Abstract
Let \(\mathcal {C}(S^{m})\) denote the set of continuous maps from the unit sphere \(S^{m}\) in the Euclidean space \(\mathbb {R}^{m+1}\) 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 Veerapazham et al. (Proc Am Math Soc 149(1):217–229, 2021).
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Acknowledgements
The author is supported by the National Board for Higher Mathematics, India through No: 0204/3/2021/R &D-II/7389. The author is very grateful to his mentor Professor B. V. Rajarama Bhat, and Professor B. Sury for useful discussions.
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Gopalakrishna, C. A note on iterated maps of the unit sphere. Aequat. Math. 98, 503–507 (2024). https://doi.org/10.1007/s00010-023-00979-6
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DOI: https://doi.org/10.1007/s00010-023-00979-6