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Šarkovskii-minimal orbits

Published online by Cambridge University Press:  24 October 2008

W. A. Coppel
W. A. Coppel
Affiliation:
Department of Mathematics, IAS, Australian National University, Canberra
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Extract

My first paper (5), submitted for publication by my supervisor Philip Hall, dealt with iteration of a continuous map of an interval. This subject has recently become fashionable and many interesting results are discussed in the book of Collet and Eckmann (4). Several of these results hold for special types of map, for example those with negative Schwarzian derivative. However here, as in my original paper, we will be concerned with arbitrary continuous maps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 397 - 408
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Block, L.Simple periodic orbits of mappings of the interval. Trans. Amer. Math. Soc. 254 (1979), 391–398.Google Scholar
(2)Block, L., Guckenheimer, J., Misiuerwicz, M. and Young, L. S. Periodic points and topological entropy of one dimensional maps. In Global theory of dynamical systems, pp. 18–34. Lecture Notes in Mathematics, no. 819 (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar
(3)Burkart, U.Interval mapping graphs and periodic points of continuous functions. J. Combin. Theory Ser. B 32 (1982), 57–68.Google Scholar
(4)Collet, P. and Eckmann, J.-P.Iterated maps on the interval as dynamical systems (Progress in Physics 1, Birkhäuser, Basel, Boston, Stuttgart, 1980).Google Scholar
(5)Coppel, W. A.The solution of equations by iteration. Proc. Cambridge Philos. Soc. 51 (1955), 41–43.Google Scholar
(6)Cvetković, D. M., Doob, M. and Sachs, H.Spectra of graphs (Academic Press, New York, San Francisco, London, 1980).Google Scholar
(7)Ho, C.-W. and Morris, C.A graph theoretic proof of Sharkovsky's theorem on the periodic points of continuous functions. Pacific J. Math. 96 (1981), 361–370.CrossRefGoogle Scholar
(8)Huppert, B.Endliche Gruppen I, Die Grundlehren der mathematischen Wissenschaften Band 134 (Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
(9)Šarkovskii, A. N.Necessary and sufficient conditions for convergence of one-dimensional. iterative processes (Russian). Ukrain. Mat. Ž. 12 (1960), 484–489.Google Scholar
(10)Šarkovskii, A. N.Coexistence of cycles of a continuous mapping of the line into itself (Russian). Ukrain. Mat. Ž. 16 (1964), no. 1, 61–71.Google Scholar
(11)Šarkovskii, A. N.On cycles and the structure of a continuous mapping (Russian). Ukrain. Mat. Ž. 17 (1965), no. 3, 104–111.Google Scholar
(12)Štefan, P.A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Comm. Math. Phys. 54 (1977), 237–248.CrossRefGoogle Scholar
(13)Straffin, P. D.Periodic points of continuous functions. Math. Mag. 51 (1978), 99–105.Google Scholar