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On the path properties of a lacunary power series


Authors: Gerd Jensen, Christian Pommerenke and Jorge M. Ramírez
Journal: Proc. Amer. Math. Soc. 142 (2014), 1591-1606
MSC (2010): Primary 30B10, 60G17, 60J65
DOI: https://doi.org/10.1090/S0002-9939-2014-12077-6
Published electronically: February 10, 2014
MathSciNet review: 3168466
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Abstract: A power series $ f(z)$ which converges in $ \mathbb{D}=\{\vert z\vert<1\}$ maps the radii $ [0,\zeta )$ onto paths $ \Gamma (\zeta )$, $ \zeta \in \mathbb{T}=\partial \mathbb{D}$. These are studied under several aspects in the case of the special lacunary series $ f(z)=z+z^2+z^4+z^8+\ldots \,$. First, the $ \Gamma (\zeta )$ are considered as random functions on the probability space $ (\mathbb{T},\mathscr {B},\mathrm {mes}/2\pi )$, where $ \mathscr {B}$ is the $ \sigma $-algebra of Borel sets and $ \mathrm {mes}$ the Lebesgue measure. Then analytical properties of the $ \Gamma (\zeta )$ are discussed which hold on subsets $ A$ of $ \mathbb{T}$ with Hausdorff dimension 1 in spite of $ \mathrm {mes}{A}=0$. Furthermore, estimates of the derivative of $ f$ and of the arc length of sections of the $ \Gamma (\zeta )$ are given. Finally, these results are used to derive connections between the distribution of critical points of $ f$ and the overall behaviour of the paths.


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Additional Information

Gerd Jensen
Affiliation: Sensburger Allee 22a, D-14055 Berlin, Germany
Email: cg.jensen@arcor.de

Christian Pommerenke
Affiliation: Institut für Mathematik, Technische Universität, D-10623 Berlin, Germany
Email: pommeren@math.tu-berlin.de

Jorge M. Ramírez
Affiliation: Universidad Nacional de Colombia, Medellín, Colombia
Email: jmramirezo@unal.edu.co

DOI: https://doi.org/10.1090/S0002-9939-2014-12077-6
Keywords: Lacunary series, Brownian motion, path properties, critical points, Bloch functions
Received by editor(s): May 22, 2012
Received by editor(s) in revised form: June 1, 2012
Published electronically: February 10, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.