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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
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show all references

References:
[1]

Inverse Probl. Imaging., 7 (2013), 27-46. doi: 10.3934/ipi.2013.7.27.  Google Scholar

[2]

J. Inverse Ill-Posed Probl., 5 (1997), 487-490. doi: 10.1515/jiip.1997.5.6.487.  Google Scholar

[3]

Proc. Steklov Inst. Math., 90 (1967), 209pp.  Google Scholar

[4]

Uspekhi Mat. Nauk, 22 (1967), 107-172.  Google Scholar

[5]

Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.  Google Scholar

[6]

Nonlinearity, 15 (2002), 281-314. doi: 10.1088/0951-7715/15/2/305.  Google Scholar

[7]

Topology, 37 (1998), 1265-1273. doi: 10.1016/S0040-9383(97)00086-4.  Google Scholar

[8]

Comm. Math. Phys., 269 (2007), 533-543. doi: 10.1007/s00220-006-0117-y.  Google Scholar

[9]

Ergod. Th. & Dynam. Sys., 28 (2008), 707-737. doi: 10.1017/S0143385707000612.  Google Scholar

[10]

Math. Res. Lett., 12 (2005), 719-729. doi: 10.4310/MRL.2005.v12.n5.a9.  Google Scholar

[11]

Mat. Contemp., 34 (2008), 155-193.  Google Scholar

[12]

Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.  Google Scholar

[13]

Ergod. Th. & Dynam. Sys., 23 (2003), 59-74. doi: 10.1017/S0143385702000822.  Google Scholar

[14]

Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-2034-3.  Google Scholar

[15]

Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[16]

(French), Ergod. Th. & Dynam. Sys., 4 (1984), 67-80. doi: 10.1017/S0143385700002273.  Google Scholar

[17]

Topology, 19 (1980), 301-312. doi: 10.1016/0040-9383(80)90015-4.  Google Scholar

[18]

Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge UK, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[19]

Discrete Contin. Dyn. Syst., 24 (2009), 471-487. doi: 10.3934/dcds.2009.24.471.  Google Scholar

[20]

Mat. Zametki, 10 (1971), 555-564.  Google Scholar

[21]

Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320.  Google Scholar

[22]

Ann. of Math., 123 (1986), 537-611. doi: 10.2307/1971334.  Google Scholar

[23]

(French), Invent. Math., 65 (1981), 71-83. doi: 10.1007/BF01389295.  Google Scholar

[24]

(Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.  Google Scholar

[25]

Geom. Funct. Anal., 22 (2012), 1460-1489. doi: 10.1007/s00039-012-0183-6.  Google Scholar

[26]

Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.  Google Scholar

[27]

J. Differential Geom., 98 (2014), 147-181.  Google Scholar

[28]

Ann. of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.  Google Scholar

[29]

Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.  Google Scholar

[30]

Invent. Math., 81 (1985), 413-426. doi: 10.1007/BF01388579.  Google Scholar

[31]

J. Diff. Geom., 39 (1994), 457-489.  Google Scholar

[32]

J. Diff. Geom., 25 (1987), 99-116.  Google Scholar

[33]

J. Statist. Phys., 85 (1996), 1-23. doi: 10.1007/BF02175553.  Google Scholar

[34]

Ergodic Theory Dynam. Systems, 28 (2008), 613-631. doi: 10.1017/S0143385707000260.  Google Scholar

[35]

J. Diff. Geom., 56 (2000), 93-110.  Google Scholar

[36]

Undergrad. Texts Math. Springer-Verlag, New York, 1967.  Google Scholar

[37]

Fund. Math., 163 (2000), 177-191.  Google Scholar

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