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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Strange attractors of uniform flows


Author: Ittai Kan
Journal: Trans. Amer. Math. Soc. 293 (1986), 135-159
MSC: Primary 58F12
MathSciNet review: 814917
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Abstract: Consider orbitally stable attractors of those flows on the open solid torus $ {D^2} \times {S^1}$ which have uniform velocity in the $ {S^1}$ direction (uniform flows). It is found that any such attractor is the frontier of a strictly nested sequence of positively invariant open solid tori. Necessary and sufficient conditions related to these tori are derived for an arbitrary set to be an orbitally stable attractor. When the cross-section of an orbitally stable attractor is a Cantor set, the first return map is found to be conjugate to an irrational rotation on a certain compact abelian group. New examples are constructed of orbitally stable attractors of uniform $ {C^\infty }$ flows whose cross-sections have uncountably many components (one of these attractors has positive $ 3$-dimensional Lebesgue measure).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0814917-4
PII: S 0002-9947(1986)0814917-4
Keywords: Attractors, strange attractors, Hausdorff dimension
Article copyright: © Copyright 1986 American Mathematical Society