Invariant measures for non-autonomous dynamical systems.

动力系统是数学的一个令人兴奋的分支。他们使用了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了应用。它们也被认为是重要的：2014年国际数学大会上四枚菲尔兹奖牌中的两枚和2018年国际数学大会上的一枚授予了动力学研究人员。这项研究的主要目的是研究非自治动力系统的行为并预测其未来。一个标准的(自治)动力系统是空间X上保持某种度量(平稳分布)的映射T。我们研究T的迭代以及它们如何改变一些初始分布。例如，在人口动力学中，给定人口的初始分布和支配其在一年内移动的规则(对应于图T)，我们可以研究多年后的变化(T的迭代)，如果我们能够确定平稳分布，我们就可以了解未来人口的行为。非自治动力系统是通过在第n步应用Tn作用于空间的映射族Tn。点x在X中的轨迹是x，t1(X)，t2(t1(X))，t3(t2(t1(X)，…对于自然科学和社会科学中的过程来说，非自治模型要现实得多，因为它们的参数一直都有轻微的变化。由于我们组成的映射是不同的，平稳分布的标准概念不再适合描述系统的长期行为。我们所能期望的最好结果是某种“渐近几乎不变”的度量，其中这个概念有不同的定义。我要探讨的另一个问题是土星环内粒子的混沌运动。这些环引起了数百年来的研究兴趣。J·弗罗兰于1992年提出了一个动力学系统模型，到目前为止还没有关于它的理论工作。计算机实验表明，该系统至少有11个不变区域。这相当于11个戒指。该系统是理论上仍未解决的三体问题的特例，它的任何进展都将是令人感兴趣的。我要研究的第三个问题是具有大量周期性运动的“岛”的二维动力系统。在每个岛上，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，在这个现象中，看似周期的轨道实际上只是近似周期的，并且它的每个点都不是一个点，而是一小群点。人们强烈怀疑，围绕太阳运行的行星的轨道就是这样的。在我的研究中，我将与我们康科迪亚大学的动力研讨会小组大力合作，该小组由A·博亚斯基博士、H·普罗普博士、我自己和我们的研究生组成。在我的指导下，研究生将完全投入到建议的研究中，以帮助实现建议的目标。

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.