The Graph Structure of a Class of Permutation Maps over Ring $\mathbb{Z}_{p^k}$

Understanding the periodic and structural properties of permutation maps over residue rings such as $\mathbb{Z}_{p^k}$ is a foundational challenge in algebraic dynamics and pseudorandom sequence analysis. Despite notable progress in characterizing global periods, a critical bottleneck remains: the lack of explicit tools to analyze local cycle structures and their evolution with increasing arithmetic precision. In this work, we propose a unified analytical framework to systematically derive the distribution of cycle lengths for a class of permutation maps over $\mathbb{Z}_{p^k}$. The approach combines techniques from generating functions, minimal polynomials, and lifting theory to track how the cycle structure adapts as the modulus $p^k$ changes. To validate the generality and effectiveness of our method, we apply it to the well-known Cat map as a canonical example, revealing the exact patterns governing its cycle formation and transition. This analysis not only provides rigorous explanations for experimentally observed regularities in fixed-point implementations of such maps but also lays a theoretical foundation for evaluating the randomness and dynamical behavior of pseudorandom number sequences generated by other nonlinear maps. The results have broad implications for secure system design, computational number theory, and symbolic dynamics.