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Dominated splitting and Pesin's entropy formula
Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems
1.  School of Mathematical Sciences, Peking University, Beijing 100871, China 
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.
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F. Abdenur and M. Viana, Flavors of partial hyperbolicity, preprint, 2008. 
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J. Alves and V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 55515569. doi: 10.1090/S000299470804484X. 
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A. Avila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 27172725. doi: 10.1088/09517715/19/11/011. 
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J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 271297. 
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R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203204. doi: 10.1007/BF01389849. 
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R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, SpringerVerlag, BerlinNew York, 1975. 
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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), 816. doi: 10.1007/BF01078168. 
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J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170212. 
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M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316. 
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K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math. (2), 171 (2010), 451489. 
[11] 
K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927942. 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), 3360. 
[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), 301308. doi: 10.1090/S00029947197102838123. 
[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), 18391849. doi: 10.1017/S0143385707000272. 
[17] 
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, SpringerVerlag, BerlinNew York, 1977. 
[18] 
V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a nonergodic one, Topology, 40 (2001), 259278. doi: 10.1016/S00409383(99)000609. 
[19] 
C. Pugh and M. Shub, Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc., 2 (2000), 152. doi: 10.1007/s100970050013. 
[20] 
C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159176. 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," 3587, 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), 811818. doi: 10.3934/dcds.2006.15.811. 
show all references
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), 55515569. doi: 10.1090/S000299470804484X. 
[3] 
A. Avila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 27172725. doi: 10.1088/09517715/19/11/011. 
[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), 271297. 
[5] 
R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203204. doi: 10.1007/BF01389849. 
[6] 
R. Bowen, "Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms," Lecture Notes in Mathematics, 470, SpringerVerlag, BerlinNew 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), 816. doi: 10.1007/BF01078168. 
[8] 
J. Pesin and M. Brin, Partially hyperbolic dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170212. 
[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), 451489. 
[11] 
K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys., 108 (2002), 927942. 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), 3360. 
[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), 301308. doi: 10.1090/S00029947197102838123. 
[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), 18391849. doi: 10.1017/S0143385707000272. 
[17] 
M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, SpringerVerlag, BerlinNew York, 1977. 
[18] 
V. Niţică and A. Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a nonergodic one, Topology, 40 (2001), 259278. doi: 10.1016/S00409383(99)000609. 
[19] 
C. Pugh and M. Shub, Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc., 2 (2000), 152. doi: 10.1007/s100970050013. 
[20] 
C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159176. 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," 3587, 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), 811818. doi: 10.3934/dcds.2006.15.811. 
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