On the Connected Components of a Certain Class of Blaschke Products in the Uniform Topology

Consider the space \(\mathcal {F}\) $\mathcal{F}$ of all inner functions on the unit open disk under the uniform topology, which is a metric topology induced by the \(H^{\infty }\) $H^{\infty }$ -norm. In the present paper, a class of Blaschke products, denoted by \(\mathcal {H}_{SC}\) ${\mathcal{H}}_{\mathit{SC}}$ , is introduced. We prove that for each \(B\in \mathcal {H}_{SC}\) $B\in {\mathcal{H}}_{\mathit{SC}}$ , B and zB belong to the same path-connected component of \(\mathcal {F}\) $\mathcal{F}$ . It plays an important role of a method to select a suitable subsequence of zeros. As a byproduct, we obtain that each Blaschke product in \(\mathcal {H}_{SC}\) ${\mathcal{H}}_{\mathit{SC}}$ has an interpolating and one-component factor.