# American Institute of Mathematical Sciences

2007, 17(3): 553-560. doi: 10.3934/dcds.2007.17.553

## Strong stable manifolds for sectional-hyperbolic sets

 1 Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received  January 2006 Revised  August 2006 Published  December 2006

The sectional-hyperbolic sets constitute a class of partially hyperbolic sets introduced in [20] to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor [9]). Here we prove that a transitive sectional-hyperbolic set with singularities contains no local strong stable manifold through any of its points. Hence a transitive, isolated, sectional-hyperbolic set containing a local strong stable manifold is a hyperbolic saddle-type repeller. In addition, a proper transitive sectional-hyperbolic set on a compact $n$-manifold has empty interior and topological dimension $\leq n-1$. It follows that a singular-hyperbolic attractor with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is expanding, i.e., its topological dimension coincides with the dimension of its central subbundle. These results apply to the robustly transitive sets considered in [22], [17] and also to the Lorenz attractor in the Lorenz equation [25].
Citation: Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553
 [1] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [2] Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 [3] Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 [4] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 [5] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 [6] R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403 [7] Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195 [8] Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27 [9] 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 [10] Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193 [11] Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245 [12] Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469 [13] Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747 [14] Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114 [15] S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 [16] Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70. [17] Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229 [18] 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 - A, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75 [19] 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 [20] Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

2016 Impact Factor: 1.099