Abstract
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.
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Matteo Levi is partially supported by the 2015 PRIN Grant Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis of the Italian Ministry of Education (MIUR). Matteo Levi, Artur Nicolau, and Odí Soler i Gibert are supported in part by the Generalitat de Catalunya (Grant 2017 SGR 395) and the Spanish Ministerio de Ciencia e Innovación (Projects MTM2014-51824-P, MTM2017-85666-P).
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Levi, M., Nicolau, A. & Soler i Gibert, O. Distortion and Distribution of Sets Under Inner Functions. J Geom Anal 30, 4166–4177 (2020). https://doi.org/10.1007/s12220-019-00236-w
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DOI: https://doi.org/10.1007/s12220-019-00236-w