Centralizers of immersions of the circle

Abstract

We prove here that for every element f f of an open and dense subset of immersions of the circle S 1 {S^1} , either the centralizer Z ( f ) Z\left ( f \right ) of f f is trivial (i.e. f f only commutes with its own powers) or f f is topologically conjugate to a map f n : S 1 → S 1 {f_n}:{S^1} \to {S^1} given by f n ( z ) = z n {f_n}\left ( z \right ) = {z^n} and, in this case, if h h is the conjugacy between f f and f n {f_n} then Z ( f ) Z\left ( f \right ) is a subgroup of { h − 1 ∘ ω f m ∘ h ; m ∈ N  and  ω n − 1 = 1 } \left \{ {{h^{ - 1}} \circ \omega {f_m} \circ h;m \in {\mathbf {N}}{\text { and }}{\omega ^{n - 1}} = 1} \right \} .