Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-31T16:34:56.802Z Has data issue: false hasContentIssue false
  • English
  • Français

Knaster-like continua and complex dynamics

Published online by Cambridge University Press:  19 September 2008

Robert L. Devaney
Robert L. Devaney
Affiliation:
Department of Mathematics, Boston University, Boston MA 02215, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we discuss the topology and dynamics of Eλ(z) = λez when λ is real and λ > 1/e. It is known that the Julia set of Eλ is the entire plane in this case. Our goal is to show that there are certain natural invariant subsets for Eλ which are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 4 , December 1993 , pp. 627 - 634
Copyright
Copyright © Cambridge University Press 1993

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[AO]Aarts, J. & Oversteegen, L.. The Geometry of Julia Sets. Preprint.Google Scholar
[B]Barge, M.. Horseshoe maps and inverse limits. Pacific J. Math. 121 (1986), 29–39.CrossRefGoogle Scholar
[C]Currey, S.. One-dimensional nonseparating plane continua with disjoint ε-dense subcontinua. Topol. and its Appl. 39 (1991), 145–151.CrossRefGoogle Scholar
[DG]Devaney, R. L. & Goldberg, L.. Uniformization of attracting basins for exponential maps. Duke Math. J. 55 (1987), 253–266.CrossRefGoogle Scholar
[DGH]Devaney, R. L., Goldberg, L. & Hubbard, J.. A dynamical approximation to the exponential map by polynomials. Preprint.Google Scholar
[DK]Devaney, R. L. & Krych, M.. Dynamics of Exp(z). Ergod. Th. & Dynam. Sys. 4 (1984), 35–52.CrossRefGoogle Scholar
[DT]Devaney, R. L. & Tangerman, F.. Dynamics of entire functions near the essential singularity. Ergod. Th. & Dynam. Sys. 6 (1986), 489–503.CrossRefGoogle Scholar
[DoG]Douady, A. & Goldberg, L.. The nonconjugacy of certain exponential functions. Holomorphic Functions and Moduli. MSRI, Springer: Berlin, 1988. pp. 1–8.Google Scholar
[EL]Eremenko, A. & Lyubich, M. Yu.. Iterates of entire functions. Dokl. Akad. Nauk SSSR 279 (1984), 25–27. English translation in Sov. Math. Dokl. 30 (1984), 592–594.Google Scholar
[GK]Goldberg, L. R. & Keen, L.. A finiteness theorem for a dynamical class of entire functions. Ergod. Th. & Dynam. Sys. 6 (1986), 183–192.CrossRefGoogle Scholar
[K]Kuratowski, K.. Topology. Vol. 2. Academic: New York, 1968.Google Scholar
[L]Lyubich, M.. Measurable dynamics of the exponential. Sov. Math. Dokl. 35 (1987), 223–226.Google Scholar
[Ma]Mayer, J.. An explosion point for the set of endpoints of the Julia set of λ exp(z). Ergod. Th. & Dynam. Sys. 10 (1990), 177–184.CrossRefGoogle Scholar
[Mi]Misiurewicz, M.. On iterates of ez. Egod. Th. & Dynam. Sys. 1 (1981), 103–106.CrossRefGoogle Scholar
[MR]Mayer, J. & Rogers, J. T.. Indecomposable continua and the Julia sets of polynomials. Preprint.Google Scholar
[Sm]Smale, S.. Diffeomorphisms with many periodic points. Differential and Combinatorial Topology. Princeton University Press: Princeton, 1964. pp 63–80.Google Scholar