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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]

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions. [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]

Abstract: We give a cocycle expansivity result for multimodal interval maps with non-flat critical points. It extends the Mañé hyperbolicity theorem to also describe orbits which pass near critical points. To cite this article: N. Dobbs, C. R. Acad. Sci. Paris, Ser. I 345 (2007). [Copyright &y& Elsevier]