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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.

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
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