On the path properties of a lacunary power series
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- by Gerd Jensen, Christian Pommerenke and Jorge M. Ramírez PDF
- Proc. Amer. Math. Soc. 142 (2014), 1591-1606 Request permission
Abstract:
A power series $f(z)$ which converges in $\mathbb {D}=\{|z|<1\}$ 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.References
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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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3168466