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May  2013, 12(3): 1183-1200. doi: 10.3934/cpaa.2013.12.1183

The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows

1. 

Department of Mathematics, Brigham Young University, Provo, UT 84602

Received  July 2011 Revised  January 2012 Published  September 2012

The nature and the classification of equilibrium-free flows on compact manifolds without boundary that possess nontrivial generalized symmetries are investigated. Such flows are shown to be rare in the sense that the set of those flows not possessing a generalized symmetry is residual. An equilibrium-free flow on the $2$-torus that possesses nontrivial generalized symmetries is classified as topologically conjugate to a minimal flow. A generalized symmetry is shown to be nontrivial when its Lyapunov exponent in the direction of the flow is nonzero. Conditions are given by which the multiplier of a nontrivial generalized symmetry is a real algebraic number of norm $\pm 1$. A set of conditions, which includes the Katok-Spatzier conjecture, is given by which an equilibrium-free flow on $n$-torus that possesses nontrivial generalized symmetries is shown to be projectively conjugate to an irrational flow of Koch type.
Citation: L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1183-1200. doi: 10.3934/cpaa.2013.12.1183
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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