A note on iterated maps of the unit sphere

TL;DR

This paper demonstrates that the set of iterated continuous maps on the unit sphere is not dense in the space of all such maps, showing that periodic points of the iteration operator are not dense and providing an alternative proof of non-chaotic behavior.

Contribution

It proves the non-density of iterated maps in the space of continuous maps on the sphere, offering a new proof regarding the non-chaotic nature of the iteration operator.

Findings

Iterated maps are not dense in the space of continuous maps on the sphere.

Periodic points of the iteration operator are not dense.

Provides an alternative proof of non-chaotic dynamics on the space.

Abstract

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].