Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-02T18:02:36.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Stratification of the space of unimodal interval maps

Published online by Cambridge University Press:  19 September 2008

Louis Block  and
David Hart
Louis Block
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, USA
David Hart
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, 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.

The space of ‘quadratic-like’ (unimodal) maps of a compact interval to itself is shown to decompose in a ‘nice’ way (stratify) according to a dynamical property of such maps (the existence of a homoclinic periodic orbit with given period). This decomposition is refined by that discovered by Sarkovskii. Orbit structure and bifurcation properties are also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 4 , December 1983 , pp. 533 - 539
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Block, L.. Homoclinic points of mappings of the interval. Proc. Amer. Math. Soc. 72 (1978), 576–580.CrossRefGoogle Scholar
[2]Block, L.. Simple periodic orbits of mappings of the interval. Trans. Amer. Math. Soc. 254 (1979), 391–398.Google Scholar
[3]Block, L.. Stability of periodic orbits in the theorem of Sarkovskii. Proc. Amer. Math. Soc. 81 (1981), 333–336.Google Scholar
[4]Block, L. & Hart, D.. The bifurcation of periodic orbits of one-dimensional maps. Ergod. Th. & Dynam. Syst. 2 (1982), 125–129.CrossRefGoogle Scholar
[5]Block, L. & Hart, D.. The bifurcation of homoclinic orbits of maps of the interval. Ergod. Th. & Dynam. Syst. 2 (1982), 131–138.CrossRefGoogle Scholar
[6]Block, L., Guckenheimer, J., Misiurewicz, M. & Young, L.-S.. Periodic points and topological entropy of one dimensional maps. Lect. Notes Math. 819 (Springer, 1980), 18–34.CrossRefGoogle Scholar
[7]Collet, P. & Eckmann, J.-P.. Iterated maps on the interval as dynamical systems, Prog. Phy. 1 (Birkhauser, 1980).Google Scholar
[8]Ho, C.-W.. On the structure of the minimum orbits of periodic points for maps of the real line. Preprint.Google Scholar
[9]Jonker, L.. Periodic orbits and kneading invariants. Proc. London Math. Soc. 39 (1979), 428–450.CrossRefGoogle Scholar
[10]Lanford, O.. Smooth transformations of intervals, Lect. Notes Math. 901 Springer (1981), 36–54.CrossRefGoogle Scholar
[11]Misiurewicz, M.. Horseshoes for mappings of the interval. Bull. Acad. Polon. Sci. 27 (1979), 167–169.Google Scholar
[12]Misiurewicz, M.. Structure of mappings of an interval with zero entropy, Publ. Math. IHES, to appear.Google Scholar
[13]Sarkovskii, A. N.. Coexistence of cycles of a continuous map of the line into itself, (Russian) Ukr. Mat. Z. 16 (1964), 61–71.Google Scholar
[14]Singer, D.. Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260–267.CrossRefGoogle Scholar
[15]Stefan, P.. A theorem of Sarkovskii on the coexistence of periodic orbits of continuous endomorphisms of the real line. Comm. Math. Phys. 54 (1977), 237–248.CrossRefGoogle Scholar