Descriptive theory of determined chaos

• Date: 2023
• Main Authors: Sharkovs’kyi , О. М., Шарковський, О. М.
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2023
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6515
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.14 Descriptive theory of sets – a classical branch of mathematics that arose at the beginning of the last century. This article offers the basics of descriptive chaos theory. It is shown that a dynamic system, if the topological entropy is positive: 1) has many different trajectory attractors, namely, a continuum of attractors; 2) the basins of most attractors have an overly complex structure, namely, are sets of the third class according to the terminology of descriptive set theory; 3) the basins of different attractors are too strongly intertwined and cannot be separated from each other by any open or closed sets, but only by sets of the second complexity class and 4) the set of all attractors of the dynamical system forms an attractor grid (network) in the space of closed sets of the state space (with the Hausdorff metric), the cells of which are created by Cantor sets from the attractors themselves. \end{enumerate}