November  2019, 39(11): 6441-6465. doi: 10.3934/dcds.2019279

SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received  December 2018 Revised  April 2019 Published  August 2019

Fund Project: Zeya Mi was partially supported by NSFC 11801278 and The Startup Foundation for Introducing Talent of NUIST(Grant No. 2017r070).

In this paper, we provide a technical result on the existence of Gibbs $ cu $-states for diffeomorphisms with dominated splittings. More precisely, for given $ C^2 $ diffeomorphim $ f $ with dominated splitting $ T_{\Lambda}M = E\oplus F $ on an attractor $ \Lambda $, by considering some suitable random perturbation of $ f $, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $ E $, then its ergodic components contain Gibbs $ cu $-states associated to $ E $. With this technique, we show the existence of SRB measures and physical measures for some systems exhibiting dominated splittings and weak hyperbolicity.

Citation: Zeya Mi. SRB measures for some diffeomorphisms with dominated splittings as zero noise limits. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6441-6465. doi: 10.3934/dcds.2019279
References:
[1]

J. F. AlvesV. Araújo and C. H. Vásquez, Stochastic stability of non-uniformly hyperbolic diffeomorphisms, Stochastics and Dynamics, 7 (2007), 299-333.  doi: 10.1142/S0219493707002049.  Google Scholar

[2]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[3]

J. F. AlvesC. L. DiasS. Luzzatto and V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding, J. Eur. Math. Soc. (JEMS), 19 (2017), 2911-2946.  doi: 10.4171/JEMS/731.  Google Scholar

[4]

J. F. Alves, Statistical Analysis of Non-Uniformly Expanding Dynamical Systems, IMPA, Rio De Janeiro, 2003.  Google Scholar

[5]

M. Andersson and C. H. Vásquez, On mostly expanding diffeomorphisms, Ergodic Theory Dynam. Systems, 38 (2018), 2838-2859.  doi: 10.1017/etds.2017.17.  Google Scholar

[6]

V. Araújo, Attractors and time averages for random maps, Ann. Inst. H. Poincar Anal. Non Linaire., 17 (2000), 307-369.  doi: 10.1016/S0294-1449(00)00112-8.  Google Scholar

[7]

V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308.  Google Scholar

[8]

A. Avila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Amer. Math. Soc., 364 (2012), 2883-2907.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar

[9]

L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Ser., 23. American Mathematical Society, Providence RI, 2002.  Google Scholar

[10]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, A Global Geometric and Probabilistic Perspective, Encyclopaedia of Math Sci., 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005.  Google Scholar

[11]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.  Google Scholar

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[13]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.   Google Scholar

[14]

Y. L. Cao and D. W. Yang, On pesin's entropy formula for dominated splittings without mixed behavior, J. Differ. Equ., 261 (2016), 3964-3986.  doi: 10.1016/j.jde.2016.06.012.  Google Scholar

[15]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergod. Th. & Dynam. Sys., 25 (2005), 1115-1138.  doi: 10.1017/S0143385704000604.  Google Scholar

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J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.  Google Scholar

[17]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), (1970), 133–163.  Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. Math., 122 (1985), 509-539.  doi: 10.2307/1971328.  Google Scholar

[20]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Characterization of measures satisfying Pesin's entropy formula, Ann. Math., 122 (1985), 540-574.  doi: 10.2307/1971329.  Google Scholar

[21]

P.-D Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094308.  Google Scholar

[22]

P.-D. Liu and K. N. Lu, A Note on partially hyperbolic attractors: Entropy conjecture and SRB measures, Discrete Contin. Dyn. Syst., 35 (2015), 341-352.  doi: 10.3934/dcds.2015.35.341.  Google Scholar

[23]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8. Springer-Verlag, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[24]

Z. Y. MiY. L. Cao and D. W. Yang, SRB measures for attractors with continuous invariant splittings, Math. Z., 288 (2018), 135-165.  doi: 10.1007/s00209-017-1883-2.  Google Scholar

[25]

Z. Y. MiY. L. Cao and D. W. Yang, A note on partially hyperbolic systems with mostly expanding centers, Proc. Amer. Math. Soc., 145 (2017), 5299-5313.  doi: 10.1090/proc/13701.  Google Scholar

[26]

J. Palis, A global view of Dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (1999), 339-351.   Google Scholar

[27]

Y. B. Pesin and Y. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[28]

V. A. Rokhlin, On the fundamental ideas of measure theorey, A. M. S. Translations, 1952 (1952), 55 pp.  Google Scholar

[29]

D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[30]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.   Google Scholar

[31]

C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergod. Th. & Dynam. Sys., 27 (2007), 253-283.  doi: 10.1017/S0143385706000721.  Google Scholar

[32]

M. Viana, Stochastic Dynamics of Deterministic Systems, Lect. Notes XXI Braz. Math Colloq., IMPA, 1997. Google Scholar

[33]

M. Viana, Dynamics: A probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians (Berlin, 1998), Doc. Math., 1 (1998), 557–578.  Google Scholar

[34]

M. Viana, Lecture Notes on Attractors and Physical Measures, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999.  Google Scholar

[35]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[36] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, 151. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316422601.  Google Scholar
[37]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

[38]

Y. T. ZangD. W. Yang and Y. L. Cao, The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy, Nonlinearity, 30 (2017), 3076-3087.  doi: 10.1088/1361-6544/aa773c.  Google Scholar

show all references

References:
[1]

J. F. AlvesV. Araújo and C. H. Vásquez, Stochastic stability of non-uniformly hyperbolic diffeomorphisms, Stochastics and Dynamics, 7 (2007), 299-333.  doi: 10.1142/S0219493707002049.  Google Scholar

[2]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[3]

J. F. AlvesC. L. DiasS. Luzzatto and V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding, J. Eur. Math. Soc. (JEMS), 19 (2017), 2911-2946.  doi: 10.4171/JEMS/731.  Google Scholar

[4]

J. F. Alves, Statistical Analysis of Non-Uniformly Expanding Dynamical Systems, IMPA, Rio De Janeiro, 2003.  Google Scholar

[5]

M. Andersson and C. H. Vásquez, On mostly expanding diffeomorphisms, Ergodic Theory Dynam. Systems, 38 (2018), 2838-2859.  doi: 10.1017/etds.2017.17.  Google Scholar

[6]

V. Araújo, Attractors and time averages for random maps, Ann. Inst. H. Poincar Anal. Non Linaire., 17 (2000), 307-369.  doi: 10.1016/S0294-1449(00)00112-8.  Google Scholar

[7]

V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308.  Google Scholar

[8]

A. Avila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Amer. Math. Soc., 364 (2012), 2883-2907.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar

[9]

L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Ser., 23. American Mathematical Society, Providence RI, 2002.  Google Scholar

[10]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, A Global Geometric and Probabilistic Perspective, Encyclopaedia of Math Sci., 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005.  Google Scholar

[11]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.  Google Scholar

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[13]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.   Google Scholar

[14]

Y. L. Cao and D. W. Yang, On pesin's entropy formula for dominated splittings without mixed behavior, J. Differ. Equ., 261 (2016), 3964-3986.  doi: 10.1016/j.jde.2016.06.012.  Google Scholar

[15]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergod. Th. & Dynam. Sys., 25 (2005), 1115-1138.  doi: 10.1017/S0143385704000604.  Google Scholar

[16]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.  doi: 10.1103/RevModPhys.57.617.  Google Scholar

[17]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), (1970), 133–163.  Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. Math., 122 (1985), 509-539.  doi: 10.2307/1971328.  Google Scholar

[20]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Characterization of measures satisfying Pesin's entropy formula, Ann. Math., 122 (1985), 540-574.  doi: 10.2307/1971329.  Google Scholar

[21]

P.-D Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094308.  Google Scholar

[22]

P.-D. Liu and K. N. Lu, A Note on partially hyperbolic attractors: Entropy conjecture and SRB measures, Discrete Contin. Dyn. Syst., 35 (2015), 341-352.  doi: 10.3934/dcds.2015.35.341.  Google Scholar

[23]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8. Springer-Verlag, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[24]

Z. Y. MiY. L. Cao and D. W. Yang, SRB measures for attractors with continuous invariant splittings, Math. Z., 288 (2018), 135-165.  doi: 10.1007/s00209-017-1883-2.  Google Scholar

[25]

Z. Y. MiY. L. Cao and D. W. Yang, A note on partially hyperbolic systems with mostly expanding centers, Proc. Amer. Math. Soc., 145 (2017), 5299-5313.  doi: 10.1090/proc/13701.  Google Scholar

[26]

J. Palis, A global view of Dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (1999), 339-351.   Google Scholar

[27]

Y. B. Pesin and Y. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[28]

V. A. Rokhlin, On the fundamental ideas of measure theorey, A. M. S. Translations, 1952 (1952), 55 pp.  Google Scholar

[29]

D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[30]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.   Google Scholar

[31]

C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergod. Th. & Dynam. Sys., 27 (2007), 253-283.  doi: 10.1017/S0143385706000721.  Google Scholar

[32]

M. Viana, Stochastic Dynamics of Deterministic Systems, Lect. Notes XXI Braz. Math Colloq., IMPA, 1997. Google Scholar

[33]

M. Viana, Dynamics: A probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians (Berlin, 1998), Doc. Math., 1 (1998), 557–578.  Google Scholar

[34]

M. Viana, Lecture Notes on Attractors and Physical Measures, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999.  Google Scholar

[35]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[36] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, 151. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316422601.  Google Scholar
[37]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

[38]

Y. T. ZangD. W. Yang and Y. L. Cao, The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy, Nonlinearity, 30 (2017), 3076-3087.  doi: 10.1088/1361-6544/aa773c.  Google Scholar

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