A note on iterated maps of the unit sphere

Let $\mathcal{C}(S^m)$ denote the set of continuous maps from the unit sphere $S^m$ in ${\mathbb{R}}^{m+1}$ into itself endowed with the supremum norm. We prove that the set $\{f^n:f\in \mathcal{C}(S^m)\text{ and }n\ge 2\}$ of iterated maps is not dense in $\mathcal{C}(S^m)$. This, in particular, proves that the periodic points of the iteration operator of order $n$ are not dense in $\mathcal{C}(S^m)$ for all $n\ge 2$, providing an alternative proof of the result that these operators are not Devaney chaotic on $\mathcal{C}(S^m)$ proved in [M. Veerapazham, C. Gopalakrishna, W. Zhang, Dynamics of the iteration operator on the space of continuous self-maps, Proc. Amer. Math. Soc., 149(1) (2021), 217–229].