Geodesic
Generic power series on subsets of the unit disk
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 637-652
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We examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda $ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a nonreal boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb {C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.
We examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda $ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a nonreal boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb {C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.
DOI : 10.21136/CMJ.2022.0021-21
Classification : 28A05, 30B30, 54H05
Keywords: complex power series; boundary behaviour; Baire category 
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Maga, Balázs; Maga, Péter. Generic power series on subsets of the unit disk. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 637-652. doi: 10.21136/CMJ.2022.0021-21

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