Coexistence of cycles of а continuous mapping of the line into itself

Abstract

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:  $$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec\ldots \prec 3\cdot 2 \prec 5 \cdot 2 \prec \ldots \prec 3 \cdot 2^2 \prec 5 \cdot 2^2$$ $$\prec\ldots \prec 2^3 \prec 2^2 \prec 2 \prec 1.$$