Iterates of Blaschke products and Peano curves
Juan Jes\'us Donaire, Artur Nicolau

TL;DR
This paper studies the boundary behavior of sums of iterates of finite Blaschke products, showing how they can approximate any complex number and under certain conditions, their images form sets with interior.
Contribution
It provides new results on the convergence and image properties of series involving iterates of Blaschke products, linking complex dynamics with boundary value problems.
Findings
Any complex number can be approximated by sums of iterates at some boundary point.
The image of the unit circle can have a non-empty interior under certain convergence conditions.
The proofs use inductive constructions exploiting the dynamics of Blaschke products.
Abstract
Let be a finite Blaschke product with which is not a rotation and let be its -th iterate. Given a sequence of complex numbers consider . If tends to but , we prove that for any complex number there exists a point in the unit circle such that converges and its sum is . If and the convergence is slow enough in a certain precise sense, then the image of the unit circle by has a non empty interior. The proofs are based on inductive constructions which use the beautiful interplay between the dynamics of as a selfmapping of the unit circle and those as a selfmapping of the unit disc.
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