Collatz-Type Dynamics on Gaussian Integers: Fractal Stability Islands, Long Cycles, and Dyadic Rational Structure

We introduce and study a family of Collatz-type dynamical systems defined on the Gaussian integers Z[i]. Each map partitions Z[i] into four residue classes modulo (1 + i) and applies a distinct affine rule to each class, mirroring the parity-based structure of the classical 3n + 1 problem. Among the variants we investigate, Variant E—defined by a fourfold contraction on doubly-even inputs and linear expansion on the remaining three classes—exhibits the richest behavior. Our computational investigation, carried out with exact rational arithmetic to eliminate floating-point error, reveals three principal findings: (i) a stable periodic orbit of length 40, verified exactly; every element of this orbit lies in Z[1/2][i] and denominators (as powers of 2) reach hundreds of digits in magnitude; (ii) the natural ambient space of Variant E is not Z[i] but the ring of dyadic Gaussian rationals Z[1/2][i], a consequence of the z/4 contraction rule; we prove that denominators are always powers of 2; (iii) the boundary of the stability islands is fractal, with rigorously estimated box-counting dimension D ≈ 1.70 ± 0.05 (R² = 0.983), placing it well above the Koch snowflake (D ≈ 1.262) and approaching the complexity of the Mandelbrot boundary. We also study a complementary Variant C. Lyapunov exponent analysis over 1000 orbits yields a mean exponent λ ≈ −0.66, with an entirely negative distribution, providing strong numerical evidence for global convergence to zero. The stability threshold is identified at k > √2, consistent with our heuristic analysis. We state several conjectures and open questions for rigorous proof.