The longitudinal KAM-cocycle of a magnetic flow

Longitudinal KAM-cocycles and action spectra of magnetic flows

Let $M$ be a closed oriented surface and let $Ω$ be a non-exact 2-form. Suppose that the magnetic flow $ϕ$ of the pair $(g,Ω)$ is Anosov. We show that the longitudinal KAM-cocycle of $ϕ$ is a coboundary if and only the Gaussian curvature is constant and $Ω$ is a constant multiple of the area form thus extending the results in \cite{P2}. We also show infinitesimal rigidity of the action spectrum of $ϕ$ with respect to variations of $Ω$. Both results are obtained by showing that if $G:M\to\mathbb R$ is any smooth function and $ω$ is any smooth 1-form on $M$ such that $G(x)+ω_{x}(v)$ integrates t…

A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies

We develop a new KAM scheme that applies to SL(2,R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's Theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

无KAM的可约性

Reducibility without KAM

本文在有限正则性与光滑条件下证明单频近常值拟周期SL(2,R)上闭链的旋转可约性，并应用于拟周期薛定谔算子。

Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed tha…