On holomorphic self-mappings of the unit disk

On Holomorphic Isometric Embeddings of the Unit Disk into Polydisks

We study the classification of holomorphic isometric embeddings of the unit disk into polydisks. As a corollary of our results, we can give a complete classification when the target is the 2-disk and the 3-disk. We also prove that the holomorphic isometric embeddings between polydisks are induced by those of the unit disk into polydisks.

Ribbon R-trees and holomorphic dynamics on the unit disk

Proper Holomorphic Mappings of the Spectral Unit Ball

We prove an Alexander type theorem for the spectral unit ball Ωn showing that there are no non-trivial proper holomorphic mappings in Ωn, n ≥ 2. Let Mn denote the space of n× n complex matrices. In order to avoid some trivialities and ambiguities we assume in the whole paper that n ≥ 2. Let ρ(A) := max{|λ| : λ ∈ Spec(A)} be the spectral radius of A ∈ Mn. Denote also by Spec(A) := {λ ∈ C : det(A...

Random Holomorphic Iterations and Degenerate Subdomains of the Unit Disk

Given a random sequence of holomorphic maps f1, f2, f3, . . . of the unit disk ∆ to a subdomain X, we consider the compositions Fn = f1 ◦ f2 ◦ . . . fn−1 ◦ fn. The sequence {Fn} is called the iterated function system coming from the sequence f1, f2, f3, . . . . We prove that a sufficient condition on the domain X for all limit functions of any {Fn} to be constant is also necessary. We prove the...

Regularity on the boundary in spaces of holomorphic functions on the unit disk

We review some results on regularity on the boundary in spaces of analytic functions on the unit disk connected with backward shift invariant subspaces in H.