Distortion and Distribution of Sets Under Inner Functions

Matteo Levi, Artur Nicolau, Odí Soler i Gibert

Research output: Contribution to journalArticleResearch


© 2019, Mathematica Josephina, Inc. It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied by Fernández and Pestana. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. We use our results to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half-plane.
Original languageEnglish
JournalJournal of Geometric Analysis
Publication statusPublished - 1 Jan 2019


  • Angular derivatives
  • Boundary fixed points
  • Hausdorff contents
  • Inner functions


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