The non-iterates are dense in the space of continuous self-maps

TL;DR

This paper shows that in certain spaces, most continuous self-maps are not iterative roots, meaning they cannot be expressed as repeated compositions of other functions, and such non-iterates are dense in the space.

Contribution

The paper introduces a new tool to identify functions without iterative roots and proves the density of non-iterates in spaces like [0,1]^m, R^m, and S^1.

Findings

Non-iterates form a dense subset in the space of continuous self-maps.

Every open set contains functions without iterative roots of any order.

The set of functions that are iterative roots is nowhere dense in these spaces.

Abstract

In this paper we develop a tool to identify functions which have no iterative roots of any order. Using this, we prove that when $X$ is $[0,1]^m$, $\mathbb{R}^m$ or $S^1$, every non-empty open set of the space $\mathcal{C}(X)$ of continuous self-maps on $X$ endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order $n\ge 2$. This, in particular, proves that the complement of $\{f^n: f\in \mathcal{C}(X)~\text{and}~n\ge 2\}$, the set of non-iterates, is dense in $\mathcal{C}(X)$ for these $X$.