Paths of inner-related functions

We characterize the connected components of the subset $\text{cni}$ of $H^{\mathrm{\infty }}$ formed by the products $bh$, where $b$ is Carleson-Newman Blaschke product and $h\in H^{\mathrm{\infty }}$ is an invertible function. We use this result to show that, except for finite Blaschke products, no inner function in the little Bloch space is in the closure of one of these components. Our main result says that every inner function can be connected with an element of $\text{cni}$ within the set of products $uh$, where $u$ is inner and $h$ is invertible. We also study some of these issues in the context of Douglas algebras.