Invariant measures for non-autonomous dynamical systems.

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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.