New criteria for ergodicity and nonuniform hyperbolicity.

*(English)*Zbl 1290.37011In the mid-nineties of the past century, Pugh and Shub conjectured that stable ergodicity is dense among partial hyperbolic dynamical systems. This paper is a remarkable contribution to this conjecture for the discrete-time case.

We recall that for \(\alpha>0\), a \(C^{1+\alpha}\) volume-preserving diffeomorphism is said to be stably ergodic if the exists a \(C^1\)-neighborhood on which every \(C^{1+\alpha}\) volume-preserving diffeomorphism is ergodic. Apart of the long-time known Anosov diffeomorphisms, the first examples of stably ergodic diffeomorphisms outside the hyperbolic world were given in [M. Grayson et al., Ann. Math. (2) 140, No. 2, 295–329 (1994; Zbl 0824.58032)] and were defined as the time-one of a geodesic flow on a negative curvature surface. In this example the neutral flow direction generates a non-hyperbolic one-dimensional manifold (central manifold). The contributions towards the previous conjecture always assumed a “one-dimensionality” hypothesis on the central bundle. Indeed, we have the result [F. Rodriguez Hertz et al., Invent. Math. 172, No. 2, 353–381 (2008; Zbl 1136.37020)] and also [K. Burns and A. Wilkinson, Ann. Math. (2) 171, No. 1, 451–489 (2010; Zbl 1196.37057)] where it is considered that the central bundle displays the “bunching condition” (in rough terms, the central dynamics is conformal, in particular, trivially satisfied if the central bundle is one-dimensional).

In the present paper it is proved that (Theorem B), within the volume-preserving discrete-time setting, stable ergodicity is \(C^1\)-dense among partial hyperbolic dynamical systems with a two-dimensional central bundle. In order to obtain this result, the authors establish a new criterion for ergodicity and for the absence of zero Lyapunov exponents (Theorem A). This criterion assures, in brief terms, ergodicity and non-uniform hyperbolicity under the hypothesis of positive measure of stable/unstable-saturated subsets associated to ergodic homoclinic classes. It is interesting to observe that another concept used along the arguments to obtain stable ergodicity is the concept of “blenders” that was originally introduced to obtain robust transitivity.

We recall that for \(\alpha>0\), a \(C^{1+\alpha}\) volume-preserving diffeomorphism is said to be stably ergodic if the exists a \(C^1\)-neighborhood on which every \(C^{1+\alpha}\) volume-preserving diffeomorphism is ergodic. Apart of the long-time known Anosov diffeomorphisms, the first examples of stably ergodic diffeomorphisms outside the hyperbolic world were given in [M. Grayson et al., Ann. Math. (2) 140, No. 2, 295–329 (1994; Zbl 0824.58032)] and were defined as the time-one of a geodesic flow on a negative curvature surface. In this example the neutral flow direction generates a non-hyperbolic one-dimensional manifold (central manifold). The contributions towards the previous conjecture always assumed a “one-dimensionality” hypothesis on the central bundle. Indeed, we have the result [F. Rodriguez Hertz et al., Invent. Math. 172, No. 2, 353–381 (2008; Zbl 1136.37020)] and also [K. Burns and A. Wilkinson, Ann. Math. (2) 171, No. 1, 451–489 (2010; Zbl 1196.37057)] where it is considered that the central bundle displays the “bunching condition” (in rough terms, the central dynamics is conformal, in particular, trivially satisfied if the central bundle is one-dimensional).

In the present paper it is proved that (Theorem B), within the volume-preserving discrete-time setting, stable ergodicity is \(C^1\)-dense among partial hyperbolic dynamical systems with a two-dimensional central bundle. In order to obtain this result, the authors establish a new criterion for ergodicity and for the absence of zero Lyapunov exponents (Theorem A). This criterion assures, in brief terms, ergodicity and non-uniform hyperbolicity under the hypothesis of positive measure of stable/unstable-saturated subsets associated to ergodic homoclinic classes. It is interesting to observe that another concept used along the arguments to obtain stable ergodicity is the concept of “blenders” that was originally introduced to obtain robust transitivity.

Reviewer: Helder Vilarinho (Covilhã)

##### MSC:

37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |

37D30 | Partially hyperbolic systems and dominated splittings |

37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |

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\textit{F. Rodriguez Hertz} et al., Duke Math. J. 160, No. 3, 599--629 (2011; Zbl 1290.37011)

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##### References:

[1] | D. V. Anosov, Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature (in Russian), Tr. Mat. Inst. Steklova 90 , Russ. Acad. Sci., Moscow, 1967; English translation in Proceedings of the Steklov Institute of Mathematics 90 , Amer. Math. Soc., Providence, 1969. · Zbl 0176.19101 |

[2] | D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory: Ergodic diffeomorphisms (in Russian), Tr. Mosk. Mat. Obs. 23 (1970), 3-36; English translation in Trans. Moscow Math. Soc. 23 (1970), 1-35. · Zbl 0255.58007 |

[3] | D. V. Anosov and Ya. G. Sinai, Certain smooth ergodic systems (in Russian), Uspekhi Mat. Nauk 22 (1967), 107-172; English translation in Russian Math. Surveys 22 (1967), 103-167. · Zbl 0177.42002 |

[4] | M.-C. Arnaud, Le “closing lemma” en topologie C 1 , Mém. Soc. Math. Fr. (N.S.) 74 (1998), 1-120. · Zbl 0920.58039 |

[5] | A. Avila, On the regularization of conservative maps , Acta Math. 205 (2010), 5-18. · Zbl 1211.37029 |

[6] | A. Avila, J. Bochi, and A. Wilkinson, Nonuniform center bunching and the genericity of ergodicity among C 1 partially hyperbolic symplectomorphisms , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 931-979. · Zbl 1191.37017 |

[7] | A. T. Baraviera and C. Bonatti, Removing zero Lyapunov exponents , Ergodic Theory Dynam. Systems 23 (2003), 1655-1670. · Zbl 1048.37026 |

[8] | L. Barreira and Ya. Pesin, “Lectures on Lyapunov exponents and smooth ergodic theory,” with appendices by M. Brin and by D. Dolgopyat, H. Hu, and Ya. Pesin, in Smooth Ergodic Theory and its Applications (Seattle, 1999) , Proc. Sympos. Pure Math. 69 , Amer. Math. Soc., Providence, 2001, 3-106. · Zbl 0996.37001 |

[9] | L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory , Univ. Lecture Ser. 23 , Amer. Math. Soc., Providence, 2002. · Zbl 0996.37001 |

[10] | J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps , Ann. of Math. (2) 161 (2005), 1423-1485. · Zbl 1101.37039 |

[11] | C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms , Ann. of Math. (2) 143 (1996), 357-396. · Zbl 0852.58066 |

[12] | C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and C 1 - generic dynamics , J. Inst. Math. Jussieu 7 (2008), 469-525. · Zbl 1156.37004 |

[13] | C. Bonatti, L. J. Díaz, and E. R. Pujals, A C 1 - generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources , Ann. of Math. (2) 158 (2003), 355-418. · Zbl 1049.37011 |

[14] | C. Bonatti, L. J. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective , Encyclopaedia Math. Sci. 102 , Springer, Berlin, 2005. · Zbl 1060.37020 |

[15] | M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170-212; English translation in Math. USSR-Isv. 8 (1974), 177-218. · Zbl 0304.58017 |

[16] | K. Burns, D. Dolgopyat, and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity , J. Stat. Phys. 108 (2002), 927-942. · Zbl 1124.37308 |

[17] | K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems , Ann. of Math. (2) 171 (2010), 451-489. · Zbl 1196.37057 |

[18] | D. Dolgopyat and A. Wilkinson, “Stable accessibility is C 1 dense” in Geometric Methods in Dynamics, II , Astérisque 287 , Soc. Math. France, Paris, 2003, 33-60. · Zbl 1213.37053 |

[19] | M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms , Ann. of Math. (2) 140 (1994), 295-329. · Zbl 0824.58032 |

[20] | E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung , Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261-304. · Zbl 0024.08003 |

[21] | A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms , Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137-173. · Zbl 0445.58015 |

[22] | A. Katok and B. Haasselblatt, Introduction to the modern theory of dynamical systems , Encyclopedia Math. Appl. 54 , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0878.58020 |

[23] | F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, I: Characterization of measures satisfying Pesin’s entropy formula , Ann. of Math. (2) 122 (1985), 509-539. · Zbl 0605.58028 |

[24] | C. Liang, W. Sun, and J. Yang, Some results on perturbations to Lyapunov exponents , preprint, [math.DS] |

[25] | R. Mañé, Ergodic theory and differentiable dynamics , Ergeb. Math. Grenzgeb. (3) 8 , Springer, Berlin, 1987. · Zbl 0616.28007 |

[26] | Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory (in Russian), Uspekhi Mat. Nauk 32 (1977), 55-112; English translation in Russian Math. Surveys 32 (1977), 55-114. · Zbl 0383.58011 |

[27] | C. Pugh and M. Shub, Ergodic attractors , Trans. Amer. Math. Soc. 312 , no. 1 (1989), 1-54. · Zbl 0684.58008 |

[28] | C. Pugh and M. Shub, “Stable ergodicity and partial hyperbolicity” in International Conference on Dynamical Systems (Montevideo, 1995) , Pitman Res. Notes Math. Ser. 362 , Longman, Harlow, 1996, 182-187. · Zbl 0867.58049 |

[29] | C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality , J. Eur. Math. Soc. (JEMS) 2 (2000), 1-52. · Zbl 0964.37017 |

[30] | F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, Creation of blenders in the conservative setting , Nonlinearity 23 (2010), 201-223. · Zbl 1191.37014 |

[31] | F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, “A survey of partially hyperbolic dynamics” in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow (Toronto, 2006) , Fields Inst. Commun. 51 , Amer. Math. Soc., Providence, 2007, 35-87. · Zbl 1149.37021 |

[32] | F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1 D-center bundle , Invent. Math. 172 (2008), 353-381. · Zbl 1136.37020 |

[33] | V. A. Rohlin, On the Fundamental Ideas of Measure Theory , Amer. Math. Soc. Transl. Ser. 2 1952 , Amer. Math. Soc., Providence, 1952. |

[34] | M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents , Invent. Math. 139 (2000), 495-508. · Zbl 0976.37013 |

[35] | S. Smale, Differentiable dynamical systems , Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747-817. · Zbl 0202.55202 |

[36] | A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic , Israel J. Math. 142 (2004), 315-344. · Zbl 1052.37019 |

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