A note on composition operators on model spaces

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $\phi\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $\Psi$ satisfying the functional equation $\Psi\circ\phi=\tau\Psi$, where $\tau$ is a unimodular constant. We prove that this problem has a solution if and only if $\phi$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $\tau=1$. If in addition, $\phi$ is parabolic, we prove that the problem has a solution $\Psi$ for $any$ unimodular $\tau$. Finally, we show that if $\phi$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $\tau$ satisfy $f\circ\phi=\tau f$.