A class of \(C^ \infty\)-stable foliations.

*(English)*Zbl 0801.58038The group of \(C^ \infty\)-diffeomorphisms of a compact manifold acts on the space of \(C^ \infty\)-codimension \(q\) foliations. This allows to define the concept of \(C^ \infty\)-stable foliation that is stronger than the notion of structural stability. Due to the particularity of the \(C^ \infty\) setting the methods of proving the structural stability of a foliation are no longer valid in order to prove the \(C^ \infty\)- stability.

In an unpublished paper of R. Hamilton [Deformation theory of foliations, preprint of the Cornell University, New York (1978)] a criterion of stability is stated. This condition is used in order to prove the stability of a class of foliations obtained as the suspension of a linear foliation on the \(n\)-dimensional torus by means of a linear Anosov diffeomorphism keeping the linear foliation invariant. The construction of these foliations starts with a diagonalizable matrix \(A\in SL(n,\mathbb{Z})\) having all its eigenvalues positive. Under some conditions on this matrix concerning its eigenvalues and eigenvectors, the associated diffeomorphism on the torus is an Anosov diffeomorphism and the associated foliation is shown to be \(C^ \infty\)-stable.

Some examples of matrices fulfilling such conditions are given, including, as a particular case, an example that can be found in the previous paper of E. Ghys and V. Sergiescu [Topology 19, 179- 197 (1980; Zbl 0478.57017)].

In an unpublished paper of R. Hamilton [Deformation theory of foliations, preprint of the Cornell University, New York (1978)] a criterion of stability is stated. This condition is used in order to prove the stability of a class of foliations obtained as the suspension of a linear foliation on the \(n\)-dimensional torus by means of a linear Anosov diffeomorphism keeping the linear foliation invariant. The construction of these foliations starts with a diagonalizable matrix \(A\in SL(n,\mathbb{Z})\) having all its eigenvalues positive. Under some conditions on this matrix concerning its eigenvalues and eigenvectors, the associated diffeomorphism on the torus is an Anosov diffeomorphism and the associated foliation is shown to be \(C^ \infty\)-stable.

Some examples of matrices fulfilling such conditions are given, including, as a particular case, an example that can be found in the previous paper of E. Ghys and V. Sergiescu [Topology 19, 179- 197 (1980; Zbl 0478.57017)].

Reviewer: J.Monterde (Burjasot)

##### MSC:

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

53C12 | Foliations (differential geometric aspects) |

57R30 | Foliations in differential topology; geometric theory |

PDF
BibTeX
XML
Cite

\textit{A. e. K. Alaoui} and \textit{M. Nicolau}, Ergodic Theory Dyn. Syst. 13, No. 4, 697--704 (1993; Zbl 0801.58038)

Full Text:
DOI

##### References:

[1] | DOI: 10.1016/0040-9383(80)90005-1 · Zbl 0478.57017 |

[2] | Ghys, Ann. Sci. Éc. Norm. Sup. 20 pp 251– (1987) |

[3] | DOI: 10.1007/BF01388867 · Zbl 0577.57010 |

[4] | Schmidt, Diophantine Approximation. Springer Lecture Notes in Mathematics 785 (1980) |

[5] | DOI: 10.1016/0040-9383(77)90034-9 · Zbl 0346.57009 |

[6] | Hamilton, Preprint (1978) |

[7] | DOI: 10.2307/2946593 · Zbl 0754.58029 |

[8] | DOI: 10.1090/S0273-0979-1990-15914-2 · Zbl 0713.57022 |

[9] | Hurder, Publ. Math. IHES 72 pp 5– (1990) · Zbl 0725.58034 |

[10] | DOI: 10.1090/S0273-0979-1982-15004-2 · Zbl 0499.58003 |

[11] | DOI: 10.1007/BF02776025 · Zbl 0785.22012 |

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.