we understand the behaviour of the population in the future. A non-autonomous dynamical system is a family of maps Tn which acts on space by application of Tn on the n-th step. The trajectory of a point x in X is x, T1 (x), T2 (T1 (x)), T3(T2(T1(x))),... The non-autonomous model is much more realistic for processes in nature and social sciences, as their parameters change slightly all the time. Since the maps we compose are different the standard notion of a stationary distribution is no longer appropriate to describe the long time behaviour of the system. The best we can hope for is some "asymptoticly almost invariant" measure, where this notion is defined in different ways. Another problem I am going to explore is the chaotic motion of particles within the rings of Saturn. These rings interested researches for hundreds of years. A dynamical systems model for the dynamics was developed in 1992 by J. Froyland and no theoretical work has been done on it so far. Computer experiments suggest that the system has at least 11 invariant regions.  This corresponds to 11 rings. The system is a special case of theoretically still unsolved 3-body problem and any progress on it would be of interest. A third problem I will study is a two-dimensional dynamical system with a large number of periodically moving "islands". Within each island the points move chaotically under an iteration of the map. This is an example of a general phenomenon known as "weak chaos", where seemingly periodic orbits are actually only approximately periodic and every point of it is not a point but a small cluster of points. There is strong suspicion that orbits of the planets around the Sun are of this character. In my research I am going to strongly cooperate with our Dynamical Seminar Group at Concordia University consisting of Dr. A. Boyarsky, Dr. H. Proppe, myself and our graduate students. The graduate students under my supervision will be fully engaged in the proposed research to help to achieve the goals of the proposal.