JMD
Lipschitz continuous invariant forms for algebraic Anosov systems
Patrick Foulon Boris Hasselblatt
We prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz continuous invariant $1$-form. This has corollaries for rigidity of time-changes, and we give a particular application to geometric rigidity of quasiconformal Anosov flows.
   Several features of the reasoning are interesting; namely, the use of exterior calculus for Lipschitz continuous forms, the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
keywords: Lipschitz regularity Anosov flow invariant forms smooth rigidity.
ERA-MS
Zermelo deformation of finsler metrics by killing vector fields
Patrick Foulon Vladimir S. Matveev

We show how geodesics, Jacobi vector fields, and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.

keywords: Finsler Zermelo deformation Killing vector fields Jacobi fields curvature symmetric spaces
JMD
Zygmund strong foliations in higher dimension
Yong Fang Patrick Foulon Boris Hasselblatt
For a compact Riemannian manifold $M$, $k\ge2$ and a uniformly quasiconformal transversely symplectic $C^k$ Anosov flow $\varphi$:$\R\times M\to M$ we define the longitudinal KAM-cocycle and use it to prove a rigidity result: $E^u\oplus E^s$ is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that $E^u\oplus E^s$ is Lipschitz-continuous. Results proved elsewhere then imply that the flow is smoothly conjugate to an algebraic one.
keywords: Zygmund regularity quasiconformal geometric rigidity smooth rigidity Anosov flow foliations. strong invariant subbundles
ERA-MS
Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows
Yong Fang Patrick Foulon Boris Hasselblatt
In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin, and we prove new results of this smooth rigidity type for a class of flows.
    For a transversely symplectic uniformly quasiconformal $C^2$ Anosov flow on a compact Riemannian manifold 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 KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form.
    We obtain a pertinent geometric rigidity result: Uniformly quasiconformal magnetic flows are geodesic flows of hyperbolic metrics.
    Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
keywords: Zygmund regularity smooth rigidity quasiconformal geometric rigidity. Anosov flow strong invariant subbundles

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