Ergodic complex plane and cylinder transformation on a periodic time scale

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. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/04/abstr.html