• Previous Article
    Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients
  • CPAA Home
  • This Issue
  • Next Article
    Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data
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 & 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. 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, (2003), 68. Google Scholar

[3]

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

[4]

L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type,, Colloq. Math., 112 (2008), 291. 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. 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, (2002). Google Scholar

[7]

D. Berend, Multi-invariant sets on tori, , Trans. Amer. Math. Soc., 280 (1983), 509. 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. 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. Google Scholar

[10]

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

[11]

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

[12]

A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori,, J. Mod. Dyn., 2 (2008), 645. 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. 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 $C\sp\infty$ perturbation has a periodic orbit on an open set of energy surfaces], 312 (1991), 989. Google Scholar

[15]

S. Hurder, Rigidity of Anosov actions of higher rank lattices,, Ann. of Math., 135 (1992), 361. 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. 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. 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. 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, (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. 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. 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. 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. 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. Google Scholar

[25]

R. de la LLave, Invariants of smooth conjugacy of hyperbolic dynamical systems II,, Comm. Math. Phys., 109 (1987), 369. 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. 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. 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, (1992). Google Scholar

[29]

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

[30]

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

[31]

L. Perko, "Differential Equations and Dynamical Systems,'', Texts in Applied Mathematics, (1991). Google Scholar

[32]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'', 2$^{nd}$ edition, (1999). Google Scholar

[33]

F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms,, J. Mod. Dyn., 1 (2007), 425. doi: 10.3934/jmd.2007.1.425. Google Scholar

[34]

P. R. Sad, Centralizers of vector fields,, Topology, 18 (1979), 97. 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, (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. 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, (1990). Google Scholar

[38]

F. W. Wilson, Jr., On the minimal sets of non-singular vector fields,, Ann. of Math., 84 (1966), 529. 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. 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, (2003), 68. Google Scholar

[3]

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

[4]

L. F. Bakker, Rigidity of projective conjugacy of quasiperiodic flows of Koch type,, Colloq. Math., 112 (2008), 291. 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. 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, (2002). Google Scholar

[7]

D. Berend, Multi-invariant sets on tori, , Trans. Amer. Math. Soc., 280 (1983), 509. 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. 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. Google Scholar

[10]

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

[11]

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

[12]

A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori,, J. Mod. Dyn., 2 (2008), 645. 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. 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 $C\sp\infty$ perturbation has a periodic orbit on an open set of energy surfaces], 312 (1991), 989. Google Scholar

[15]

S. Hurder, Rigidity of Anosov actions of higher rank lattices,, Ann. of Math., 135 (1992), 361. 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. 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. 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. 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, (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. 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. 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. 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. 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. Google Scholar

[25]

R. de la LLave, Invariants of smooth conjugacy of hyperbolic dynamical systems II,, Comm. Math. Phys., 109 (1987), 369. 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. 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. 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, (1992). Google Scholar

[29]

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

[30]

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

[31]

L. Perko, "Differential Equations and Dynamical Systems,'', Texts in Applied Mathematics, (1991). Google Scholar

[32]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,'', 2$^{nd}$ edition, (1999). Google Scholar

[33]

F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms,, J. Mod. Dyn., 1 (2007), 425. doi: 10.3934/jmd.2007.1.425. Google Scholar

[34]

P. R. Sad, Centralizers of vector fields,, Topology, 18 (1979), 97. 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, (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. 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, (1990). Google Scholar

[38]

F. W. Wilson, Jr., On the minimal sets of non-singular vector fields,, Ann. of Math., 84 (1966), 529. doi: 10.2307/1970458. Google Scholar

[1]

Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1

[2]

L. Bakker, G. Conner. A class of generalized symmetries of smooth flows. Communications on Pure & Applied Analysis, 2004, 3 (2) : 183-195. doi: 10.3934/cpaa.2004.3.183

[3]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837

[4]

Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765

[5]

Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1799-1819. doi: 10.3934/dcds.2019078

[6]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[7]

Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593

[8]

Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271

[9]

Enoch Humberto Apaza Calla, Bulmer Mejia Garcia, Carlos Arnoldo Morales Rojas. Topological properties of sectional-Anosov flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4735-4741. doi: 10.3934/dcds.2015.35.4735

[10]

Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839

[11]

Oliver Butterley, Carlangelo Liverani. Smooth Anosov flows: Correlation spectra and stability. Journal of Modern Dynamics, 2007, 1 (2) : 301-322. doi: 10.3934/jmd.2007.1.301

[12]

Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599

[13]

Andrey Gogolev. Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori. Journal of Modern Dynamics, 2008, 2 (4) : 645-700. doi: 10.3934/jmd.2008.2.645

[14]

Andrey Gogolev, Misha Guysinsky. $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 183-200. doi: 10.3934/dcds.2008.22.183

[15]

Zemer Kosloff. On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms. Journal of Modern Dynamics, 2018, 13: 251-270. doi: 10.3934/jmd.2018020

[16]

Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471

[17]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403

[18]

M. S. Bruzón, M. L. Gandarias, J. C. Camacho. Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation. Conference Publications, 2015, 2015 (special) : 151-158. doi: 10.3934/proc.2015.0151

[19]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035

[20]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]