Finite Realization of Periodic-Point Tables
Abstract
A finite cyclic action realizes a divisor table exactly when its Mobius data count whole cycles.
Theorem 1.1 (The finite divisor-table criterion).
Proof. Machine-checked in Lean as D5/S1/Dynamics/PeriodicOrbits/FiniteRealization.finite_realization_iff (✓ std3). ∎
Citation. Yash Puri and Thomas Ward (2001). Arithmetic and growth of periodic orbits. URL: https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.tex.
Commentary.
Fix a positive natural M and a natural-valued table a. There is a permutation of Fin (a M), with Mth power the identity and fixed count a d for every divisor d of M, exactly when the integer Mobius transform at each such d is nonnegative and divisible by d. In the formula A(sigma,d) is the actual fixedCount, and delta is the integer factor-pair Mobius transform defined in OrbitCounting.
The converse takes delta(a,d) divided by d copies of the cyclic set ZMod d and advances every copy by one. Mobius inversion gives the required fixed counts. At time M every point returns, so the total carrier has cardinality a M and can be relabeled by Fin (a M). This is a finite restriction of the classical exact-realization criterion.
The table’s value at zero and its values away from divisors of M are unused. The empty carrier, M equal to one, and singleton cycles are included. The permutation’s order divides M; equality of the order with M and transitivity are not asserted. Positive-time fixed counts extend along the greatest common divisor with M, so arbitrary prescribed outside-divisor values cannot be inferred.