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**Stable ergodicity of certain linear automorphisms of the torus.**
*(English)*
Zbl 1098.37028

For a linear automorphism of a higher-dimensional torus, ergodicity amounts to that no root of unity is an eigenvalue of the defining matrix. Stable ergodicity means small perturbations are still ergodic. The author calls a linear automorphism of a torus pseudo-Anosov if it is ergodic and the characteristic polynomial of its matrix is irreducible over the integers and is not a polynomial in a higher power of its variable. Then by his main theorems, pseudo-Anosov toral automorphisms are \(C^5\)-stably ergodic in dimensions greater than 5 if the center space is two-dimensional, and, in dimension 4, pseudo-Anosov automorphisms are \(C^{22}\)-stably ergodic. It follows that all ergodic linear toral automorphisms are stably ergodic in dimensions less than 6. A prominent notion in the proof is the essential accessibility property of an automorphism, which means that the set of points pathwise reachable from a given point via stable and unstable leaves of the invariant foliations, has null or full Lebesgue measure. By a result of Pugh and Shub, this property is sufficient for ergodicity if certain additional conditions are met. In smoothing conjugations, KAM theory comes into play.

Reviewer: Dieter Erle (Dortmund)

### MSC:

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37D30 | Partially hyperbolic systems and dominated splittings |

37A99 | Ergodic theory |

37C75 | Stability theory for smooth dynamical systems |

37L50 | Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems |