One-sided invertibility of binomial functional operators with a shift on rearrangementinvariant spaces

Let (gamma) be an oriented Jordan smooth curve and (alpha) a diffeomorphism of (gamma) sonto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator A=aI-bW where a and b are continuous functions, I is the identity operator,W is the shift operator,Wf=f (ring operator) (alpha), on a reflexive rearrangement-invariant spaceX(gamma) with Boyd indices (alpha)X , (beta)X and Zippin indices p x,q x satisfying inequalities 0 < (alpha)x = px <= qx = (beta)x < 1.