Monotonic cocycles
By Artur Avila, R. Krikorian. 2013
51 citations recorded by the source corpus.
We develop a “local theory” of multidimensional quasiperiodic $${\mathrm {SL}}(2,{\mathbb R})$$SL(2,R) cocycles which are not homotopic to a constant. It describes a $$C^1$$C1-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of $${\mathrm {SL}}(2,{\mathbb R})$$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 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 Hölder 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 “à 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 $${\mathrm {SL}}(2,{\mathbb R})$$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.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/e390abbeda3f4f91599e7b96b046c9442fb59d48
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In Geometry and Topology
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles (2013, result): “ as in [GoSc]. We remark that Theorem 2 has an analog as in [JiKoSch], that is, we can prove continuity of LE with respect to the C2 cos-type of potentials. For other related results, Avila-Krikorian [AK] recently studied so-called monotonic cocycles which are a class of smooth or analytic cocycles ”