Singular integral operators for an unlimited contour

Let Γ   be a closed or unclosed unlimited contour, a shift α(t)α(t) maps homeomorphically the contour Γ onto itself with preserving or reversing the direction on Γ   and also satisfies the conditions for some natural number n⩾2n⩾2, αn(t)≅tαn(t)≅t, and αj(t)≇tαj(t)≇t for 1⩽j