Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T11:00:38.312Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitivity and the centre for maps of the circle

Published online by Cambridge University Press:  19 September 2008

Ethan M. Coven  and
Irene Mulvey
Ethan M. Coven
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06457, USA
Irene Mulvey
Affiliation:
Department of Mathematics, Swarthmore College, Swarthmore, PA 19081, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 1 , March 1986 , pp. 1 - 8
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Auslander, J. & Katznelson, Y.. Continuous maps of the circle without periodic points. Israel J. Math. 32 (1979), 375–381.CrossRefGoogle Scholar
[2]Barge, M. & Martin, J.. Chaos, periodicity and snakelike continua. Preprint, 1984.CrossRefGoogle Scholar
[3]Barge, M. & Martin, J.. Dense orbits on the interval. Preprint, 1984.Google Scholar
[4]Block, L.. Periodic orbits of continuous mappings of the circle. Trans. Amer. Math. Soc. 260 (1980), 555–562.CrossRefGoogle Scholar
[5]Block, L., Coven, E., Mulvey, I. & Nitecki, Z.. Homoclinic and nonwandering points for maps of the circle. Ergod. Th. & Dynam. Sys. 3 (1983), 521–532.CrossRefGoogle Scholar
[6]Coppel, A.. Continuous maps of an interval. Xeroxed notes, 1984.Google Scholar
[7]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
[8]Sarkovskǐi, A. N.. Fixed points and the center of a continuous mapping of the line into itself. (Ukrainian, Russian and English summaries.) Dopovidi Akad. Nauk. Ukrain. RSR 1964, 865–868.Google Scholar
[9]Xiong, J.-C.. for every continuous map f of the interval. Kexue Tongbao (English Ed.) 28 (1983), 21–23.Google Scholar