Centralizers of immersions of the circle

Abstract:

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