# American Institute of Mathematical Sciences

April  2012, 32(4): 1435-1447. doi: 10.3934/dcds.2012.32.1435

## Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  June 2010 Revised  August 2011 Published  October 2011

In [23] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
We give several applications of the dynamical density to partially hyperbolic maps that preserve some $acip$. We show that if $f$ is essentially accessible and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible and center bunched, then either $f$ preserves a smooth measure or there is no $acip$ at all.
Citation: Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
##### References:
 [1] F. Abdenur and M. Viana, Flavors of partial hyperbolicity, preprint, 2008. [2] J. Alves and V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 5551-5569. doi: 10.1090/S0002-9947-08-04484-X. [3] A. Avila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 2717-2725. doi: 10.1088/0951-7715/19/11/011. [4] J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 271-297. [5] R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204. doi: 10.1007/BF01389849. [6] R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. [7] M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Functional Anal. Appl., 9 (1975), 8-16. doi: 10.1007/BF01078168. [8] J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. [9] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316. [10] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. (2), 171 (2010), 451-489. [11] K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351. [12] D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense, in "Geometric Methods in Dynamics (II)", Astérisque 287 (2003), 33-60. [13] T. Fisher, "On the Structure of Hyperbolic Sets," Ph.D thesis, Northwestern University, 2004. [14] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. [15] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. [16] N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynam. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272. [17] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. [18] V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology, 40 (2001), 259-278. doi: 10.1016/S0040-9383(99)00060-9. [19] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013. [20] C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176. doi: 10.1007/BF01390119. [21] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., 2007. [22] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, \arXiv{0809.4862}., (). [23] Z. Xia, Hyperbolic invariant sets with positive measures, Discrete Contin. Dyn. Syst., 15 (2006), 811-818. doi: 10.3934/dcds.2006.15.811.

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##### References:
 [1] F. Abdenur and M. Viana, Flavors of partial hyperbolicity, preprint, 2008. [2] J. Alves and V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 5551-5569. doi: 10.1090/S0002-9947-08-04484-X. [3] A. Avila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 2717-2725. doi: 10.1088/0951-7715/19/11/011. [4] J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 271-297. [5] R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204. doi: 10.1007/BF01389849. [6] R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. [7] M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Functional Anal. Appl., 9 (1975), 8-16. doi: 10.1007/BF01078168. [8] J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. [9] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316. [10] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. (2), 171 (2010), 451-489. [11] K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351. [12] D. Dolgopyat and A Wilkinson, Stable accessibility is $C^1$ dense, in "Geometric Methods in Dynamics (II)", Astérisque 287 (2003), 33-60. [13] T. Fisher, "On the Structure of Hyperbolic Sets," Ph.D thesis, Northwestern University, 2004. [14] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. [15] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. [16] N. Gourmelon, Adapted metrics for dominated splittings, Ergod. Th. Dynam. Sys., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272. [17] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. [18] V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology, 40 (2001), 259-278. doi: 10.1016/S0040-9383(99)00060-9. [19] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013. [20] C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176. doi: 10.1007/BF01390119. [21] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in "Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow," 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., 2007. [22] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, \arXiv{0809.4862}., (). [23] Z. Xia, Hyperbolic invariant sets with positive measures, Discrete Contin. Dyn. Syst., 15 (2006), 811-818. doi: 10.3934/dcds.2006.15.811.
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