Stably ergodic dynamical systems and partial hyperbolicity.

*(English)*Zbl 0883.58025The authors study partially hyperbolic diffeomorphisms \(f:M \to M\) of a compact, connected, boundaryless manifold \(M\). A diffeomorphism \(f\) is partially hyperbolic if \(Tf:TM\to TM\) leaves invariant a continuous splitting \(TM= E^u \oplus E^c \oplus E^s\), where \(Tf\) expands \(E^u\), \(Tf\) contracts \(E^s\) and \(\forall p\in M\) \(\sup |T_pf |_{E^s} |< \inf m(T_p |_{E^c})\) and \(\sup|T_p f|_{E^c} |< \inf m(T_p f|_{E^u})\), where \(m(T)= \inf \{|T\nu|:|\nu |= 1\}\). Subbundles \((E,F)\) are said to have the accessibility property if every pair of points in \(M\) can be joined by a piecewise \(C^1\) path. It is said that a diffeomorphism \(f\) has sufficiently Hölder invariant bundles if the Hölder exponents of the bundles \(E^u\), \(E^c\), \(E^s\) are greater than some constant, depending on \(\dim M\), and satisfying certain conditions. If the spectra of \(T^uf\), \(T^cf\) and \(T^sf\) lie in thin, well separated annuli, it is said that \(f\) has a bunched spectrum.

Some main results of the paper:

1. If a \(C^2\), volume preserving diffeomorphism \(f:M \to M\) is partially hyperbolic, dynamically coherent, has the essential accessibility property, and its invariant bundles are sufficiently Hölder, then \(f\) is ergodic.

2. If in addition to the preceding conditions \(f\) has the accessibility property, the invariant bundles of \(f\) are \(C^1\), and the spectrum of \(Tf\) is sufficiently bunched, then \(f\) is stably ergodic, i.e., \(f\) is ergodic and so is every volume preserving diffeomorphism of \(M\) that \(C^2\) approximates it.

3. The time-one map of the geodesic flow on the unit tangent bundle \(M\) of a compact Riemannian manifold of constant negative curvature is stably ergodic.

Some main results of the paper:

1. If a \(C^2\), volume preserving diffeomorphism \(f:M \to M\) is partially hyperbolic, dynamically coherent, has the essential accessibility property, and its invariant bundles are sufficiently Hölder, then \(f\) is ergodic.

2. If in addition to the preceding conditions \(f\) has the accessibility property, the invariant bundles of \(f\) are \(C^1\), and the spectrum of \(Tf\) is sufficiently bunched, then \(f\) is stably ergodic, i.e., \(f\) is ergodic and so is every volume preserving diffeomorphism of \(M\) that \(C^2\) approximates it.

3. The time-one map of the geodesic flow on the unit tangent bundle \(M\) of a compact Riemannian manifold of constant negative curvature is stably ergodic.

Reviewer: Victor Sharapov (Volgograd)

PDF
BibTeX
XML
Cite

\textit{C. Pugh} and \textit{M. Shub}, J. Complexity 13, No. 1, 125--179 (1997; Zbl 0883.58025)

Full Text:
DOI

##### References:

[1] | C. Bonatti, L. Dias, 1994, Persistent nonhyperbolic transitive diffeomorphisms |

[2] | J. Brezin, M. Shub, Stable ergodicity in homogeneous spaces · Zbl 0896.58040 |

[3] | Brin, M.I.; Pesin, J.B., Partially hyperbolic dynamical systems, Math. USSR izv., 8, 177-218, (1974) · Zbl 0309.58017 |

[4] | Grasse, K.A., On accessibility and normal accessibility: the openness of controllability in the fineC0, J. differential equations, 53, 387-414, (1984) · Zbl 0553.93012 |

[5] | Grayson, M.; Pugh, C.; Shub, M., Stably ergodic diffeomorphisms, Ann. math., 140, 295-329, (1994) · Zbl 0824.58032 |

[6] | M. Gromov, 1995, Carnot-Caratheodory spaces seen from within · Zbl 0864.53025 |

[7] | Hirsch, M.; Pugh, C.; Shub, M., Invariant manifolds, Springer lecture notes in mathematics, 583, (1977), Springer-Verlag Heidelberg |

[8] | A. Katok, A. Kononenko, 1996, Cocycle stability for partially hyperbolic systems · Zbl 0853.58082 |

[9] | A. Kolmogorov, 1954, Proceedings of the International Congress of Mathematicians, 1, 315, 313 |

[10] | Lobry, C., Une propriété générique des couples de champs de vecteurs, Czechoslovak math. J., 22, 230-237, (1972) · Zbl 0242.58007 |

[11] | C. Lobry, 1973, Dynamical polysystems and control theory, Geometric Methods in Systems Theory, Reidel, Boston · Zbl 0279.93012 |

[12] | Moore, C.C., Ergodicity of flows on homogeneous spaces, Amer. J. math., 88, 154-178, (1966) · Zbl 0148.37902 |

[13] | Parry, W., Dynamical systems on nil-manifolds, Bull. London math soc., 2, 37-40, (1970) · Zbl 0194.05601 |

[14] | Pugh, C.; Shub, M., Ergodicity of Anosov actions, Inventiones math., 15, 1-23, (1972) · Zbl 0236.58007 |

[15] | C. Pugh, M. Shub, A. Wilkinson, Hölder foliations, Duke J. Math. [PSW] |

[16] | Shub, M., Global stability of dynamical systems, (1987), Springer-Verlag New York |

[17] | Sussman, H., Some properties of vector field systems that are not altered by small perturbations, J. differential equations, 20, 292-315, (1976) · Zbl 0346.49036 |

[18] | A. Wilkinson, 1995, Stable Ergodicity of the Time One Map of a Geodesic Flow, University of California at Berkeley |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.