John M. Davis, Billy J. Jackson, Dylan Poulsen
Abstract:
We introduce the ergodic complex plane, a global analogue of Hilger's local complex
plane on time scales, which simultaneously encodes exponential growth and frequency.
Averaging the cylinder transformation on a periodic time scale leads to the notions
of ergodic growth rate and ergodic frequency, unifying local and global stability
perspectives. This yields the ergodic cylinder transformation, a univalent map inducing
an orthogonal curvilinear coordinate system on the regressive complex plane. Within
this framework, we develop a decomposition analogous to Hilger's real and imaginary
parts, and define the box plus operation, extending the circle plus operation globally.
Submitted November 10, 2025. Published January 12, 2026.
Math Subject Classifications: 93D05, 93D30, 37B25, 34N05, 26E70.
Key Words: Time scales; frequency; Hilger complex plane; stability.
DOI: 10.58997/ejde.2026.04
Show me the PDF file (1468 KB), TEX file for this article.
| John M. Davis Department of Mathematics Baylor University Waco, TX 76798, USA email: John_M_Davis@baylor.edu | |
| Billy J. Jackson Department of Mathematics, Statistics, and Computer Science University of Illinois Chicago, IL 60607, USA email: bjjack06@uic.edu | |
 |
Dylan Poulsen Department of Mathematics and Computer Science Washington College Chestertown, MD 21620, USA email: dpoulsen2@washcoll.edu |
Return to the EJDE web page