Portrait
The functional graph of f on 𝔽_p. Each connected component has a unique cycle (periodic orbit) with trees hanging off each cycle node.
● Periodic (on cycle)
● Preperiodic (tails)
★ Fixed point (period 1)
Theory
Fixed pts: solutions to f(x)=x in 𝔽_p.
Periodic orbit of period n: f^n(x)=x, f^k(x)≠x for k<n.
Portrait: digraph Γ_f with edges x→f(x). Rho-shaped components.
For f(x)=x²: fixed pts are 0,1 (and -1 if p≡3 mod 4 gives √(-1) issues). Period divides ord(2) in (ℤ/p)*.