22 problems

Let $\zeta$ be a primitive irreducible unimodular Pisot substitution over an alphabet ${\mathcal{A}}$. It satisfies the strong coincidence condition if every pair of letters in…

Strong coincidence conjecture. If a substitution is irreducible Pisot then it is strongly coincident.

Let $\varphi$ be a Pisot substitution on $n$ letters. Let $\tilde\varphi$ be a rewriting of $\varphi$ such that, for some letters $b,e$ in the rewritten alphabet,…

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $c$ be a positive integer. A minimal speedup with orbit number $c$ means a minimal bounded speedup $(X,S)$ of…

Let $\theta$ be a constant-length left proper substitution and let $c>1$ satisfy $\gcd(c,|\theta|)=1$. If $l$ is a factor of $c$ with $1\leq l<c$, then $(X_\theta,T^l)$ and…

Delone-set and overlapping-substitution conjecture. For an appropriate choice of $F$ and $A$, this set is a Delone set of $\Omega$ with inflation symmetry, and there is a correspon…

Pisot inflation conjecture. The system $cS$ has pure point powder diffraction, and consequently the main theorem applies to its associated point process.

Let $C_a(X)$ denote the characteristic function associated with the first letter $a$ of the Fibonacci substitution. The quantities $_n$ measure the left endpoints of intervals on w…

Let $C_a(X)$ be the characteristic function associated with the first letter $a$ of the Fibonacci substitution. The preceding results show that $C_a(X)$ has no zeroes on…

BDL representation conjecture. Such a geometric representation with BDL property exists if and only if at least one eigenvalue of $M_{\psi}$ has modulus strictly smaller than $1$.

Let $X$ be the minimal dynamical system generated by the substreetution tree ${\mathfrak J}$. For $\mathfrak{A}\in X$, define … Let $R_n$ be the number of patches of size $n$ appea…

Let $X$ be the minimal dynamical system generated by the fixed substreetution tree ${\mathfrak J}$, and let $p(n,\mathfrak{A})$ denote the number of $n$-step preimages of…

Guo–Han–Wu's conjecture. Fixed points of substitutions of non-constant length on $\{0,1\}$ cannot be apwenian.

Let $\mathcal A$ be an ordered alphabet, and let $\varphi_0$ and $\varphi_1$ be primitive morphisms with the same incidence matrix. Let ${\bf u}_0$ and ${\bf u}_1$ be one-sided fix…

Let … and … be the two Pisot substitutions. Sellami–Sirvent's conjecture. The balanced pair algorithm fails for the pair $(\varphi_0,\varphi_1)$. This is a specific example concern…

Infinite diagonal types conjecture. There are infinitely many types of squares and rectangles on the diagonal.

Measurable toral-translation conjecture. The dynamical system $(\Omega_{\sigma},S)$ is measurably equivalent to a toral translation.

Let $\mathbf w$ and $\mathbf w'$ be fixed points of an irreducible Pisot substitution $\varphi$. A letter $a$ and factorizations … are such that the finite prefixes $p$ and $p'$ ar…

Irrationality conjecture for $\alpha_{**}$. The number $\alpha_{**}$ is irrational.

Let $\sigma$ be a substitution over a finite alphabet $\mathcal{A}$. The substitution is of Pisot type if its incidence matrix has a dominant Pisot eigenvalue satisfying the hypoth…

Independent-substitution conjecture. Then $x$ is ultimately periodic.

Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers, let $A$ be a finite alphabet, and let ${\bf x}\in A^{\mathbb{N}}$ be a sequence. Hansel's conjecture. T…