TL;DR
This paper investigates the density properties of exponential of exponential transformations of lines in the complex plane, revealing conditions under which these transformations are dense or not dense.
Contribution
It characterizes the density and non-density of exponential of exponential of lines in the complex plane based on their angles and Hausdorff dimension.
Findings
Exponential of exponential of almost every line is dense in the complex plane.
For lines through a point, certain sets of angles with Hausdorff dimension one lead to non-dense images.
The results connect geometric properties of lines with complex exponential transformations.
Abstract
Exponential of exponential of almost every line in the complex plane is dense in the plane. On the other hand, for lines through any point, for a set of angles of Hausdorff dimension one, exponential of exponential of a line with angle from that set is not dense in the plane.
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