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May  2015, 35(5): 1801-1816. doi: 10.3934/dcds.2015.35.1801

The magnetic ray transform on Anosov surfaces

Gareth Ainsworth 1,

1. 

Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, CB3 0WB, United Kingdom

Received  May 2013 Revised  October 2013 Published  December 2014

Assume (M,g,$\Omega$) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final section of the paper we give an application to the entropy production of magnetic flows perturbed by symmetric 2-tensors.
Keywords: X-ray transform, Anosov surface, geodesics..
Mathematics Subject Classification: 53C25, 53C21, 53C2.
Citation: Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
References:
[1]

G. Ainsworth, The attenuated magnetic ray transforms on surfaces, Inverse Probl. Imaging., 7 (2013), 27-46. doi: 10.3934/ipi.2013.7.27.

[2]

Yu. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., 5 (1997), 487-490. doi: 10.1515/jiip.1997.5.6.487.

[3]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967), 209pp.

[4]

D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Uspekhi Mat. Nauk, 22 (1967), 107-172.

[5]

V. I. Arnold, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.

[6]

K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314. doi: 10.1088/0951-7715/15/2/305.

[7]

C. Croke and V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifolds, Topology, 37 (1998), 1265-1273. doi: 10.1016/S0040-9383(97)00086-4.

[8]

N. S. Dairbekov and G. P. Paternain, Entropy production in Gaussian thermostats, Comm. Math. Phys., 269 (2007), 533-543. doi: 10.1007/s00220-006-0117-y.

[9]

N. S. Dairbekov and G. P. Paternain, Rigidity properties of Anosov optical hypersurfaces, Ergod. Th. & Dynam. Sys., 28 (2008), 707-737. doi: 10.1017/S0143385707000612.

[10]

N. S. Dairbekov and G. P. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows, Math. Res. Lett., 12 (2005), 719-729. doi: 10.4310/MRL.2005.v12.n5.a9.

[11]

N. S. Dairbekov and G. P. Paternain, On the cohomological equation of magnetic flows, Mat. Contemp., 34 (2008), 155-193.

[12]

N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.

[13]

N. S. Dairbekov and V. A. Sharafutdinov, Some problems of integral geometry on Anosov manifolds, Ergod. Th. & Dynam. Sys., 23 (2003), 59-74. doi: 10.1017/S0143385702000822.

[14]

H. M. Farkas and I. Kra, Riemann Surfaces, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-2034-3.

[15]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[16]

E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, (French), Ergod. Th. & Dynam. Sys., 4 (1984), 67-80. doi: 10.1017/S0143385700002273.

[17]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved (2)-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4.

[18]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge UK, 1995. doi: 10.1017/CBO9780511809187.

[19]

D. Jane and G. P. Paternain, On the injectivity of the X-ray transform for Anosov thermostats, Discrete Contin. Dyn. Syst., 24 (2009), 471-487. doi: 10.3934/dcds.2009.24.471.

[20]

A. N. Livsic, Certain properties of the homology of Y-systems, Mat. Zametki, 10 (1971), 555-564.

[21]

A. N. Livsic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320.

[22]

R. de la Llave, J. M. Marco and R. Moriyon, Canonical pertubation theory of Anosov systems and regularity results for the Livsic cohomology equation, Ann. of Math., 123 (1986), 537-611. doi: 10.2307/1971334.

[23]

R. Michel, Sur la rigidité imposée par la longeur des géodésiques, (French), Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.

[24]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, (Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.

[25]

G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields, Geom. Funct. Anal., 22 (2012), 1460-1489. doi: 10.1007/s00039-012-0183-6.

[26]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[27]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces, J. Differential Geom., 98 (2014), 147-181.

[28]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[29]

J. Plante and W. Thurston, Anosov flows and the fundamental group, Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.

[30]

M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426. doi: 10.1007/BF01388579.

[31]

M. Pollicott, Derivatives of topological entropy for Anosov and geodesic flows, J. Diff. Geom., 39 (1994), 457-489.

[32]

D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom., 25 (1987), 99-116.

[33]

D. Ruelle, Postivity of entropy production in non-equilibrium statistical mechanics, J. Statist. Phys., 85 (1996), 1-23. doi: 10.1007/BF02175553.

[34]

D. Ruelle, Differentiation of SRB states for hyperbolic flows, Ergodic Theory Dynam. Systems, 28 (2008), 613-631. doi: 10.1017/S0143385707000260.

[35]

V. A. Sharafutdinova and G. Uhlmann, On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points, J. Diff. Geom., 56 (2000), 93-110.

[36]

I. Singer and J. Thorpe, Lecture Notes on Elementary Topology and Geometry, Undergrad. Texts Math. Springer-Verlag, New York, 1967.

[37]

M. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fund. Math., 163 (2000), 177-191.

show all references

References:
[1]

G. Ainsworth, The attenuated magnetic ray transforms on surfaces, Inverse Probl. Imaging., 7 (2013), 27-46. doi: 10.3934/ipi.2013.7.27.

[2]

Yu. Anikonov and V. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl., 5 (1997), 487-490. doi: 10.1515/jiip.1997.5.6.487.

[3]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967), 209pp.

[4]

D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Uspekhi Mat. Nauk, 22 (1967), 107-172.

[5]

V. I. Arnold, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.

[6]

K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity, 15 (2002), 281-314. doi: 10.1088/0951-7715/15/2/305.

[7]

C. Croke and V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifolds, Topology, 37 (1998), 1265-1273. doi: 10.1016/S0040-9383(97)00086-4.

[8]

N. S. Dairbekov and G. P. Paternain, Entropy production in Gaussian thermostats, Comm. Math. Phys., 269 (2007), 533-543. doi: 10.1007/s00220-006-0117-y.

[9]

N. S. Dairbekov and G. P. Paternain, Rigidity properties of Anosov optical hypersurfaces, Ergod. Th. & Dynam. Sys., 28 (2008), 707-737. doi: 10.1017/S0143385707000612.

[10]

N. S. Dairbekov and G. P. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows, Math. Res. Lett., 12 (2005), 719-729. doi: 10.4310/MRL.2005.v12.n5.a9.

[11]

N. S. Dairbekov and G. P. Paternain, On the cohomological equation of magnetic flows, Mat. Contemp., 34 (2008), 155-193.

[12]

N. S. Dairbekov, G. P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.

[13]

N. S. Dairbekov and V. A. Sharafutdinov, Some problems of integral geometry on Anosov manifolds, Ergod. Th. & Dynam. Sys., 23 (2003), 59-74. doi: 10.1017/S0143385702000822.

[14]

H. M. Farkas and I. Kra, Riemann Surfaces, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-2034-3.

[15]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[16]

E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, (French), Ergod. Th. & Dynam. Sys., 4 (1984), 67-80. doi: 10.1017/S0143385700002273.

[17]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved (2)-manifolds, Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4.

[18]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge UK, 1995. doi: 10.1017/CBO9780511809187.

[19]

D. Jane and G. P. Paternain, On the injectivity of the X-ray transform for Anosov thermostats, Discrete Contin. Dyn. Syst., 24 (2009), 471-487. doi: 10.3934/dcds.2009.24.471.

[20]

A. N. Livsic, Certain properties of the homology of Y-systems, Mat. Zametki, 10 (1971), 555-564.

[21]

A. N. Livsic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320.

[22]

R. de la Llave, J. M. Marco and R. Moriyon, Canonical pertubation theory of Anosov systems and regularity results for the Livsic cohomology equation, Ann. of Math., 123 (1986), 537-611. doi: 10.2307/1971334.

[23]

R. Michel, Sur la rigidité imposée par la longeur des géodésiques, (French), Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.

[24]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, (Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.

[25]

G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields, Geom. Funct. Anal., 22 (2012), 1460-1489. doi: 10.1007/s00039-012-0183-6.

[26]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[27]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces, J. Differential Geom., 98 (2014), 147-181.

[28]

L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[29]

J. Plante and W. Thurston, Anosov flows and the fundamental group, Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.

[30]

M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426. doi: 10.1007/BF01388579.

[31]

M. Pollicott, Derivatives of topological entropy for Anosov and geodesic flows, J. Diff. Geom., 39 (1994), 457-489.

[32]

D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom., 25 (1987), 99-116.

[33]

D. Ruelle, Postivity of entropy production in non-equilibrium statistical mechanics, J. Statist. Phys., 85 (1996), 1-23. doi: 10.1007/BF02175553.

[34]

D. Ruelle, Differentiation of SRB states for hyperbolic flows, Ergodic Theory Dynam. Systems, 28 (2008), 613-631. doi: 10.1017/S0143385707000260.

[35]

V. A. Sharafutdinova and G. Uhlmann, On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points, J. Diff. Geom., 56 (2000), 93-110.

[36]

I. Singer and J. Thorpe, Lecture Notes on Elementary Topology and Geometry, Undergrad. Texts Math. Springer-Verlag, New York, 1967.

[37]

M. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fund. Math., 163 (2000), 177-191.

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