We develop a new KAM scheme that applies to SL(2, R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc. 23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi (Asterisque 49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere (J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author (Ergod. Th. & Dynam. Sys. 22 (2002) 437–468; Proc. Amer. Math. Soc. 130 (2002), 103–109).

We give an example of a strictly positive analytic reparametrization (or time change) of an irrational flow on T 3 that is mixing. As an immediate application we obtain perturbations of completely integrable Hamiltonian systems that display many invariant tori on which the restricted dynamics is mixing.