In this paper we will discuss recent work of the authors (Downey, Greenberg and Weber  and Downey and Greenberg [6, 7]) devoted to understanding some new naturally definable degree classes which capture the dynamics of various natural constructions arising from disparate areas of classical computability theory.
It is quite rare in computability theory to find a single class of degrees which capture precisely the underlying dynamics of a wide class of apparently similar constructions, demonstrating that they all give the same class of degrees. A good example of this phenomenon is work pioneered by Martin  who identified the high c.e. degrees as the ones arising from dense simple, maximal, hh-simple and other similar kinds of c.e. sets constructions. Another example would be the example of the promptly simple degrees by Ambos-Spies, Jockusch, Shore and Soare . Another more recent example of current great interest is the class of K-trivial reals of Downey, Hirscheldt, Nies and Stephan , and Nies [23, 24].