Strange attractors of uniform flows

Author:
Ittai Kan

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider orbitally stable attractors of those flows on the open solid torus which have uniform velocity in the 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 flows whose cross-sections have uncountably many components (one of these attractors has positive -dimensional Lebesgue measure).

**[R]**Rufus Bowen,*On Axiom A diffeomorphisms*, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 35. MR**0482842****[R]**Rufus Bowen and John Franks,*The periodic points of maps of the disk and the interval*, Topology**15**(1976), no. 4, 337–342. MR**0431282**, https://doi.org/10.1016/0040-9383(76)90026-4**[G]**Duffing [**1918**],*Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz*, Braunschweig.**[C]**C. H. Edwards Jr.,*Concentric solid tori in the 3-sphere*, Trans. Amer. Math. Soc.**102**(1962), 1–17. MR**0140091**, https://doi.org/10.1090/S0002-9947-1962-0140091-X**[J]**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****[J]**John G. Hocking and Gail S. Young,*Topology*, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR**0125557****[J]**Jean-Pierre Kahane and Raphaël Salem,*Ensembles parfaits et séries trigonométriques*, Actualités Sci. Indust., No. 1301, Hermann, Paris, 1963 (French). MR**0160065****[S]**Sheldon E. Newhouse,*Lectures on dynamical systems*, Dynamical systems (Bressanone, 1978) Liguori, Naples, 1980, pp. 209–312. MR**660646****[R]**Plykin [**1974**],*Sources andd sinks of**-diffeomorphisms of surfaces*, USSR Math.-Sb.**23**, pp. 233-253.**[S]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, https://doi.org/10.1090/S0002-9904-1967-11798-1**[K]**Yoneyama [**1917**],*Theory of continuous sets of points*, Tôhoku Math. J.**12**, pp. 43-158.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F12

Retrieve articles in all journals with MSC: 58F12

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0814917-4

Keywords:
Attractors,
strange attractors,
Hausdorff dimension

Article copyright:
© Copyright 1986
American Mathematical Society