Descriptive Theory of Deterministic Chaos

Descriptive set theory is a classical branch of mathematics formed at the beginning of the last century. We offer the basics of descriptive chaos theory. It is shown that if the topological entropy is positive, then a dynamical system:(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 they 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 network (grid) 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.