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Propriétés des attracteurs de Birkhoff. (Properties of Birkhoff attractors). (French) Zbl 0657.58009
Birkhoff’s ideas on instability rings, which occur only in smooth area preserving two-dimensional maps, are reexamined and expressed in a ‘more modern’ terminology. Some of the properties are proven ‘more completely and rigorously’. The additional insight so obtained by the author is applied to the study of weakly dissipative twist maps of an annulus, i.e. to what may possibly happen to instability rings when the condition of area preservation is removed, and replaced by a smooth weak dissipation. The latter renders possibly the existence of what the author calls Birkhoff attractors. The main tool of the study consists of a particular generalization of the Poincaré notion of a rotation number. Several well known results are rediscovered, e.g. that: 1) an infinity of former centres may survive a sufficiently weak dissipative perturbation, and become ‘sinks’ (i.e. attractive foci), and 2) the resulting attractor structure does not depend ‘continuously’ on the dissipative perturbation. An analogy with Cremona maps is briefly discussed. The references are highly selective, and all non classical ones belong to the same school of thought.
Reviewer: I.Gumowski

58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
57R35 Differentiable mappings in differential topology
Full Text: DOI
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