Monotonic cocycles

Abstract

We develop a "local theory" of multidimensional quasiperiodic $\SL(2,\R)$ cocycles which are not homotopic to a constant. It describes a $C^1$-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of $\SL(2,\R)$ cocycles homotopic to a constant. Our analysis is centered around the notion of monotonicity with respect to some dynamical variable. For such {\it monotonic cocycles}, we obtain a sharp rigidity result, minimality of the projective action, typical nonuniform hyperbolicity, and a surprising result of smoothness of the Lyapunov exponent (while no better than H\"older can be obtained in the case of cocycles homotopic to a constant, and only under arithmetic restrictions). Our work is based on complexification ideas, extended "\`a la Lyubich" to the smooth setting (through the use of asymptotically holomorphic extensions). We also develop a counterpart of this theory centered around the notion of monotonicity with respect to a parameter variable, which applies to the analysis of $\SL(2,\R)$ cocycles over more general dynamical systems and generalizes key aspects of Kotani Theory. We conclude with a more detailed discussion of one-dimensional monotonic cocycles, for which results about rigidity and typical nonuniform hyperbolicity can be globalized using a new result about convergence of renormalization.