Coexistence of the cycles of a continuous mapping of the line into itself
The basic result of this investigation may be formulated as follows. Consider a set of natural numbers in which the following relationship 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 a cycle of order $n_2$ follows from the existence of a cycle of order The following theorem holds.
Theorem. The introduced relationship transforms the set of natural numbers into an ordered set, ordered in the following way:
$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec ... \prec 3 \cdot 2 \prec 5 \cdot 2 \prec ... \prec 3 \cdot 2^2 \prec 5 \cdot 2^2 \prec ... \prec 2^3 \prec 2^2 \prec 2 \prec 1$
Citation Example: Sharkovsky O. M. Coexistence of the cycles of a continuous mapping of the line into itself // Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 61-71.