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Monotonic cocycles. (English) Zbl 1356.37070

One of the purpose of this paper is to develop a local theory of cocycles non-homotopic to a constant, covering perturbations of the simplest cocycles arising in such a case, and which are the \(\mathrm{SO}(2,\mathbb{R})\)-valued ones. The claim is that the theory so displayed is more robust than the other ones. In a first part one deals successively with the regularity of the Lyapunov exponent, the rigidity arising from zero Lyapunov exponent, and the minimality of the associated projective action. In a second part, one analyzes the one-dimensional quasi-periodic cocycles non-homotopic to a constant from the global point of view. A convergence of renormalization result is stated, which ensures that, under some conditions simple to be realized, renormalization turns to be monotonic.

MSC:

37H05 General theory of random and stochastic dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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