Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T13:43:34.476Z Has data issue: false hasContentIssue false
  • English
  • Français

The bifurcation of periodic orbits of one-dimensional 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 bifurcation of C1-continuous families of maps of the interval or circle is studied. It is shown, for example, that period-tripling cannot occur. This yields topological properties of the stratification of C1(I, I) induced by the Sarkovskii order, and corresponding bifurcation properties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 2 , June 1982 , pp. 125 - 129
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Block, L.. Stability of periodic orbits in the theorem of Sarkovskii. Proc. Amer. Math. Soc. 81 (1981), 333–336.Google Scholar
[2]Block, L.. Periods of periodic points of maps of the circle which have a fixed point. Proc. Amer. Math. Soc. 82 (1981), 481–486.Google Scholar
[3]Block, L.. Simple periodic orbits of mappings of the interval. Trans. Amer. Math. Soc. 254 (1979), 391–398.Google Scholar
[4]Block, L.. Homoclinic points of mappings of the interval. Proc. Amer. Math. Soc. 72 (1978), 576–580.Google Scholar
[5]Block, L., Guckenheimer, J., Misiurewicz, M., & Young, L.-S.. Periodic points and topological entropy of one-dimensional maps. Lecture Notes in Math. No. 819, Springer: Berlin, 1980, pp. 18–34.Google Scholar
[6]Collet, P. & Eckmann, J. P.. Iterated maps on the interval as dynamical systems. Progr. in Phys. Phys. 1 (Birkhauser, 1980).Google Scholar
[7]Feigenbaum, M.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.Google Scholar
[8]Misiurewicz, M.. Structure of mappings of an interval with zero entropy. Publ. Math. I.H.E.S., to appear.Google Scholar
[9]Misiurewicz, M.. Horseshoes for mappings of the interval. Bull. Acad. Polon. Sci. 27 (1979), 167–169.Google Scholar
[10]Nitecki, Z.. Topological dynamics on the interval. Ergodic Theory and Dynamical Systems, Vol. II (Proceedings of the Special Year, Maryland, 1979–1980) Progr. in Math. Birkhäuser, 1981, to appear.Google Scholar
[11]Nitecki, Z.. Maps of the interval with closed period set. Proc. Amer. Math. Soc. 85 (1982) 451–456.Google Scholar
[12]Sarkovskii, A.. Coexistence of cycles of a continuous map of the line into itself. Ukr. Mat. Z. 16 (1964), 61–71. (In Russian.)Google Scholar