Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T21:37:00.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudocircles, diffeomorphisms and perturbable dynamical systems*

Published online by Cambridge University Press:  14 October 2010

Judy A. Kennedy  and
James A. Yorke
Judy A. Kennedy
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE19716, USA
James A. Yorke
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct an example of a C∞ diffeomorphism on a 7-manifold which has an invariant set with an uncountable number of pseudocircle components. Furthermore, any diffeomorphism which is sufficiently close (in the C1 metric) to the constructed map has a similar invariant set. We also discuss the topological nature of the invariant set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 1031 - 1057
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B]Bing, R. H.. Concerning hereditarily indecomposable continua. Pacific J. Math. 1 (1951), 43–51.Google Scholar
[Fl]Fearnley, L.. The pseudo-circle is not homogeneous. Bull. Amer. Math. Soc. 75 (1969), 554–558.CrossRefGoogle Scholar
[F2]Fearnley, L.. The pseudo-circle is unique. Bull. Amer. Math. Soc. 75 (1969), 398–401.CrossRefGoogle Scholar
[Ha]Handel, M.. A pathological area-preserving C∞ diffeomorphism of the plane. Proc. Amer. Math. Soc. 86 (1982), 163–168.Google Scholar
[He]Herman, M.. Construction of some curious diffeomorphisms of the Riemann sphere. J. London Math. Soc. 34 (1986), 375–384.CrossRefGoogle Scholar
[KR]Kennedy, J. and Rogers, J. T. Jr.Orbits of the pseudocircle. Trans. Amer. Math. Soc. 296 (1986), 327–340.Google Scholar
[KY]Kennedy, J. and Yorke, J. A.. Pseudocircles in dynamical systems. Trans. Amer. Math. Soc. 343 (1994), 349–366.Google Scholar
[KY2]Kennedy, J. and Yorke, J. A.. The forced damped pendulum and the Wada property. Continuum Theory and Dynamical Systems (Lecture Notes in Pure and Applied Mathematics 149). Ed. T. West. 1993, pp. 157–182.Google Scholar
[K]Krasinkiewicz, J.. Mapping properties of hereditarily indecomposable continua. Bull. Acad. Pol. Soc. 25 (1977), 507–513.Google Scholar
[KM]Krasinkiewicz, J. and Mine, P.. Mappings onto indecomposable continua. Bull. Acad. Pol. Soc. 25 (1977), 675–680.Google Scholar
[M]Moekel, R.. Rotations of the closures of some simply connected domains. Complex Variables 4 (1985), 285–294.Google Scholar
[OT]Oversteegen, L. G. and Tymchatyn, E. D.. On hereditarily indecomposable continua. Geometric and Algebraic Topology (Banach Centre Publ. 18). PWN, Warsaw, 1986, pp 403–413.Google Scholar
[PR]Pommerenke, C. and Rodin, B.. Intrinsic rotations of simply connected regions II. Complex Variables 4 (1985), 223–232.Google Scholar
[R]Rogers, J. T. Jr.The pseudocircle is not homogeneous. Trans. Amer. Math. Soc. 148 (1970), 417–428.Google Scholar