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We develop a new KAM scheme that applies to SL(2, $${{\mathbb R}}$$) 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. [ABSTRACT FROM AUTHOR]

Let $\mathfrak{F}_{\lambda}^{n}$ be the $\mathop {\mathfrak {osp}}\nolimits \,(n|2)$-module of weighted densities on ℝ of weight λ. We compute the cohomology spaces $\mathrm{H}^{k}_{\mathrm{diff}}\left(\mathop {\mathfrak {osp}}\nolimits \,(n|2),\mathfrak{F}_{\lambda}^{n}\right)$, where k=1 and n=0,1,2 or k=2 and n=0,1. We explicitly give cocycles spanning these cohomology spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C 1 persistently no small angles if and only if f has a dominated splitting of index i on the C 1 i-preperiodic set P ( f). To prove this, we mainly use some important conceptions and techniques developed by Christian Bonatti. In the last section, we also give a characterization of the finest dominated splitting for linear cocycles. [ABSTRACT FROM AUTHOR]

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W t − I ∈ S 2 (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations. [ABSTRACT FROM AUTHOR]