A note on iterated maps of the unit sphere.

Let C (S m) denote the set of continuous maps from the unit sphere S m in the Euclidean space R m + 1 into itself endowed with the supremum norm. We prove that the set { f n : f ∈ C (S m) and n ≥ 2 } of iterated maps is not dense in C (S m) . This, in particular, proves that the periodic points of the iteration operator of order n are not dense in C (S m) for all n ≥ 2 , providing an alternative proof of the result that these operators are not Devaney chaotic on C (S m) proved in Veerapazham et al. (Proc Am Math Soc 149(1):217–229, 2021). [ABSTRACT FROM AUTHOR]