Research Article

A Digit-Based Discrete Dynamical System Generated by a Figure- Changing Transformation

Abstract: Digit-based iterative processes often give rise to rich dynamical behavior despite being defined by simple arithmetic rules. In this paper, we investigate a figure-changing transformation acting on four-digit numbers, where each iteration is obtained by shifting digits and appending the digital root of a partial digit sum. We model this procedure as a discrete dynamical system on a finite state space and prove that every orbit eventually becomes periodic. Computational methods implemented in Python are used to perform an exhaustive analysis of all 104 four-digit states. The computation identifies exactly 168 distinct periodic cycles with cycle lengths 1, 2, 6, 8, 18, 24, and 72, and determines the corresponding basin structures. The proposed transformation illustrates how simple digit operations generate highly structured finite dynamical systems, providing a framework for further investigations of arithmetic transformations and their dynamical properties.