-
Previous Article
Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements
- DCDS Home
- This Issue
-
Next Article
On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
The magnetic ray transform on Anosov surfaces
1. | Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, CB3 0WB, United Kingdom |
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. |
[1] |
Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471 |
[2] |
François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems and Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713 |
[3] |
Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems and Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020 |
[4] |
Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089 |
[5] |
Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems and Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 |
[6] |
Nuutti Hyvönen, Martti Kalke, Matti Lassas, Henri Setälä, Samuli Siltanen. Three-dimensional dental X-ray imaging by combination of panoramic and projection data. Inverse Problems and Imaging, 2010, 4 (2) : 257-271. doi: 10.3934/ipi.2010.4.257 |
[7] |
Arun K. Kulshreshth, Andreas Alpers, Gabor T. Herman, Erik Knudsen, Lajos Rodek, Henning F. Poulsen. A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data. Inverse Problems and Imaging, 2009, 3 (1) : 69-85. doi: 10.3934/ipi.2009.3.69 |
[8] |
Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems and Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015 |
[9] |
Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems and Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431 |
[10] |
Weihao Shen, Wenbo Xu, Hongyang Zhang, Zexin Sun, Jianxiong Ma, Xinlong Ma, Shoujun Zhou, Shijie Guo, Yuanquan Wang. Automatic segmentation of the femur and tibia bones from X-ray images based on pure dilated residual U-Net. Inverse Problems and Imaging, 2021, 15 (6) : 1333-1346. doi: 10.3934/ipi.2020057 |
[11] |
Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc. Inverse Problems and Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 |
[12] |
Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27 |
[13] |
Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems and Imaging, 2020, 14 (1) : 1-26. doi: 10.3934/ipi.2019061 |
[14] |
Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems and Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009 |
[15] |
Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems and Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 |
[16] |
Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems and Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013 |
[17] |
Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619 |
[18] |
Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems and Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453 |
[19] |
Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems and Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143 |
[20] |
Wenzhi Luo, Zeév Rudnick, Peter Sarnak. The variance of arithmetic measures associated to closed geodesics on the modular surface. Journal of Modern Dynamics, 2009, 3 (2) : 271-309. doi: 10.3934/jmd.2009.3.271 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]