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# Continuous Self-maps of the Circle

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http://dx.doi.org/10.4153/CMB-1997-013-7
Canad. Math. Bull. 40(1997), 108-116
Published:1997-03-01
Printed: Mar 1997
• J. Schaer
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## Abstract

Given a continuous map $\delta$ from the circle $S$ to itself we want to find all self-maps $\sigma\colon S\to S$ for which $\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not zero, the transformations $\sigma$ form a subgroup of the cyclic group $C_r$. If $r=0$, all such invertible transformations form a group isomorphic either to a cyclic group $C_n$ or to a dihedral group $D_n$ depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: {\it Let $\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and $\lim_{x\to -\infty}\Delta(x)=-\infty$, $\lim_{x\to+\infty}\Delta (x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd R$ such that $\Delta\circ\Sigma=\Delta$ is the identity $\Sigma=\id_{\bbd R}$.

MSC Classifications:  53A04 - Curves in Euclidean space 55M25 - Degree, winding number 55M35 - Finite groups of transformations (including Smith theory) [See also 57S17]
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