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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

L. Bakker 1,

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.
Keywords: Anosov Diffeomorphisms, equilibrium-free flows., generalized symmetries.
Mathematics Subject Classification: Primary: 37C55, 37C85; Secondary: 11R9.
Citation: L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure & 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.  Google Scholar

[2]

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[3]

L. F. Bakker, Structure of group invariants of a quasiperiodic flow, Electron. J. Differential Equations, 39 (2004), 1-14.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,'' University Lecture Series, 23, American Mathematical Society, 2002.  Google Scholar

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D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc., 280 (1983), 509-532. doi: 10.1090/S0002-9947-1983-0716835-6.  Google Scholar

[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.  Google Scholar

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K. Dekimpe, What is an infra-nilmanifold endomorphism?, Notices Amer. Math. Soc., 58 (2011), 688-689.  Google Scholar

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B. R. Fayad, Weak mixing for reparameterized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201. doi: 10.1017/S0143385702000081.  Google Scholar

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B. R. Fayad, Analytic mixing reparameterizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

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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.  Google Scholar

[13]

A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35. doi: 10.3934/jmd.2007.1.1.  Google Scholar

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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.  Google Scholar

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S. Hurder, Rigidity of Anosov actions of higher rank lattices, Ann. of Math., 135 (1992), 361-410. doi: 10.2307/2946593.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[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.  Google Scholar

[29]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[30]

J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup., 22 (1989), 99-108.  Google Scholar

[31]

L. Perko, "Differential Equations and Dynamical Systems,'' Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1991.  Google Scholar

[32]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' 2nd edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[34]

P. R. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104. doi: 10.1016/0040-9383(79)90027-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,'' Texts in Applied Mathematics, 2, Springer-Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. F. Bakker, Structure of group invariants of a quasiperiodic flow, Electron. J. Differential Equations, 39 (2004), 1-14.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,'' University Lecture Series, 23, American Mathematical Society, 2002.  Google Scholar

[7]

D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc., 280 (1983), 509-532. doi: 10.1090/S0002-9947-1983-0716835-6.  Google Scholar

[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.  Google Scholar

[9]

K. Dekimpe, What is an infra-nilmanifold endomorphism?, Notices Amer. Math. Soc., 58 (2011), 688-689.  Google Scholar

[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.  Google Scholar

[11]

B. R. Fayad, Analytic mixing reparameterizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

[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.  Google Scholar

[13]

A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35. doi: 10.3934/jmd.2007.1.1.  Google Scholar

[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.  Google Scholar

[15]

S. Hurder, Rigidity of Anosov actions of higher rank lattices, Ann. of Math., 135 (1992), 361-410. doi: 10.2307/2946593.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[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.  Google Scholar

[29]

S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770. doi: 10.2307/2373372.  Google Scholar

[30]

J. Palis and J. C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup., 22 (1989), 99-108.  Google Scholar

[31]

L. Perko, "Differential Equations and Dynamical Systems,'' Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1991.  Google Scholar

[32]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'' 2nd edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[34]

P. R. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104. doi: 10.1016/0040-9383(79)90027-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,'' Texts in Applied Mathematics, 2, Springer-Verlag, New York, 1990.  Google Scholar

[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.  Google Scholar

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