# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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, Amer. Math. Soc., Providence, RI, 2002. [2] L. A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos, 11 (2001), 802-808. doi: 10.1063/1.1418763. [3] C.-Q. Cheng and Y.-S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/1990), 275-292. [4] D. Dolgopyat, H. Hu and Ya. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components, Appendix to "Lectures on Lyapunov Exponents and Smooth Ergodic Theory'' in the book "Smooth Ergodic Theory and Its Applications'', Proc. Sympos. Pure Math., (2001), 95-106. Available from: http://www.math.psu.edu/pesin/papers_www/dhp.pdf. [5] V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergod. Th. Dynam. Syst., 8 (1988), 531-553. [6] V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic, Comm. Math. Phys. 135 (1991), 267-302. [7] P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Lineaire, 11 (1994), 359-409. [8] P. Duarte, Elliptic Isles in families of area preserving maps, Ergod. Th. Dynam. Syst., 28 (2008), 1781-1813. doi: 10.1017/S0143385707000983. [9] A. Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys. 309 (2012), no. 1, 155-192. [10] M. Herman, Stabilité Topologique des systémes dynamiques conservatifs, [Topological stability of conservative dynamical systems], preprint, (1990). [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Springer-Verlag, Berlin-New York, 1977. [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: http://www.math.psu.edu/pesin/papers_www/hpt1.pdf. [13] H. Hu and A. Talitskaya, A hyperbolic diffeomorphism with countably many ergodic components near identity, preprint, (2002). Available from: http://www.math.msu.edu/~hhu/paper/iiif.pdf. [14] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, "Ergodic Theory," Springer-Verlag, New York, 1982. [15] C. Liverani, Birth of an elliptic island in a chaotic sea, Math. Phys. Electron. J., 10 (2004), Paper 1, 13 pp. (electronic). [16] Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112. doi: 10.1070/RM1977v032n04ABEH001639. [17] Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regul. Chaotic Dyn., 12 (2007), 476-489. doi: 10.1134/S1560354707050024. [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), 439-463. [19] C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13 (1997), 125-179. [20] Ya. Sinai, "Topics in Ergodic Theory," Princeton University Press, Princeton, NJ., 1994. [21] J.-M. Strelcyn, The "coexistence problem" for conservative dynamical systems: a review, Colloq. Math. 62 (1991), 331-345. [22] M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behavior, Comm. Math. Phys., 80 (1981), 453-464. doi: 10.1007/BF01941656. [23] M. Wojtkowski, On the ergodic properties of piecewise linear perturbations of the twist map, Ergod. Th. Dynam. Syst., 2 (1982), 525-542. [24] Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergod. Th. Dynam. Syst., 12 (1992), 621-631. doi: 10.1017/S0143385700006969. [25] J.-C. Yoccoz, Travaux de Herman sur les tores invariants, (French) [Works of Herman on invariant tori], Astérisque, 206 (1992), 311-344. Séminaire Bourbaki, Vol. 1991/92.

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##### References:
 [1] L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," Univ. Lect. Series, 23, Amer. Math. Soc., Providence, RI, 2002. [2] L. A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos, 11 (2001), 802-808. doi: 10.1063/1.1418763. [3] C.-Q. Cheng and Y.-S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/1990), 275-292. [4] D. Dolgopyat, H. Hu and Ya. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components, Appendix to "Lectures on Lyapunov Exponents and Smooth Ergodic Theory'' in the book "Smooth Ergodic Theory and Its Applications'', Proc. Sympos. Pure Math., (2001), 95-106. Available from: http://www.math.psu.edu/pesin/papers_www/dhp.pdf. [5] V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergod. Th. Dynam. Syst., 8 (1988), 531-553. [6] V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic, Comm. Math. Phys. 135 (1991), 267-302. [7] P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Lineaire, 11 (1994), 359-409. [8] P. Duarte, Elliptic Isles in families of area preserving maps, Ergod. Th. Dynam. Syst., 28 (2008), 1781-1813. doi: 10.1017/S0143385707000983. [9] A. Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys. 309 (2012), no. 1, 155-192. [10] M. Herman, Stabilité Topologique des systémes dynamiques conservatifs, [Topological stability of conservative dynamical systems], preprint, (1990). [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Springer-Verlag, Berlin-New York, 1977. [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: http://www.math.psu.edu/pesin/papers_www/hpt1.pdf. [13] H. Hu and A. Talitskaya, A hyperbolic diffeomorphism with countably many ergodic components near identity, preprint, (2002). Available from: http://www.math.msu.edu/~hhu/paper/iiif.pdf. [14] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, "Ergodic Theory," Springer-Verlag, New York, 1982. [15] C. Liverani, Birth of an elliptic island in a chaotic sea, Math. Phys. Electron. J., 10 (2004), Paper 1, 13 pp. (electronic). [16] Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112. doi: 10.1070/RM1977v032n04ABEH001639. [17] Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regul. Chaotic Dyn., 12 (2007), 476-489. doi: 10.1134/S1560354707050024. [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), 439-463. [19] C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13 (1997), 125-179. [20] Ya. Sinai, "Topics in Ergodic Theory," Princeton University Press, Princeton, NJ., 1994. [21] J.-M. Strelcyn, The "coexistence problem" for conservative dynamical systems: a review, Colloq. Math. 62 (1991), 331-345. [22] M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behavior, Comm. Math. Phys., 80 (1981), 453-464. doi: 10.1007/BF01941656. [23] M. Wojtkowski, On the ergodic properties of piecewise linear perturbations of the twist map, Ergod. Th. Dynam. Syst., 2 (1982), 525-542. [24] Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergod. Th. Dynam. Syst., 12 (1992), 621-631. doi: 10.1017/S0143385700006969. [25] J.-C. Yoccoz, Travaux de Herman sur les tores invariants, (French) [Works of Herman on invariant tori], Astérisque, 206 (1992), 311-344. Séminaire Bourbaki, Vol. 1991/92.
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