Coexistence of cycles of а continuous mapping of the line into itself
UDC 517.9 Our main result can be formulated as follows:   Consider the set of natural numbers in which the following relation is introduced: $n_1$ precedes $n_2$ $(n_1 \preceq n_2)$ if, for any continuous mappings of the real line into itself, the existence of а cycle...
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| Дата: | 2024 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2024
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/8026 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | UDC 517.9
Our main result can be formulated as follows:   Consider the set of natural numbers in which the following relation is introduced: $n_1$ precedes $n_2$ $(n_1 \preceq n_2)$ if, for any continuous mappings of the real line into itself, the existence of а cycle of order $n_2$ follows from the existence of а cycle of order $n_1.$  The following theorem is true:
Theorem. The introduced relation transforms the set of natural numbers into an ordered set with  the following ordering: $$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec\ldots \prec 3\cdot 2 \prec 5 \cdot 2 \prec \ldots \prec 3 \cdot 2^2 \prec 5 \cdot 2^2$$ $$\prec\ldots \prec 2^3 \prec 2^2 \prec 2 \prec 1.$$ |
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| DOI: | 10.3842/umzh.v76i1.8026 |