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Continuous Self-maps of the Circle

 Printed: Mar 1997
  • J. Schaer
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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, 55M25, 55M35 show english descriptions Curves in Euclidean space
Degree, winding number
Finite groups of transformations (including Smith theory) [See also 57S17]
53A04 - Curves in Euclidean space
55M25 - Degree, winding number
55M35 - Finite groups of transformations (including Smith theory) [See also 57S17]

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