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

The basic result of this investigation may be formulated as follows. Consider the set of natural numbers in which the following relationship is introduced: n 1 precedes n 2 (n 1 ≼ n 2 ) if for any continuous mapping of the real line into itself the existence of a cycle of order n 2 follows from the existence of a cycle of order n 1 . The following theorem holds.