Universal supercritical behavior for some skew-product maps

Abstract

<p style='text-indent:20px;'>We consider skew-product maps over circle rotations <inline-formula><tex-math id="M1">\begin{document}$ x\mapsto x+\alpha\; $\end{document}</tex-math></inline-formula> with factors that take values in <inline-formula><tex-math id="M2">\begin{document}$ {{{\rm{SL}}}}(2,{\mathbb{R}}) $\end{document}</tex-math></inline-formula>. In numerical experiments with <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> the inverse golden mean, Fibonacci iterates of almost Mathieu maps with rotation number <inline-formula><tex-math id="M4">\begin{document}$ 1/4 $\end{document}</tex-math></inline-formula> and positive Lyapunov exponent exhibit asymptotic scaling behavior. We prove the existence of such asymptotic scaling for "periodic" rotation numbers and for large Lyapunov exponent. The phenomenon is universal, in the sense that it holds for open sets of maps, with the scaling limit being independent of the maps. The set of maps with a given periodic rotation number is a real analytic manifold of codimension <inline-formula><tex-math id="M5">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> in a suitable space of maps.</p>