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The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows
1. | Department of Mathematics, Brigham Young University, Provo, UT 84602 |
References:
[1] |
R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225.
doi: 10.1090/S0002-9939-1965-0193181-8. |
[2] |
L. F. Bakker, A reducible representation of the generalized symmetry group of a quasiperiodic flow, in "Dynamical Systems and Differential Equations'' (W. Feng, S. Hu and X. Lu eds.), Discrete Contin. Dyn. Syst., suppl. (2003), 68-77. |
[3] |
L. F. Bakker, Structure of group invariants of a quasiperiodic flow, Electron. J. Differential Equations, 39 (2004), 1-14. |
[4] |
L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type, Colloq. Math., 112 (2008), 291-312.
doi: 10.4064/cm112-2-6. |
[5] |
L. F. Bakker and G. Conner, A class of generalized symmetries of smooth flows, Commun. Pure Appl. Anal., 3 (2004), 183-195.
doi: 10.3934/cpaa.2004.3.183. |
[6] |
L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,'' University Lecture Series, 23, American Mathematical Society, 2002. |
[7] |
D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc., 280 (1983), 509-532.
doi: 10.1090/S0002-9947-1983-0716835-6. |
[8] |
D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. Kam method and $Z^k$ actions on the torus, Ann. of Math., 172 (2010), 1805-1858.
doi: 10.4007/annals.2010.172.1805. |
[9] |
K. Dekimpe, What is an infra-nilmanifold endomorphism?, Notices Amer. Math. Soc., 58 (2011), 688-689. |
[10] |
B. R. Fayad, Weak mixing for reparameterized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201.
doi: 10.1017/S0143385702000081. |
[11] |
B. R. Fayad, Analytic mixing reparameterizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468.
doi: 10.1017/S0143385702000214. |
[12] |
A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori, J. Mod. Dyn., 2 (2008), 645-700.
doi: 10.3934/jmd.2008.2.645. |
[13] |
A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35.
doi: 10.3934/jmd.2007.1.1. |
[14] |
M. R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites périodiques sur un ouvert de surfaces d'énergies, (French) [Examples of Hamiltonian flows such that no $C^\infty$ perturbation has a periodic orbit on an open set of energy surfaces], C.R. Acad. Sci. Paris Sér. I Math., 312 (1991), 989-994. |
[15] |
S. Hurder, Rigidity of Anosov actions of higher rank lattices, Ann. of Math., 135 (1992), 361-410.
doi: 10.2307/2946593. |
[16] |
B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, J. Mod. Dyn., 1 (2007), 123-146.
doi: 10.3934/jmd.2007.1.123. |
[17] |
B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $ Z^k$ actions, Geom. Topol., 10 (2006), 929-954.
doi: 10.2140/gt.2006.10.929. |
[18] |
B. Kalinin and R. Spatzier, On the classification of Cartan Actions, Geom. Funct. Anal., 17 (2007), 468-490.
doi: 10.1007/s00039-007-0602-2. |
[19] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[20] |
A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structures for $Z^d$-actions by automorphims of a torus, Comment. Math. Helv., 77 (2002), 718-745.
doi: 10.1007/PL00012439. |
[21] |
A. Katok and J. W. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math., 75 (1991), 203-241.
doi: 10.1007/BF02776025. |
[22] |
A. Katok and J. W. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions, Israel J. Math., 93 (1996), 253-280.
doi: 10.1007/BF02776025. |
[23] |
H. Koch, A renormalization group for Hamiltonians, with applications to KAM tori, Ergodic Theory Dynam. Systems, 19 (1999), 475-521.
doi: 10.1017/S0143385799130128. |
[24] |
A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766. |
[25] |
R. de la LLave, Invariants of smooth conjugacy of hyperbolic dynamical systems II, Comm. Math. Phys., 109 (1987), 369-378.
doi: 10.1007/BF01206141. |
[26] |
J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector, Nonlinearity, 15 (2002), 647-664.
doi: 10.1088/0951-7715/15/3/307. |
[27] |
A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.
doi: 10.2307/2373551. |
[28] |
K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem,'' Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. |
[29] |
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
[30] |
J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup., 22 (1989), 99-108. |
[31] |
L. Perko, "Differential Equations and Dynamical Systems,'' Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1991. |
[32] |
C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' 2nd edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, 1999. |
[33] |
F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442.
doi: 10.3934/jmd.2007.1.425. |
[34] |
P. R. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104.
doi: 10.1016/0040-9383(79)90027-2. |
[35] |
H. P. F. Swinnerton-Dyer, "A Brief Guide to Algebraic Number Theory,'' London Mathematical Society, 50, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9781139173360. |
[36] |
D. I. Wallace, Conjugacy classes of hyperbolic matrices in $SL(n, Z)$ and ideal classes in an order, Trans. Amer. Math. Soc., 283 (1984), 177-184.
doi: 10.1090/S0002-9947-1984-0735415-0. |
[37] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,'' Texts in Applied Mathematics, 2, Springer-Verlag, New York, 1990. |
[38] |
F. W. Wilson, Jr., On the minimal sets of non-singular vector fields, Ann. of Math., 84 (1966), 529-536.
doi: 10.2307/1970458. |
show all references
References:
[1] |
R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225.
doi: 10.1090/S0002-9939-1965-0193181-8. |
[2] |
L. F. Bakker, A reducible representation of the generalized symmetry group of a quasiperiodic flow, in "Dynamical Systems and Differential Equations'' (W. Feng, S. Hu and X. Lu eds.), Discrete Contin. Dyn. Syst., suppl. (2003), 68-77. |
[3] |
L. F. Bakker, Structure of group invariants of a quasiperiodic flow, Electron. J. Differential Equations, 39 (2004), 1-14. |
[4] |
L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type, Colloq. Math., 112 (2008), 291-312.
doi: 10.4064/cm112-2-6. |
[5] |
L. F. Bakker and G. Conner, A class of generalized symmetries of smooth flows, Commun. Pure Appl. Anal., 3 (2004), 183-195.
doi: 10.3934/cpaa.2004.3.183. |
[6] |
L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,'' University Lecture Series, 23, American Mathematical Society, 2002. |
[7] |
D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc., 280 (1983), 509-532.
doi: 10.1090/S0002-9947-1983-0716835-6. |
[8] |
D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. Kam method and $Z^k$ actions on the torus, Ann. of Math., 172 (2010), 1805-1858.
doi: 10.4007/annals.2010.172.1805. |
[9] |
K. Dekimpe, What is an infra-nilmanifold endomorphism?, Notices Amer. Math. Soc., 58 (2011), 688-689. |
[10] |
B. R. Fayad, Weak mixing for reparameterized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201.
doi: 10.1017/S0143385702000081. |
[11] |
B. R. Fayad, Analytic mixing reparameterizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468.
doi: 10.1017/S0143385702000214. |
[12] |
A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori, J. Mod. Dyn., 2 (2008), 645-700.
doi: 10.3934/jmd.2008.2.645. |
[13] |
A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35.
doi: 10.3934/jmd.2007.1.1. |
[14] |
M. R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites périodiques sur un ouvert de surfaces d'énergies, (French) [Examples of Hamiltonian flows such that no $C^\infty$ perturbation has a periodic orbit on an open set of energy surfaces], C.R. Acad. Sci. Paris Sér. I Math., 312 (1991), 989-994. |
[15] |
S. Hurder, Rigidity of Anosov actions of higher rank lattices, Ann. of Math., 135 (1992), 361-410.
doi: 10.2307/2946593. |
[16] |
B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, J. Mod. Dyn., 1 (2007), 123-146.
doi: 10.3934/jmd.2007.1.123. |
[17] |
B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $ Z^k$ actions, Geom. Topol., 10 (2006), 929-954.
doi: 10.2140/gt.2006.10.929. |
[18] |
B. Kalinin and R. Spatzier, On the classification of Cartan Actions, Geom. Funct. Anal., 17 (2007), 468-490.
doi: 10.1007/s00039-007-0602-2. |
[19] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[20] |
A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structures for $Z^d$-actions by automorphims of a torus, Comment. Math. Helv., 77 (2002), 718-745.
doi: 10.1007/PL00012439. |
[21] |
A. Katok and J. W. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math., 75 (1991), 203-241.
doi: 10.1007/BF02776025. |
[22] |
A. Katok and J. W. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions, Israel J. Math., 93 (1996), 253-280.
doi: 10.1007/BF02776025. |
[23] |
H. Koch, A renormalization group for Hamiltonians, with applications to KAM tori, Ergodic Theory Dynam. Systems, 19 (1999), 475-521.
doi: 10.1017/S0143385799130128. |
[24] |
A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766. |
[25] |
R. de la LLave, Invariants of smooth conjugacy of hyperbolic dynamical systems II, Comm. Math. Phys., 109 (1987), 369-378.
doi: 10.1007/BF01206141. |
[26] |
J. Lopes Dias, Renormalization of flows on the multidimensional torus close to a KT frequency vector, Nonlinearity, 15 (2002), 647-664.
doi: 10.1088/0951-7715/15/3/307. |
[27] |
A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.
doi: 10.2307/2373551. |
[28] |
K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem,'' Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. |
[29] |
S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
[30] |
J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup., 22 (1989), 99-108. |
[31] |
L. Perko, "Differential Equations and Dynamical Systems,'' Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1991. |
[32] |
C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' 2nd edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, 1999. |
[33] |
F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442.
doi: 10.3934/jmd.2007.1.425. |
[34] |
P. R. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104.
doi: 10.1016/0040-9383(79)90027-2. |
[35] |
H. P. F. Swinnerton-Dyer, "A Brief Guide to Algebraic Number Theory,'' London Mathematical Society, 50, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9781139173360. |
[36] |
D. I. Wallace, Conjugacy classes of hyperbolic matrices in $SL(n, Z)$ and ideal classes in an order, Trans. Amer. Math. Soc., 283 (1984), 177-184.
doi: 10.1090/S0002-9947-1984-0735415-0. |
[37] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,'' Texts in Applied Mathematics, 2, Springer-Verlag, New York, 1990. |
[38] |
F. W. Wilson, Jr., On the minimal sets of non-singular vector fields, Ann. of Math., 84 (1966), 529-536.
doi: 10.2307/1970458. |
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