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Preimages under linear combinations of iterates of finite Blaschke products

Spyridon Kakaroumpas, Odí Soler i Gibert

Abstract

Consider a finite Blaschke product $f$ with $f(0) = 0$ which is not a rotation and denote by $f^n$ its $n$-th iterate. Given a sequence $\{a_n\}$ of complex numbers, consider the series $F(z) = \sum_n a_n f^n(z).$ We show that for any $w \in \mathbb{C},$ if $\{a_n\}$ tends to zero but $\sum_n |a_n| = \infty,$ then the set of points $ξ$ in the unit circle for which the series $F$ converges to $w$ has Hausdorff dimension $1.$ Moreover, we prove that this result is optimal in the sense that the conclusion does not hold in general if one considers Hausdorff measures given by any measure function more restrictive than the power functions $t^δ,$ $0 < δ< 1.$

Preimages under linear combinations of iterates of finite Blaschke products

Abstract

Consider a finite Blaschke product with which is not a rotation and denote by its -th iterate. Given a sequence of complex numbers, consider the series We show that for any if tends to zero but then the set of points in the unit circle for which the series converges to has Hausdorff dimension Moreover, we prove that this result is optimal in the sense that the conclusion does not hold in general if one considers Hausdorff measures given by any measure function more restrictive than the power functions
Paper Structure (6 sections, 18 theorems, 205 equations)

This paper contains 6 sections, 18 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Preliminary results
  4. The construction of Cantor like sets
  5. Proof of main results
  6. Theorem \ref{['thm:PaleyWeissDimensionThm']} is optimal

Key Result

Theorem A

Let $f$ be a finite Blaschke product with $f(0) = 0$ which is not a rotation. Consider a sequence $\{a_n\}$ of complex numbers tending to $0$ and such that $\sum_n |a_n| = \infty.$ Then for any $w \in \mathbb{C}$ there exists $\xi \in \partial\mathbb{D}$ such that $\sum_n a_n f^n(\xi)$ converges and

Theorems & Definitions (30)

  • Theorem A: Donaire--Nicolau, ref:DonaireNicolau
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4: ref:NicolauConvergenceIterates
  • Corollary 5
  • proof
  • Lemma 6: ref:DonaireNicolau
  • Corollary 7: ref:DonaireNicolau
  • Corollary 8: ref:DonaireNicolau
  • ...and 20 more