7 problems

The $\mathcal{S}$-adic Pisot conjecture. The tiling condition holds, and $(X_{\boldsymbol{\sigma}}, \Sigma, \mu)$ has pure discrete spectrum. This conjecture generalizes the classi…

Let a dynamical system be defined by a Pisot substitution on $d$ letters. A dynamical system has pure discrete spectrum when its spectrum is pure discrete. Pure discrete spectrum c…

45-state automaton conjecture. For all $k\geq4$, the minimal automaton of the regular language $\overset{\circ}{\,{}^{t}\!L^{s_k}_{a,a}}$ has $45$ states.

Natural-alphabet interior conjecture. For all irreducible unit Pisot substitutions $s$, we have

Minimal-alphabet interior conjecture. For all irreducible unit Pisot matrices $M$ with spectral radius $\beta$, we have

A Pisot substitution is called homological when it satisfies the topological condition on the first rational Čech cohomology of its tiling space introduced for reducible non-unimod…

Let $\sigma$ be an irreducible Pisot substitution, and let $K_\sigma$ be its associated representation space. Self-replicating tiling conjecture. Every irreducible Pisot substituti…