Invariant measures for non-autonomous dynamical systems.

动力系统是数学中一个令人兴奋的分支。他们使用的结果和方法，从许多其他分支，并找到应用在许多分支的数学和其他科学。他们也被认为是重要的：在2014年和2018年国际数学大会上，四个菲尔兹奖章中的两个被授予动力学研究人员。 该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（自治的）动力系统是空间X上的映射T，它保持某种测度（平稳分布）。我们研究了T的迭代以及它们如何改变某些初始分布。例如，在人口动力学中，给定一个人口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），如果我们可以识别平稳分布，我们就可以理解人口在未来的行为。一个非自治动力系统是一族映射Tn，它通过在第n步应用Tn而作用于空间。X中的点x的轨迹是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中的过程更为现实，因为它们的参数一直在轻微变化。由于我们组成的地图是不同的，一个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”的测度，其中这个概念以不同的方式定义。另一个我将要探讨的问题是土星环内粒子的混沌运动。数百年来，这些光环引起了人们的研究兴趣。J.Froyland于1992年提出了一个动力学系统模型，迄今为止还没有对它进行理论研究。计算机实验表明，该系统至少有11个不变区域。 相当于11枚戒指。该系统是理论上尚未解决的三体问题的一个特例，任何进展都将是令人感兴趣的。 第三个问题，我将研究的是一个二维动力系统与大量的周期性移动的“岛屿”。在每一个岛上，点在地图的迭代下混乱地移动。这是一个被称为“弱混沌”的一般现象的例子，其中看似周期性的轨道实际上只是近似周期性的，并且它的每个点都不是一个点，而是一小簇点。人们强烈怀疑行星围绕太阳的轨道具有这种性质。在我的研究中，我将与康考迪亚大学的动态研讨会小组密切合作。Boyarsky，Dr. H. Proppe，我和我们的研究生。在我的指导下，研究生将充分参与拟议的研究，以帮助实现提案的目标。

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.