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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

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