Our main result can be formulated as follows: Consider the set of natural numbers in which the following relation is introduced: n1 precedes n2 (n1 ⪯ n2) if, for any continuous map of the real line into itself, the existence of a cycle of order n2 follows from the existence of a cycle of order n1. The following theorem is true:
Theorem. The introduced relation turns the set of natural numbers into an ordered set with the following ordering:
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A. N. Sharkovsky, Ukr. Mat. Zh., 12, No. 12, 104–109 (1960).
A. N. Sharkovsky, “The reducibility of a continuous function of a real variable and the structure of the stationary points of the corresponding iteration process,” Dokl. Akad. Nauk SSSR, 139, No. 5, 1067–1070 (1961); English translation: Soviet Math. Dokl., 2 (1961).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 5–16, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.8026.
Oleksandr Sharkovsky is deceased
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Sharkovsky, O. Coexistence of Cycles of a Continuous Map of the Real Line Into Itself. Ukr Math J 76, 3–14 (2024). https://doi.org/10.1007/s11253-024-02303-0
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DOI: https://doi.org/10.1007/s11253-024-02303-0