156 problems

Non-factorability conjecture. On a large class of groups, not every SI SFT is an SFT factor of a contractible SFT.

Second minimal odd-orbit digraph conjecture. For every integer $k\geq 3$, the digraph of any second minimal $(2k+1)$-orbit has one of $4k-3$ possible types up to inverses.

The linear repetitivity conjecture. Every aperiodic linearly repetitive Delone set is densely repetitive.

Let $\mathcal{T}_{\chi}$ and $\mathcal{T}_{\psi}$ be the tiling spaces associated with substitutions $\chi$ and $\psi$, and let $\tilde{\chi}$ and $\tilde{\psi}$ denote the associa…

Let $\mathcal{M}$ be the class of maps under consideration, let $\operatorname{invlim} f$ denote the inverse limit of $f$, let $\chi_f$ be the substitution associated with $f$, and…

Let $S$ be a finite set, let $d>1$, and let $X=S^{\mathbb{Z}^d}$. A $d$-dimensional cellular automaton is a continuous map $T:X\to X$ commuting with the shift action of…

Let ${\bf f}=f_0f_1f_2\cdots$ be a paperfolding word over $\{1,4\}$, and define ${\bf v}=v_0v_1v_2\cdots$ by … for all $n\geq 0$, with the even-position symbols of ${\bf f}$ recode…

Low-growth existence conjecture. There exists $N>1$ and a surjective cellular automaton $f$ on $N$ symbols such that $\nu(f,S_N)<N$. The surrounding discussion poses related quanti…

Let $\Sigma_N$ be the space of doubly infinite sequences over an alphabet of $N$ symbols, and let $f:\Sigma_N\to\Sigma_N$ be a surjective one-dimensional cellular automaton. Let…

Almost one-to-one coding conjecture. A Pisot automorphism with associated Pisot number $\beta$ admits an almost $1$-to-$1$ coding by $S_\beta$.

Sturmian-region conjectures. (i) If $-4\leq r-s\leq3$, then, except for $(r,s)=(4,1)$, $w_{r,s}$ is Sturmian. (ii) If $r-s=4$, then $w_{r,s}$ is Sturmian if and only if $r$ is even…

The decoration conjecture. If $q<q'$ and $w\succcurlyeq w'$, then $P_q^w$ forces $P_{q'}^{w'}$. Moreover, $P_q^w$ and $P_{q'}^{w'}$ have the same braid type if and only if $q=q'$ a…

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 $\beta$ be a Pisot unit, meaning an algebraic integer greater than $1$ whose other algebraic conjugates have modulus less than $1$ and whose norm is a unit. Call $\beta$ weakly…

Periodic orbit-closure conjecture. The orbit closure

Nivat's conjecture. Then $c$ is $1$-periodic.

Let $\mathcal A$ be a finite alphabet with at least two elements, and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a two-dimensional configuration. For $k,n\in\mathbb N$, let…

Let $w$ be a binary word. A balanced border is a border of $w$ whose numbers of $0$s and $1$s are equal; the statistics $\rho^w$ and $q^w$ are the letter-frequency quantities assoc…

Let $n\in\mathbb{Z}_{>0}$ be odd. Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the circle, and let $E_2:\mathbb{T}\to\mathbb{T}$ be the doubling map. For observables … consider measur…

Rauzy's conjecture. Except for very particular vectors of letter frequencies, there do not exist any infinite ternary words with constant abelian complexity equal to $3$.

Let $(\mathcal{T},\sigma,M)$ be a primitive substitution tiling system of $\mathbb{R}^d$ with finite local complexity. Call it conservation-forced when it satisfies the conservatio…

A substitution system is a tiling system with finitely many tile types and a substitution rule, and Algorithm is the paper's refinement procedure that augments each tile type with…

A substitution system is a tiling system equipped with tile substitutions; it preserves adjacency when, for any two tiles $t,t'$ belonging to distinct supertiles $s,s'$, the tiles…

Let $\mathcal{R}$ be the collection of all maps $\mathcal{F}:[0,1]\to[0,1]$ that serve as factor maps for first-return maps $\psi_v:R\setminus W\to R$ of the Lorenz attractor, not…

Periodic coloring conjecture. For any $V$, $N$, $\Sigma$, and $\Omega$ as above, the solution set $\operatorname{Coloring}(V,N,\Sigma,\Omega)$ is not aperiodic.