Rigidity properties of Anosov optical hypersurfaces

Dairbekov, N. S.; Paternain, Gabriel P.

doi:10.1017/s0143385707000612

Cited by 10 publications

(32 citation statements)

References 23 publications

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“…In [8], N. S. Dairbekov and G. P. Paternain proved Theorem 1.3 for the case of the magnetic flow on a Riemannian surface. The same result was proved in [6] by N. S. Dairbekov and G. P. Paternain for thermostats on Riemannian surfaces and in [7] for magnetic flows on Finsler manifolds of any dimension.…”

Section: Closed Surfacessupporting

confidence: 66%

See 2 more Smart Citations

The X-ray Transform on a General Family of Curves on Finsler Surfaces

Assylbekov

Dairbekov

2017

J Geom Anal

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For a compact oriented Finsler surface with smooth boundary, we consider the scalar and vector integral geometry problems over a general family of curves running between boundary points and parametrized by arclength. We impose a natural condition which results in the no conjugate points condition in the case when the curves in question are geodesic lines. Our main theorem generalizes Mukhometov's theorem in several directions.We also consider these problems on a closed oriented Finsler surface. In this case the integral geometry problems make sense provided that sufficiently many curves in the family are periodic. To this end, we assume that the induced flow on the unit circle bundle of the surface is Anosov. Also, we study the cohomological equation of thermostats without conjugate points.

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Section: Closed Surfacessupporting

confidence: 66%

“…Using these conditions together with the Jacobi equation, which we do not derive here since it can be done similarly as in [7], we conclude that…”

Section: Pestov Identitymentioning

confidence: 99%

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The X-ray Transform on a General Family of Curves on Finsler Surfaces

Assylbekov

Dairbekov

2017

J Geom Anal

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“…We first recall commutation formulas for horizontal and vertical derivatives [33] (see also [7,Lemma 4.1], which deals with the case of Finsler metrics). If u ∈ C ∞ (T M \ {0}), then…”

Section: Pestov Identitymentioning

confidence: 99%

“…But the results in [7,Theorem B] give a complete understanding of the cohomological equation for Anosov magnetic flows. Indeed, there is a solution of (7) iff θ is an exact form.…”

Section: Proof Of Theorem Amentioning

confidence: 99%

Entropy Production in Thermostats II

Dairbekov

Paternain

2007

J Stat Phys

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Abstract. We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfiesfor every 2-plane σ, where K w is the sectional curvature of the associated Weyl connection and E σ is the orthogonal projection of E onto σ. Then the entropy production of any SRB measure is positive unless E has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.

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Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

2015

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In the recent articles [PSU13, PSU14c], a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation. Contents

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Rigidity properties of Anosov optical hypersurfaces

Cited by 10 publications

References 23 publications

The X-ray Transform on a General Family of Curves on Finsler Surfaces

The X-ray Transform on a General Family of Curves on Finsler Surfaces

Entropy Production in Thermostats II

Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

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