Anosov diffeomorphisms on nilmanifolds up to dimension 8

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dc.creatorLauret, Jorge
dc.creatorWill, Cynthia E.
dc.date2004-06-09
dc.date2004-07-20
dc.date.accessioned2026-07-07T05:09:07Z
dc.date.available2026-07-07T05:09:07Z
dc.descriptionAfter more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the (rational) Lie algebra of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less or equal than 8, which also classify nilmanifolds admitting an Anosov diffeomorphism in those dimensions. As a corollary we obtain that if an infranilmanifold of dimension n<9 admits an Anosov diffeomorphism f and it is not a torus or a compact flat manifold (i.e. covered by a torus), then n=6 or 8 and the signature of f necessarily equals {3,3} or {4,4}, respectively. We had to study the set of all rational forms up to isomorphism for many real Lie algebras, which is a subject on its own and it is treated in a section completely independent of the rest of the paper.
dc.descriptionCorrected version: added Remarks 2.9, 2.11, 2.15, 5.1; new version of Proposition 3.3 and some minor changes in the proofs of Sections 4 and 5 (37 pages)
dc.identifierhttps://arxiv.org/abs/math/0406199
dc.identifierhttp://arxiv.org/abs/math/0406199
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/71514
dc.subjectDynamical Systems
dc.subjectDifferential Geometry
dc.titleAnosov diffeomorphisms on nilmanifolds up to dimension 8
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