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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stability properties of a class of attractors

Author: Jorge Lewowicz
Journal: Trans. Amer. Math. Soc. 185 (1973), 183-198
MSC: Primary 58F10; Secondary 34D30, 58F15
MathSciNet review: 0343315
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Abstract: Let A be an attractor of an analytical dynamical system defined in $ {R^n} \times R$. The class of attractors considered in this paper consists of those A which remain stable as invariant subsets of the complex extension of the flow to $ {C^n} \times R$. If A is a critical point or a closed orbit, these are the elementary or generic attractors. It is shown that such an A is always a submanifold of $ {R^n}$ and that there exists a Lie group acting on A and containing the given flow as a one parameter dense subgroup; as a consequence, some necessary and sufficient conditions for an analytical dynamical system to have an attracting generic periodic motion are given.

It is also shown that for any flow $ {C^1}$-close to the given one, there is a unique retraction of a neighbourhood of A onto a submanifold of $ {R^n}$ homeomorphic to A that commutes with the flow.

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Keywords: Analytical dynamical system, stable invariant set, elementary or generic attractors, complex Poincaré diffeomorphism, complex extension of the flow, retraction commuting with the flow, minimal set, one parameter subgroup, Banach manifold, contraction
Article copyright: © Copyright 1973 American Mathematical Society