On sets with recurrence properties, their topological structure and entropy

We study local versions of transitivity and weak mixing expressed in terms of properties of sets. In particular, the structure of the class of these sets in the hyperspace X2 is investigated. We introduce the definition of transitive sets of higher degrees and prove that any weakly mixing set of order 2 has a stronger form of transitivity. We also discuss weakly mixing sets and chaotic sets over fibers of a factor map with relation to entropy of considered transformations.