Cocycles, cohomology and combinatorial constructions in ergodic theory (Q2778777)

This paper is an expanded and revised version of the second part (of four) of A. Katok's major unpublished work `Constructions in ergodic theory'. An updated version of the first part, covering the method of periodic approximation and its applications to the study of generic and exceptional dynamics in various categories, is being published in Proc. Steklov Inst. Math. NEWLINENEWLINENEWLINEAs the authors say, this paper reflects to some extent the state of play twenty years ago. However, this does not diminish the importance of the methods and viewpoints in it. The main sections are: Principal Constructions (including the so-called Mackey range, cohomological equations, group extensions, induced transformations and so on); Structure of Equivalence Classes (concerning how equivalence classes of cocycles are distributed, how much a cocycle need be modified to push it into a prescribed cohomology class, cohomology to `regular' cocycles, and the sometimes dramatic effect of mildly increased smoothness conditions); Rigidity and Stability (surveying situations in which equivalence classes of cocycles have some reasonable structure); Wild cochains with tame coboundaries (well-behaved cocycles that are measurably coboundaries but only with wild transfer functions, and their role in the construction of examples of dynamical systems with very different measurable and topological properties); Non-trivial cocycles (concerning the difficult task of establishing that a given cocycle is not a coboundary even if the transfer function is allowed to be badly behaved). NEWLINENEWLINENEWLINEThis paper is a welcome survey, with valuable perspectives on the field.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].