Pisot Conjecture

Is every Pisot substitution a rotation in disguise?

A substitution on a finite alphabet A is a map σ: A → A^* sending each letter to a nonempty word, extended to words by concatenation. Its occurrence matrix M counts letters: M_ab = |σ(b)|_a. The substitution is primitive if some power of M is positive, and Pisot if the characteristic polynomial of M is irreducible over ℚ and its Perron–Frobenius eigenvalue λ is a Pisot number: an algebraic integer > 1 whose other conjugates all lie strictly inside the unit circle. A primitive substitution generates a uniquely ergodic subshift (X_σ, μ_σ, S).

The Pisot conjecture says that for every primitive Pisot substitution, (X_σ, μ_σ, S) is measurably conjugate to a rotation on a compact abelian group. Rauzy proved it for the Tribonacci substitution 0→01, 1→02, 2→0 (a rotation on 𝕋²); it is a theorem for two-letter alphabets (Host; Barge–Diamond 2002). From three letters up it is open.

A rotation is rigid (some sequence of powers S^n_k converges to the identity in measure), so a non-rigid Pisot substitution would refute the conjecture. Rigidity leaves a combinatorial fingerprint: writing p(n) for the number of words of length n in the language and q(n) for those that begin and end with the same letter, Donoso–Maass–Radić show that a rigid primitive substitution subshift has lim sup_n q(n)/p(n) = 1. The challenge: find a primitive Pisot substitution on at most 9 letters whose ratio q(n)/p(n) stays bounded away from 1. Every submission is verified exactly — primitivity, the Pisot property, and the complexity profile up to length 1000 — and scored by the largest ratio over 500 < n ≤ 1000. Lower is better.

A low score at finite n is a hint, not a proof: rigid systems approach 1 only along sparse subsequences, and every Pisot substitution known so far is believed to be rigid. The database exists to make the search legible — and to collect the complexity profiles for whoever finds the pattern.