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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.
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.
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).
|
[4] |
D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems,, Uspekhi Mat. Nauk, 22 (1967), 107.
|
[5] |
V. I. Arnold, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255.
|
[6] |
K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy,, Nonlinearity, 15 (2002), 281.
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.
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.
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.
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.
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.
|
[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.
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.
doi: 10.1017/S0143385702000822. |
[14] |
H. M. Farkas and I. Kra, Riemann Surfaces,, Second Edition, (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.
doi: 10.1215/S0012-7094-03-11932-8. |
[16] |
E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles,, (French), 4 (1984), 67.
doi: 10.1017/S0143385700002273. |
[17] |
V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved (2)-manifolds,, Topology, 19 (1980), 301.
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, (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.
doi: 10.3934/dcds.2009.24.471. |
[20] |
A. N. Livsic, Certain properties of the homology of Y-systems,, Mat. Zametki, 10 (1971), 555.
|
[21] |
A. N. Livsic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296.
|
[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.
doi: 10.2307/1971334. |
[23] |
R. Michel, Sur la rigidité imposée par la longeur des géodésiques,, (French), 65 (1981), 71.
doi: 10.1007/BF01389295. |
[24] |
R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry,, (Russian), 232 (1977), 32.
|
[25] |
G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields,, Geom. Funct. Anal., 22 (2012), 1460.
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.
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.
|
[28] |
L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093.
doi: 10.4007/annals.2005.161.1093. |
[29] |
J. Plante and W. Thurston, Anosov flows and the fundamental group,, Topology, 11 (1972), 147.
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.
doi: 10.1007/BF01388579. |
[31] |
M. Pollicott, Derivatives of topological entropy for Anosov and geodesic flows,, J. Diff. Geom., 39 (1994), 457.
|
[32] |
D. Ruelle, Resonances for Axiom A flows,, J. Diff. Geom., 25 (1987), 99.
|
[33] |
D. Ruelle, Postivity of entropy production in non-equilibrium statistical mechanics,, J. Statist. Phys., 85 (1996), 1.
doi: 10.1007/BF02175553. |
[34] |
D. Ruelle, Differentiation of SRB states for hyperbolic flows,, Ergodic Theory Dynam. Systems, 28 (2008), 613.
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.
|
[36] |
I. Singer and J. Thorpe, Lecture Notes on Elementary Topology and Geometry,, Undergrad. Texts Math. Springer-Verlag, (1967).
|
[37] |
M. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature,, Fund. Math., 163 (2000), 177.
|
show all references
References:
[1] |
G. Ainsworth, The attenuated magnetic ray transforms on surfaces,, Inverse Probl. Imaging., 7 (2013), 27.
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.
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).
|
[4] |
D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems,, Uspekhi Mat. Nauk, 22 (1967), 107.
|
[5] |
V. I. Arnold, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255.
|
[6] |
K. Burns and G. P. Paternain, Anosov magnetic flows, critical values and topological entropy,, Nonlinearity, 15 (2002), 281.
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.
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.
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.
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.
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.
|
[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.
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.
doi: 10.1017/S0143385702000822. |
[14] |
H. M. Farkas and I. Kra, Riemann Surfaces,, Second Edition, (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.
doi: 10.1215/S0012-7094-03-11932-8. |
[16] |
E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles,, (French), 4 (1984), 67.
doi: 10.1017/S0143385700002273. |
[17] |
V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved (2)-manifolds,, Topology, 19 (1980), 301.
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, (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.
doi: 10.3934/dcds.2009.24.471. |
[20] |
A. N. Livsic, Certain properties of the homology of Y-systems,, Mat. Zametki, 10 (1971), 555.
|
[21] |
A. N. Livsic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296.
|
[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.
doi: 10.2307/1971334. |
[23] |
R. Michel, Sur la rigidité imposée par la longeur des géodésiques,, (French), 65 (1981), 71.
doi: 10.1007/BF01389295. |
[24] |
R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry,, (Russian), 232 (1977), 32.
|
[25] |
G. P. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and Higgs fields,, Geom. Funct. Anal., 22 (2012), 1460.
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.
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.
|
[28] |
L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid,, Ann. of Math., 161 (2005), 1093.
doi: 10.4007/annals.2005.161.1093. |
[29] |
J. Plante and W. Thurston, Anosov flows and the fundamental group,, Topology, 11 (1972), 147.
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.
doi: 10.1007/BF01388579. |
[31] |
M. Pollicott, Derivatives of topological entropy for Anosov and geodesic flows,, J. Diff. Geom., 39 (1994), 457.
|
[32] |
D. Ruelle, Resonances for Axiom A flows,, J. Diff. Geom., 25 (1987), 99.
|
[33] |
D. Ruelle, Postivity of entropy production in non-equilibrium statistical mechanics,, J. Statist. Phys., 85 (1996), 1.
doi: 10.1007/BF02175553. |
[34] |
D. Ruelle, Differentiation of SRB states for hyperbolic flows,, Ergodic Theory Dynam. Systems, 28 (2008), 613.
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.
|
[36] |
I. Singer and J. Thorpe, Lecture Notes on Elementary Topology and Geometry,, Undergrad. Texts Math. Springer-Verlag, (1967).
|
[37] |
M. Wojtkowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature,, Fund. Math., 163 (2000), 177.
|
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