A note on composition operators on model spaces

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankhah we consider the following problem: for a given inner function ϕ ∉ Aut ( 𝔻 ) , find a non-constant inner function Ψ satisfying the functional equation Ψ ∘ ϕ = τ Ψ , where τ is a unimodular constant. We prove that this problem has a solution if and only if ϕ 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 τ = 1 . If in addition, ϕ is parabolic, we prove that the problem has a solution Ψ for any unimodular τ . Finally, we show that if ϕ is of zero hyperbolic step, then no non-constant Bloch function f and no unimodular constant τ satisfy f ∘ ϕ = τ f .