Continuous Self-Maps of the Circle

Abstract

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 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: Let Δ: ℝ → ℝ be continuous, nowhere constant, and lim_x→−∞ Δ(x) = −∞, limx→−∞ Δ(xx) = +∞; then the only continuous map Σ: R → R such that Δ o Σ = Δ is the identity Σ = id_ℝ.