Strange attractors of uniform flows
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- by Ittai Kan PDF
- Trans. Amer. Math. Soc. 293 (1986), 135-159 Request permission
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).References
- Rufus Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR 0482842
- Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4 Duffing [1918], Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz, Braunschweig.
- C. H. Edwards Jr., Concentric solid tori in the $3$-sphere, Trans. Amer. Math. Soc. 102 (1962), 1–17. MR 140091, DOI 10.1090/S0002-9947-1962-0140091-X
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
- Sheldon E. Newhouse, Lectures on dynamical systems, Dynamical systems (Bressanone, 1978) Liguori, Naples, 1980, pp. 209–312. MR 660646 Plykin [1974], Sources andd sinks of $A$-diffeomorphisms of surfaces, USSR Math.-Sb. 23, pp. 233-253.
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1 Yoneyama [1917], Theory of continuous sets of points, Tôhoku Math. J. 12, pp. 43-158.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 293 (1986), 135-159
- MSC: Primary 58F12
- DOI: https://doi.org/10.1090/S0002-9947-1986-0814917-4
- MathSciNet review: 814917