  Let $\Gamma$ be an oriented Jordan smooth curve and $\alpha$ be a
diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of
periodic points. We prove criteria for one-sided invertiblity 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 $Wf=f\circ\alpha$, in a
reflexive rearrangement-invariant space $X(\Gamma)$ with Boyd indices
$\alpha_X,\beta_X$ and Zippin indices $p_X,q_X$ satisfying inequalities \[
0<\alpha_X=p_X\le q_X=q_X<1. \]
