Iterative square roots of functions

TL;DR

This paper investigates the existence and density of continuous functions that are square roots of other functions across various topological spaces, providing new characterizations and approximation results.

Contribution

It introduces novel criteria for the non-existence of iterative square roots and demonstrates their density in spaces homeomorphic to cubes and Euclidean spaces.

Findings

Continuous self-maps with no square roots are dense in certain spaces.

Every continuous self-map with a boundary fixed point can be approximated by iterative squares.

New characterizations for detecting non-existence of square roots are established.

Abstract

An iterative square root of a function $f$ is a function $g$ such 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. This is 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 $m.$ On the other hand, we also prove that every continuous self-map of 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.