Geodesic
Continuous Self-Maps of the Circle
Canadian mathematical bulletin, Tome 40 (1997) no. 1, pp. 108-116

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Given a continuous map δ from the circle S to itself we want to find all self-maps σ: S → S for which δ o σ. If the degree r of δ is not zero, the transformations σ 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 Cn or to a dihedral group Dn 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: Let Δ: R → R be continuous, nowhere constant, and limx→−∞ Δ(x) = −∞, limx→−∞ Δ(xx) = +∞; then the only continuous map Σ: R → R such that Δ o Σ = Δ is the identity Σ = idR.
DOI : 10.4153/CMB-1997-013-7
Mots-clés : 53A04, 55M25, 55M35 
Schaer, J. Continuous Self-Maps of the Circle. Canadian mathematical bulletin, Tome 40 (1997) no. 1, pp. 108-116. doi: 10.4153/CMB-1997-013-7
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