21 problems

Let $\sigma$ be an irreducible Pisot unit substitution, let $X_\sigma$ be its associated subshift, and let the Rauzy fractal be the compact geometric representation constructed fro…

Let $Til(p/1)$ denote the rational tilings in the family with denominator $1$. In these tilings, the eigenvalues controlling fluctuations are all less than $1$, and in $Til(2)$ thi…

Let $Til(p/q)$ denote the rational tilings in the family of substitution systems described in the paper, with $p/q$ rational. The previously established case $Til(1/2)$ has triangl…

Consider the substitution tilings $Til(p/q)$ and the population matrix governing their tile-size distributions. Let the second eigenvalue mean the second eigenvalue of this matrix.…

Let $\theta$ be a primitive substitution on a finite alphabet, and suppose that none of the eigenvalues associated with $\theta$ has magnitude equal to one. Consider the $[?]$ and…

Model-set conjecture. Every unimodular substitution of Pisot type gives rise to a regular model set, though not necessarily a generic one.

Consider the substitution system and associated operator from the paper, with tile-counting function $\#_{\boldsymbol a}(n)$, recurrence from Lemma on the Catalan-like counting seq…

Let ${\boldsymbol a}=(1,9,9,9,\ldots)$, and let $\#_{\boldsymbol a}(n)$ denote its tile-counting function. Asymptotic conjecture. As $n$ tends to infinity, … The first two terms ar…

Generalization of the Pisot substitution conjecture. Then $s$ is good. The source presents this as a possible generalization of the Pisot substitution conjecture, and the supplied…

Let $\sigma$ be an irreducible Pisot unimodular substitution, and let $\sigma^\omega$ denote its associated directive sequence. The property of being good is the property asserted…

An irreducible Pisot substitution is a substitution whose incidence matrix has an irreducible characteristic polynomial with a dominant Pisot eigenvalue. Its associated subshift is…

A substitution is said to satisfy the strong coincidence condition when the corresponding iterated letter maps satisfy the strong coincidence property for non-constant-length subst…

Sharpness conjecture. The quantity

Iteration-time conjecture. The integer $n$ is close to the bound obtained in the iteration estimate.

Starting from the $n\!\times\! n$ Fourier matrix with $n>1$, consider the substitution on $n^2$ letters described above, with $n$ basic letters occurring in $n$ bar-flavours, and l…

Let $\sigma$ be a unimodular irreducible Pisot substitution, let $M_{\sigma}$ be its incidence matrix, and let $p(x)$ be the characteristic polynomial of $M_{\sigma}$. Let $q(x)$ b…

Let $\sigma$ be an irreducible unimodular Pisot substitution. For any two symbols $i,j$ in its alphabet, the strong coincidence condition on prefixes (respectively, on suffixes) me…

Let $\sigma$ be a substitution on an alphabet of $k$ symbols, with dynamical system $(\Omega_u,S)$ associated to a fixed point, and suppose that $\sigma$ is unimodular and irreduci…

Class P conjecture. If $x$ is a palindromic word fixed by a primitive substitution, then

Let $\sigma$ be a primitive aperiodic substitution, and let $X_\sigma$ be its associated subshift. The automorphism group $\operatorname{Aut}(X_\sigma)$ contains the subgroup of sh…

Cortez–Solomyak's invertibility conjecture. The NBC hypothesis may be omitted from part (2) of the theorem: the substitution map $\mathcal G$ is always invertible on the set of non…