Parole chiave

• holomorphic function
• infinite composition
• Blaschke product

Abstract Let $\mathbb{D}\subseteq\mathbb{C}$ denote the open unit disk. Composition is a natural operation on the space $H(\mathbb{D})$ of all holomorphic functions $f:\mathbb{D}\to\mathbb{D}$: given $f_1, f_2\in H(\mathbb{D})$, then also ${f_1\circ f_2\in H(\mathbb{D})}$. The same is true for infinite (left) compositions of these functions in the following sense: if for given $(f_n)\subseteq H(\mathbb{D})$ we set $F_n:=f_n\circ \dots \circ f_1$, then every nonconstant limit function of $(F_n)$ belongs to $H(\mathbb{D})$. There are several composition-invariant subsets of $H(\mathbb{D})$, and it makes sense to ask whether those are also stable under infinite compositions. A result by G. Ferreira says that this is true for the subset of inner functions whereas it fails for the subset of finite Blaschke products. In this talk we discuss yet two more composition-invariant subsets of $H(\mathbb{D})$, the sets of indestructible and of maximal Blaschke products, and show that these are stable under infinite compositions, too.

This is joint work with Daniela Kraus and Oliver Roth.