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Rotation numbers, twists and a Sharkovskii–Misiurewicz-type ordering for patterns on the interval

Published online by Cambridge University Press:  19 September 2008

A. M. Blokh
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294–2060, USA

Abstract

We introduce rotation numbers and pairs characterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[ALM]Alseda, L., Llibre, J. and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One. World Scientific: Singapore, 1993.CrossRefGoogle Scholar
[Ba]Baldwin, S.. Generalizations of a Theorem of Sharkovskii on orbits of continuous real-valued functions. Discrete Math. 67 (1987), 111127.CrossRefGoogle Scholar
[B1]Blokh, A. M.. Decomposition of dynamical systems on an interval. Russ. Math. Surv. 38 (5) (1983), 133134.CrossRefGoogle Scholar
[B2]Blokh, A. M.. The spectral decomposition for one-dimensional maps. SUNY at Stony Brook, Preprint # 1991/14, September (to appear in Dynamics Reported).Google Scholar
[B3]Blokh, A. M.. Functional rotation numbers for interval maps. Trans. Amer. Math. Soc. (1993). To appear.Google Scholar
[B4]Blokh, A. M.. Rotation numbers for unimodal maps. MSRI Preprint 058–94. (1994).Google Scholar
[I]Ito, R.. Rotation sets are closed. Math. Proc. Camb. Phil. Soc. 89 (1981), 107111.CrossRefGoogle Scholar
[LMPY]Li, T.-Y., Misiurewicz, M., Pianigiani, G. and Yorke, J.. No division implies chaos. Trans. Amer. Math. Soc. 273 (1982), 191199.CrossRefGoogle Scholar
[M1]Misiurewiczs, M.. Periodic points of maps of degree one of a circle. Ergod. Th. & Dynam. Sys. 2 (1982), 221227.CrossRefGoogle Scholar
[MN]Misiurewicz, M. and Nitecki, Z.. Combinatorial patterns for maps of the interval. Mem. Amer. Math. Soc. 456 (1990).Google Scholar
[MZ]Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Land. Math. Soc. 2 (1989), 490506.CrossRefGoogle Scholar
[NPT]Newhouse, S., Palis, J. and Takens, F.. Bifurcations and stability of families of diffeomorphisms. IHES Publ. Math. 57 (1983), 571.CrossRefGoogle Scholar
[S]Sharkovskii, A. N.. Co-existence of the cycles of a continuous mapping of the line into itself. Ukrain. Math. J. 16 (1964), 6171.Google Scholar