Coexistence of Cycles of a Continuous Map of the Real Line Into Itself

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: \(3\prec 5\prec 7\prec 9\prec 11\prec \dots \prec 3\bullet 2\prec 5\bullet 2\prec \dots \prec 3\bullet {2}^{2}\prec 5\bullet {2}^{2}\prec \dots \prec {2}^{3}\prec {2}^{2}\prec 2\prec 1.\) $$3\prec 5\prec 7\prec 9\prec 11\prec \cdots \prec 3\bullet 2\prec 5\bullet 2\prec \cdots \prec 3\bullet 2^2\prec 5\bullet 2^2\prec \cdots \prec 2^3\prec 2^2\prec 2\prec 1.$$