Arithmetic Dynamics — Iteration over Finite Fields

Iterate a rational map f : 𝔽_p → 𝔽_p over a finite field of prime order p. Every point eventually enters a cycle (by finiteness). The functional graph of f decomposes into cycles with "rho"-shaped tails — a structure analogous to Pollard's rho algorithm. Watch fixed points (period 1), cycles, and preperiodic tails.

Functional Graph

Cycle Structure

Orbit Length Histogram