math.DS — Dynamical Systems

Globally periodic rational difference equations of arbitrary order

Contributed by John Rudge

A rational difference equation of order $k$ with nonnegative coefficients is globally periodic if every solution with positive initial data repeats with a common period. Five such equations have long been known, among them the Lyness equation $x_{n+2}=(1+x_{n+1})/x_n$, and a problem of Gasull asks whether every globally periodic equation of this kind is obtained from one of the five by rescaling and unfolding. The answer was known to be yes for $k\in\{1,2,3,4,5,7,9,11\}$. Here I show that it is yes at every order. The proof rests on two ideas. The first is a tropical degeneration: global periodicity forces an associated min-plus recursion to be periodic, and this is refuted, for every choice of coefficients at once, by an unbounded solution of the min-plus recursion linearised along a periodic orbit. The second is the $\sigma$-bump, a linearised solution whose existence reduces to a combinatorial condition on residues modulo $2k$; in the hardest case that condition comes down to a lemma on three-term progressions under unit multipliers modulo $2k$. The proof, including its finite computations, has been formalised in the Lean proof assistant.