Almost no points are periodic, and the exception is exactly the polynomials you'd guess.
Reduce a polynomial modulo a prime. Over a finite field, every point is eventually periodic (the state space is finite). The question: what proportion of points are actually periodic? As the prime grows, does this proportion converge?
Complete classification (arXiv:2603.21620): for polynomials of degree ≥2, the limit inferior of the proportion of periodic points modulo primes goes to zero — unless the polynomial is linearly conjugate to a Chebyshev polynomial. Chebyshev polynomials are the unique exception. Their proportion of periodic points remains bounded away from zero because their dynamics are semiconjugate to multiplication maps on algebraic groups, which have many periodic points for structural reasons.
This resolves a 2014 question of Juul, Kurlberg, Madhu, and Tucker. The earlier partial results covered specific polynomial classes; this paper addresses the remaining cases to complete the picture.
The structural insight: the “typical” polynomial has vanishingly few periodic points modulo large primes. Periodicity is the exception, not the rule. The only polynomials that maintain a positive fraction of periodic points are those whose dynamics reduce to group multiplication — the most algebraically structured maps possible. Chaos (in the sense of generic polynomial dynamics) destroys periodicity. Structure (in the sense of algebraic group action) preserves it. The proportion of periodic points is a measure of how much hidden algebraic structure the dynamics contain.