# 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 & Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
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##### References:
 [1] Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233 [2] Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549 [3] Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63 [4] Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224 [5] Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91 [6] Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957 [7] Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102 [8] Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107 [9] Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433 [10] Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228 [11] Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, Raúl Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75 [12] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [13] Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004 [14] Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287 [15] Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61 [16] F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74 [17] Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861 [18] Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009 [19] Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619 [20] Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

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