On linear cocycles over irrational rotations with secondary collisions

Abstract

We consider a skew product F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</inf> = (σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω</inf> ,A) over irrational rotation ${\sigma _\omega }(x) = x + \omega $ of a circle ${\mathbb{T}^1}$. It is supposed that the transformation $A:{\mathbb{T}^1} \to SL(2,\mathbb{R})$, being a C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -map, has the form $A(x) = R(\varphi (x))Z(\lambda (x))$, where R(φ) is a rotation in ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> over the angle φ and Z(λ) = diag{λ,λ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> } is a diagonal matrix. Assuming that λ(x) ≥λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> > 1 with a sufficiently large constant λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> and the function φ is such that cos φ(x) possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</inf> . We apply the critical set method to show that, under some additional requirements on the derivative of the function φ, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</inf> becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.