Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows (Q600829)

We have the following theorems: I. Let \(M\) be a compact Riemannian manifold of dimension at least 5, let \(k\geq 2\), and let \(\varphi :\mathbb R\times M\to M\) be a uniformly quasiconformal transversely symplectic \(C^k\) Anosov flow. Then \( E^u\oplus E^s\) is Zygmund-regular and there is an obstruction to higher regularity that defines the cohomology class of a cocyle we call the longitudinal KAM-cocycle. This obstruction can be described geometrically as the curvature of the image of a transversal under a return map, and the following are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \( E^u\oplus E^s\) is ``little Zygmund'' (the function is said to be ``little Zygmund'' if \(\|f(x+h) +f(x-h) -2f(x)\|=o(\|h\|)\)); \item[(2)] The longitudinal KAM-cocycle is a coboundary; \item[(3)] \( E^u\oplus E^s\) is Lipschitz-continuous; \item[(4)] \(\varphi\) is up to finite covers, constant rescaling and a canonical time-change \(C^k\)-conjugate to the suspension of a symplectic Anosov automorphism of a tours or the geodesic flow of a real hyperbolic manifold. \end{itemize}} Theorem I can be used to obtain rigidity results (Zygmund-regularity, and Lipschitz regularity together with conjugacity of the flow with an algebraic one). We obtain also geometric rigidity result: Uniformly quasiconformal magnetic flows are geodesic flows of hyperbolic metrics. II. Let \((N,g)\) be an \(n\)-dimensional closed negatively curved Riemannian manifold and \(\Omega\) a \(C^{\infty}\) closed 2-form of \(N\). For small \(\lambda\in\mathbb R\), let \(\varphi ^{\lambda}\) be the magnetic Anosov flow of the pair \((g,\lambda\Omega)\). Suppose that \(n\geq 3\) and \(\varphi ^{\lambda}\) is uniformly quasiconformal. Then \(g\) has constant negative curvature and \(\lambda\Omega =0\). In particular, the longitudinal KAM-cocycle of \(\varphi ^{\lambda}\) is a coboundary.