  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].
