The non-iterates are dense in the space of continuous self-maps

B. V. Rajarama Bhat; Chaitanya Gopalakrishna

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arXiv · 2208.04093

The non-iterates are dense in the space of continuous self-maps

B. V. Rajarama Bhat·Chaitanya Gopalakrishna

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Abstract

In this paper we develop a tool to identify functions which have no iterative roots of any order. Using this, we prove that when $X$ is $[0,1]^m$, $\mathbb{R}^m$ or $S^1$, every non-empty open set of the space $\mathcal{C}(X)$ of continuous self-maps on $X$ endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order $n\ge 2$. This, in particular, proves that the complement of $\{f^n: f\in \mathcal{C}(X)~\text{and}~n\ge 2\}$, the set of non-iterates, is dense in $\mathcal{C}(X)$ for these $X$.

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BibTeX RIS

B. V. Rajarama Bhat, Chaitanya Gopalakrishna. 2022-08-08. The non-iterates are dense in the space of continuous self-maps. https://arxiv.org/abs/2208.04093

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