X-minimal patterns and a generalization of Sharkovskiĭ's theorem

Jozef Bobok; Milan Kuchta

Fundamenta Mathematicae (1998)

  • Volume: 156, Issue: 1, page 33-66
  • ISSN: 0016-2736

Abstract

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We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.

How to cite

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Bobok, Jozef, and Kuchta, Milan. "X-minimal patterns and a generalization of Sharkovskiĭ's theorem." Fundamenta Mathematicae 156.1 (1998): 33-66. <http://eudml.org/doc/212260>.

@article{Bobok1998,
abstract = {We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.},
author = {Bobok, Jozef, Kuchta, Milan},
journal = {Fundamenta Mathematicae},
keywords = {iteration; periodic orbit; cycle; pattern; minimal; forcing relation; Sharkovskiĭ s theorem; cycles; Sharkovskij's theorem},
language = {eng},
number = {1},
pages = {33-66},
title = {X-minimal patterns and a generalization of Sharkovskiĭ's theorem},
url = {http://eudml.org/doc/212260},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Bobok, Jozef
AU - Kuchta, Milan
TI - X-minimal patterns and a generalization of Sharkovskiĭ's theorem
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 1
SP - 33
EP - 66
AB - We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.
LA - eng
KW - iteration; periodic orbit; cycle; pattern; minimal; forcing relation; Sharkovskiĭ s theorem; cycles; Sharkovskij's theorem
UR - http://eudml.org/doc/212260
ER -

References

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  2. [ALS] L. Alsedà, J. Llibre and R. Serra, Mimimal periodic orbits for continuous maps of the interval, Trans. Amer. Math. Soc. 286 (1984), 595-627. Zbl0569.54040
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  9. [BK] J. Bobok and M. Kuchta, Invariant measures for maps of the interval that do not have points of some period, Ergodic Theory Dynam. Systems 14 (1994), 9-21. Zbl0805.58033
  10. [C] W. A. Coppel, Šarkovskii-minimal orbits, Math. Proc. Cambridge Philos. Soc. 93 (1983), 397-408. Zbl0521.58044
  11. [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, Oxford, 1960. 
  12. [LMPY] T.-Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc. 273 (1982), 191-199. Zbl0495.58018
  13. [S] A. N. Sharkovskiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Zh. 16 (1964), 61-71 (in Russian). 
  14. [St] P. Štefan, A theorem of Šarkovskiĭ on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248. Zbl0354.54027

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