TY - JOUR
AU - O. M. Sharkovsky
PY - 2024/02/02
Y2 - 2024/02/26
TI - Coexistence of cycles of а continuous mapping of the line into itself
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 76
IS - 1
SE - Research articles
DO - 10.3842/umzh.v76i1.8026
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/8026
AB - UDC 517.9Our 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: $$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec\ldots \prec 3\cdot 2 \prec 5 \cdot 2 \prec \ldots \prec 3 \cdot 2^2 \prec 5 \cdot 2^2$$ $$\prec\ldots \prec 2^3 \prec 2^2 \prec 2 \prec 1.$$
ER -