On iterative roots of injective functions

Abstract

In 1951 Łojasiewicz found a necessary and sufficient condition for the existence of a q-iterative root of an arbitrary bijective function g for any \(q\ge 2\). In this article we extend this result to the injective case. More precisely, a necessary and sufficient condition for the existence of an iterative root of an injective function is proved, and in the case of existence, the characterization and enumeration of all iterative roots are given. Furthermore, we devise a construction by which all iterative roots of an injective function can be constructed (provided that the considered function has at least one iterative root). As an illustration, we apply the developed theory to several results from the literature to obtain somewhat more elegant and shorter proofs of those results.