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DCDS

Let $\varphi$ be a $C^\infty$ transversely symplectic topologically mixing Anosov flow such that $dim E^{su}\geq 2$. We suppose that the weak distributions of $\varphi$ are $C^1$. If the length Hamenstädt metrics of $\varphi$ are sub-Riemannian then we prove that the weak distributions of $\varphi$ are necessarily $C^\infty$. Combined with our previous rigidity result in [5] we deduce the classification of such Anosov flows with $C^1$ weak distributions provided that the
length Hamenstädt metrics are sub-Riemannian.

DCDS

In this paper we investigate smooth conjugacy of $C^\infty$ expanding maps on certain nilmanifolds. We show that several rigidity results about expanding maps on the circle can not be generalized directly to higher dimensions. For example the following result is obtained: Let $\Gamma_1\backslash N_1$ and $\Gamma_2\backslash N_2$ be two nilmanifolds of homogeneous type. We show that for any positive integer $k$ there exist on the product nilmanifold $\Gamma_1\times \Gamma_2\backslash N_1\times N_2$ a $C^\infty$ expanding map $\varphi$ and an expanding nilendomorphism $\psi$ which are $C^k$ conjugate, but not $C^{k,lip}$ conjugate. While in the case of dimension one, it was shown by M. Shub and D. Sullivan that if two $C^\infty$ expanding maps on $\mathbb S^1$ are absolutely continuously conjugate, then they must be $C^\infty$ conjugate.

DCDS

By using a formula relating topological entropy and cohomological pressure, we obtain several rigidity results about contact Anosov flows. For example, we prove the following result: Let $\varphi$ be a $C^\infty$ contact Anosov flow. If its Anosov splitting is $C^2$ and it is $C^0$ orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of $\varphi$ coincide. This result generalizes a result of U. Hamenstädt for geodesic flows.

DCDS

This paper deals with interactions between metric quasiconformal geometry and the rigidity of Anosov flows. In the first part of this article, we study a canonical time change of Anosov flows. Then we use it to obtain the thorough classification of volume-preserving quasiconformal Anosov flows.

Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence:

Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$.

If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.

Given a $C^\infty$ Anosov flow $\varphi$, it is well-known that along the leaves of its strong unstable foliation there are two natural distances: the Hamenst$\ddot {\rm a}$dt distance and the induced Riemannian distance. In general these two distances are H$\ddot{\rm o}$lder equivalent. We prove the following rigidity result about quasisymmetric equivalence:

Let $\varphi: M\to M$ be a $C^\infty$ transversely symplectic Anosov flow with $C^1$ weak distributions. We suppose that ${\rm dim}M\geq 5$.

If over a leaf of the strong unstable foliation, the Hamenst$\ddot{\rm a}$dt distance is quasisymmetric equivalent to the Riemannian distance, then up to finite covers $\varphi$ is $C^\infty$ orbit equivalent either to the geodesic flow of a closed hyperbolic manifold, or to the suspension of an Anosov automorphism.

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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