# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2019, 39 (11) : 6441-6465. doi: 10.3934/dcds.2019279
##### References:
 [1] J. F. Alves, V. Araújo and C. H. Vásquez, Stochastic stability of non-uniformly hyperbolic diffeomorphisms, Stochastics and Dynamics, 7 (2007), 299-333.  doi: 10.1142/S0219493707002049. [2] J. F. Alves, C. 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. [3] J. F. Alves, C. L. Dias, S. 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. [4] J. F. Alves, Statistical Analysis of Non-Uniformly Expanding Dynamical Systems, IMPA, Rio De Janeiro, 2003. [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. [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. [7] V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308. [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. [9] L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Ser., 23. American Mathematical Society, Providence RI, 2002. [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. [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. [12] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975. [13] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. [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. [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. [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. [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. [18] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. [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. [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. [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. [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. [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. [24] Z. Y. Mi, Y. 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. [25] Z. Y. Mi, Y. 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. [26] J. Palis, A global view of Dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (1999), 339-351. [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. [28] V. A. Rokhlin, On the fundamental ideas of measure theorey, A. M. S. Translations, 1952 (1952), 55 pp. [29] D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810. [30] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. [31] C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergod. Th. & Dynam. Sys., 27 (2007), 253-283.  doi: 10.1017/S0143385706000721. [32] M. Viana, Stochastic Dynamics of Deterministic Systems, Lect. Notes XXI Braz. Math Colloq., IMPA, 1997. [33] M. Viana, Dynamics: A probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians (Berlin, 1998), Doc. Math., 1 (1998), 557–578. [34] M. Viana, Lecture Notes on Attractors and Physical Measures, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. [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. [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. [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. [38] Y. T. Zang, D. 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.

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##### References:
 [1] J. F. Alves, V. Araújo and C. H. Vásquez, Stochastic stability of non-uniformly hyperbolic diffeomorphisms, Stochastics and Dynamics, 7 (2007), 299-333.  doi: 10.1142/S0219493707002049. [2] J. F. Alves, C. 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. [3] J. F. Alves, C. L. Dias, S. 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. [4] J. F. Alves, Statistical Analysis of Non-Uniformly Expanding Dynamical Systems, IMPA, Rio De Janeiro, 2003. [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. [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. [7] V. Araújo, Infinitely many stochastically stable attractors, Nonlinearity, 14 (2001), 583-596.  doi: 10.1088/0951-7715/14/3/308. [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. [9] L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Ser., 23. American Mathematical Society, Providence RI, 2002. [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. [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. [12] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975. [13] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. [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. [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. [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. [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. [18] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. [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. [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. [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. [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. [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. [24] Z. Y. Mi, Y. 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. [25] Z. Y. Mi, Y. 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. [26] J. Palis, A global view of Dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (1999), 339-351. [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. [28] V. A. Rokhlin, On the fundamental ideas of measure theorey, A. M. S. Translations, 1952 (1952), 55 pp. [29] D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810. [30] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. [31] C. H. Vásquez, Statistical stability for diffeomorphisms with dominated splitting, Ergod. Th. & Dynam. Sys., 27 (2007), 253-283.  doi: 10.1017/S0143385706000721. [32] M. Viana, Stochastic Dynamics of Deterministic Systems, Lect. Notes XXI Braz. Math Colloq., IMPA, 1997. [33] M. Viana, Dynamics: A probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians (Berlin, 1998), Doc. Math., 1 (1998), 557–578. [34] M. Viana, Lecture Notes on Attractors and Physical Measures, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. [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. [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. [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. [38] Y. T. Zang, D. 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.
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