Iterates of Blaschke products and Peano curves

2021/11/18 by Donaire, Juan Jesús, Nicolau, Artur
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.09828

Abstract

Let f be a finite Blaschke product with f(0)=0 which is not a rotation and let fⁿ be its n-th iterate. Given a sequence \a_n\ of complex numbers consider F= ∑ a_n fⁿ. If \a_n\ tends to 0 but ∑ |a_n| = ∞, we prove that for any complex number w there exists a point ξ in the unit circle such that ∑ a_nfⁿ(ξ) converges and its sum is w. If ∑ |a_n| < ∞ and the convergence is slow enough in a certain precise sense, then the image of the unit circle by F has a non empty interior. The proofs are based on inductive constructions which use the beautiful interplay between the dynamics of f as a selfmapping of the unit circle and those as a selfmapping of the unit disc.

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