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JMD

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.

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.

JMD

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.
ERA-MS

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

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.

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.

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