December  2012, 32(12): 4149-4170. doi: 10.3934/dcds.2012.32.4149

On essential coexistence of zero and nonzero Lyapunov exponents

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States

Received  March 2011 Revised  May 2012 Published  August 2012

We show that there exists a $C^\infty$ volume preserving diffeomorphism $P$ of a compact smooth Riemannian manifold $\mathcal{M}$ of dimension 4, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense subset $\mathcal{G}$ of not full measure and has zero Lyapunov exponent on the complement of $\mathcal{G}$. Moreover, $P|\mathcal{G}$ has countably many disjoint open ergodic components.
Citation: Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", Univ. Lect. Series, 23 (2002).   Google Scholar

[2]

L. A. Bunimovich, Mushrooms and other billiards with divided phase space,, Chaos, 11 (2001), 802.  doi: 10.1063/1.1418763.  Google Scholar

[3]

C.-Q. Cheng and Y.-S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings,, Celestial Mech. Dynam. Astronom., 47 (): 275.   Google Scholar

[4]

D. Dolgopyat, H. Hu and Ya. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components,, Appendix to, (2001), 95.   Google Scholar

[5]

V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergod. Th. Dynam. Syst., 8 (1988), 531.   Google Scholar

[6]

V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic,, Comm. Math. Phys. 135 (1991), 135 (1991), 267.   Google Scholar

[7]

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 11 (1994), 359.   Google Scholar

[8]

P. Duarte, Elliptic Isles in families of area preserving maps,, Ergod. Th. Dynam. Syst., 28 (2008), 1781.  doi: 10.1017/S0143385707000983.  Google Scholar

[9]

A. Gorodetski, On stochastic sea of the standard map,, Comm. Math. Phys. 309 (2012), 309 (2012), 155.   Google Scholar

[10]

M. Herman, Stabilité Topologique des systémes dynamiques conservatifs,, [Topological stability of conservative dynamical systems], (1990).   Google Scholar

[11]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Springer-Verlag, (1977).   Google Scholar

[12]

H. Hu, Ya. Pesin and A. Talitskaya, A volume preserving diffeomorphism with essential coexistence of zero and nonzero Lyapunov exponents,, to appear in Comm. Math. Phys. Available from: , ().   Google Scholar

[13]

H. Hu and A. Talitskaya, A hyperbolic diffeomorphism with countably many ergodic components near identity,, preprint, (2002).   Google Scholar

[14]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, "Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

[15]

C. Liverani, Birth of an elliptic island in a chaotic sea,, Math. Phys. Electron. J., 10 (2004).   Google Scholar

[16]

Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[17]

Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents,, Regul. Chaotic Dyn., 12 (2007), 476.  doi: 10.1134/S1560354707050024.  Google Scholar

[18]

F. Przytycki, Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour,, Ergod. Th. Dynam. Syst. 2 (1982), 2 (1982), 439.   Google Scholar

[19]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity 13 (1997), 13 (1997), 125.   Google Scholar

[20]

Ya. Sinai, "Topics in Ergodic Theory,", Princeton University Press, (1994).   Google Scholar

[21]

J.-M. Strelcyn, The "coexistence problem" for conservative dynamical systems: a review,, Colloq. Math. 62 (1991), 62 (1991), 331.   Google Scholar

[22]

M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behavior,, Comm. Math. Phys., 80 (1981), 453.  doi: 10.1007/BF01941656.  Google Scholar

[23]

M. Wojtkowski, On the ergodic properties of piecewise linear perturbations of the twist map,, Ergod. Th. Dynam. Syst., 2 (1982), 525.   Google Scholar

[24]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms,, Ergod. Th. Dynam. Syst., 12 (1992), 621.  doi: 10.1017/S0143385700006969.  Google Scholar

[25]

J.-C. Yoccoz, Travaux de Herman sur les tores invariants,, (French) [Works of Herman on invariant tori], 206 (1992), 311.   Google Scholar

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", Univ. Lect. Series, 23 (2002).   Google Scholar

[2]

L. A. Bunimovich, Mushrooms and other billiards with divided phase space,, Chaos, 11 (2001), 802.  doi: 10.1063/1.1418763.  Google Scholar

[3]

C.-Q. Cheng and Y.-S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings,, Celestial Mech. Dynam. Astronom., 47 (): 275.   Google Scholar

[4]

D. Dolgopyat, H. Hu and Ya. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components,, Appendix to, (2001), 95.   Google Scholar

[5]

V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergod. Th. Dynam. Syst., 8 (1988), 531.   Google Scholar

[6]

V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic,, Comm. Math. Phys. 135 (1991), 135 (1991), 267.   Google Scholar

[7]

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 11 (1994), 359.   Google Scholar

[8]

P. Duarte, Elliptic Isles in families of area preserving maps,, Ergod. Th. Dynam. Syst., 28 (2008), 1781.  doi: 10.1017/S0143385707000983.  Google Scholar

[9]

A. Gorodetski, On stochastic sea of the standard map,, Comm. Math. Phys. 309 (2012), 309 (2012), 155.   Google Scholar

[10]

M. Herman, Stabilité Topologique des systémes dynamiques conservatifs,, [Topological stability of conservative dynamical systems], (1990).   Google Scholar

[11]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Springer-Verlag, (1977).   Google Scholar

[12]

H. Hu, Ya. Pesin and A. Talitskaya, A volume preserving diffeomorphism with essential coexistence of zero and nonzero Lyapunov exponents,, to appear in Comm. Math. Phys. Available from: , ().   Google Scholar

[13]

H. Hu and A. Talitskaya, A hyperbolic diffeomorphism with countably many ergodic components near identity,, preprint, (2002).   Google Scholar

[14]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, "Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

[15]

C. Liverani, Birth of an elliptic island in a chaotic sea,, Math. Phys. Electron. J., 10 (2004).   Google Scholar

[16]

Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[17]

Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents,, Regul. Chaotic Dyn., 12 (2007), 476.  doi: 10.1134/S1560354707050024.  Google Scholar

[18]

F. Przytycki, Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour,, Ergod. Th. Dynam. Syst. 2 (1982), 2 (1982), 439.   Google Scholar

[19]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity 13 (1997), 13 (1997), 125.   Google Scholar

[20]

Ya. Sinai, "Topics in Ergodic Theory,", Princeton University Press, (1994).   Google Scholar

[21]

J.-M. Strelcyn, The "coexistence problem" for conservative dynamical systems: a review,, Colloq. Math. 62 (1991), 62 (1991), 331.   Google Scholar

[22]

M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behavior,, Comm. Math. Phys., 80 (1981), 453.  doi: 10.1007/BF01941656.  Google Scholar

[23]

M. Wojtkowski, On the ergodic properties of piecewise linear perturbations of the twist map,, Ergod. Th. Dynam. Syst., 2 (1982), 525.   Google Scholar

[24]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms,, Ergod. Th. Dynam. Syst., 12 (1992), 621.  doi: 10.1017/S0143385700006969.  Google Scholar

[25]

J.-C. Yoccoz, Travaux de Herman sur les tores invariants,, (French) [Works of Herman on invariant tori], 206 (1992), 311.   Google Scholar

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