Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions

Abstract In 1964, Sarkovskii defined a certain linear ordering ⩽ s of the positive integers and proved that m ⩽ s n if every continuous f : R → R having an orbit of size n also has an orbit of size m . This idea is extended to get a partial (but not linear) ordering in which the pattern of the orbit is taken into account. For example if x 1 x 2 x 3 x 4 , then x 1 → x 2 → x 3 → x 4 x → 1 and x 1 → x 3 → x 2 → x 4 → x 1 are both orbits of size 4 but are considered to have distinct patterns in this paper. A combinatorial algorithm which decides the status of any two patterns with respect to the partial ordering is derived, and examples are given for patterns of small size.