Iterative square roots of functions

Abstract

AbstractAn iterative square root of a self-mapfis a self-mapgsuch that$g(g(\cdot ))=f(\cdot )$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in${\mathbb R}^{m}$and to the whole of$\mathbb {R}^{m}$for every positive integer$m.$However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in$\mathbb {R}^{m}$with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.