Pisot substitution conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $A$ be a finite alphabet. A substitution $\sigma:A\to A^+$ maps every letter to a nonempty finite word and extends by concatenation. Its incidence matrix is $M_{ab}=|\sigma(b)|_a$, the number of occurrences of $a$ in $\sigma(b)$. It is an irreducible Pisot substitution if the characteristic polynomial of $M$ is irreducible over $\mathbb Q$ and its Perron--Frobenius eigenvalue $\beta$ is a Pisot number: an algebraic integer $\beta>1$ all of whose other algebraic conjugates have modulus less than one. Such a substitution is primitive. Let $X_\sigma\subset A^{\mathbb Z}$ consist of the bi-infinite sequences whose finite words occur in iterates $\sigma^j(a)$. With the left shift $S$, it has a unique invariant probability measure $\mu_\sigma$. Pure discrete spectrum means that $L^2(X_\sigma,\mu_\sigma)$ is the closed linear span of eigenfunctions of the unitary operator $Uf=f\circ S$; an eigenfunction satisfies $Uf=\lambda f$ for some $|\lambda|=1$. No algebraic-unit or two-letter restriction is imposed.

Statement recorded in the supplied source.

Does $(X_\sigma,\mu_\sigma,S)$ have pure discrete spectrum for every irreducible Pisot substitution $\sigma$? [S2]

Short English statement

Does every irreducible Pisot substitution generate a shift whose measure-theoretic spectrum is entirely discrete?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4600053 (AMR-045-0053). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K. Nakaishi, Pisot Substitution Conjecture and Rauzy Fractals (2024), Introduction, Pisot Substitution Conjecture; Section 2, pp.1 and3. https://arxiv.org/pdf/2401.07771.