July  2013, 33(7): 2621-2629. doi: 10.3934/dcds.2013.33.2621

The local $C^1$-density of stable ergodicity

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received  June 2012 Revised  November 2012 Published  January 2013

In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems which, in a stable way, have two ergodic measures such that one has all center Lyapunov exponents non-negative and the other one has all center Lyapunov exponents non-positive.
Citation: Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621
References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

D. V. Anosov and Ya. Sinai, Certain smooth ergodic systems,, UspehiMat. Nauk, 22 (1967), 107.   Google Scholar

[3]

A. Ávila, On the regularization of conservative maps,, Acta Math., 205 (2010), 5.  doi: 10.1007/s11511-010-0050-y.  Google Scholar

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A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms,, Trans. Amer. Math. Soc., 364 (2012), 2883.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems,, Ann. of Math. (2), 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[6]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33.  doi: 10.1007/s00222-004-0368-1.  Google Scholar

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.  doi: 10.2307/2118647.  Google Scholar

[8]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.  doi: 10.1017/S1474748008000030.  Google Scholar

[9]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity,, Comment. Math. Helv., 79 (2004), 753.  doi: 10.1007/s00014-004-0819-8.  Google Scholar

[10]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity,, J. Stat. Phys., 108 (2002), 927.  doi: 10.1023/A:1019779128351.  Google Scholar

[11]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity,, in, 69 (2001), 327.   Google Scholar

[12]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[13]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Astérisque, 287 (2003), 33.   Google Scholar

[14]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

[15]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents,, preprint, ().   Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity,, in, 362 (1995), 182.   Google Scholar

[17]

F. Rodríguez Hertz, M. A. Rodríguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting,, Nonlinearity, 23 (2010), 211.  doi: 10.1088/0951-7715/23/2/001.  Google Scholar

[19]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, Duke Math. J., 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar

show all references

References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

D. V. Anosov and Ya. Sinai, Certain smooth ergodic systems,, UspehiMat. Nauk, 22 (1967), 107.   Google Scholar

[3]

A. Ávila, On the regularization of conservative maps,, Acta Math., 205 (2010), 5.  doi: 10.1007/s11511-010-0050-y.  Google Scholar

[4]

A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms,, Trans. Amer. Math. Soc., 364 (2012), 2883.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems,, Ann. of Math. (2), 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

[6]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33.  doi: 10.1007/s00222-004-0368-1.  Google Scholar

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,, Ann. of Math., 143 (1996), 357.  doi: 10.2307/2118647.  Google Scholar

[8]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.  doi: 10.1017/S1474748008000030.  Google Scholar

[9]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity,, Comment. Math. Helv., 79 (2004), 753.  doi: 10.1007/s00014-004-0819-8.  Google Scholar

[10]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity,, J. Stat. Phys., 108 (2002), 927.  doi: 10.1023/A:1019779128351.  Google Scholar

[11]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity,, in, 69 (2001), 327.   Google Scholar

[12]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems,, Ann. of Math., 171 (2010), 451.  doi: 10.4007/annals.2010.171.451.  Google Scholar

[13]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Astérisque, 287 (2003), 33.   Google Scholar

[14]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

[15]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents,, preprint, ().   Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity,, in, 362 (1995), 182.   Google Scholar

[17]

F. Rodríguez Hertz, M. A. Rodríguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting,, Nonlinearity, 23 (2010), 211.  doi: 10.1088/0951-7715/23/2/001.  Google Scholar

[19]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, Duke Math. J., 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar

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