428 problems

A topological dynamical system is a pair $(X,\tau)$, where $X$ is a compact Hausdorff space and $\tau:X\to X$ is continuous. A bounded sequence $(a_n)_{n\in\mathbb N}$ is determini…

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

Dejean's conjecture.

Let $q$ be a Wang tile set, and let $W_q$ be its tiling space. A tiling $x\in W_q$ is doubly periodic if … for some $n_0,n_1>0$. Wang's conjecture. Every Wang tile set $q$ such tha…

Periodic orbit-closure conjecture. The orbit closure

Lindenstrauss–Tsukamoto conjecture. If

Let $E$ and $F$ be essential graphs, and let $A_E$ and $A_F$ be their adjacency matrices. Write $A_E\sim_{SE}A_F$ for shift equivalence and $A_E\sim_{SSE}A_F$ for strong shift equi…

Let $G$ be a finitely generated group. The Ballier–Stein domino problem conjecture. The domino problem for $G$ is decidable if and only if $G$ is virtually free. This conjecture pr…

Let $G$ be a finitely generated group. A strongly aperiodic subshift of finite type (SFT) is an SFT with no periodic configurations. The strongly aperiodic SFT conjecture. $G$ admi…

Entropy-minimal symbolic characterization conjecture. There exists an entropy minimal subshift

Closed-orbit density conjecture. The density of closed orbit measures holds for any transitive piecewise monotonic map with positive topological entropy.

Soficity conjecture. The subshift $H$ is sofic if and only if $H^\ast$ is sofic.

A $C^r$-diffeomorphism is a diffeomorphism of class $C^r$, where $r$ is a real number satisfying $r\geq 2$. A symbolic extension of a dynamical system is a symbolic dynamical syste…

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

Let $G$ be a finitely generated group. A weakly aperiodic subshift of finite type (SFT) is an SFT with no finite-orbit requirement beyond the absence of a fully periodic configurat…

Intrinsic ergodicity conjecture. The answer to the preceding question is positive at least for binary subshifts, and every bounded density shift is intrinsically ergodic.

Let $x_{3/2}$ be the binary Thue–Morse sequence in base $3/2$, and let $\mu[w]$ denote the frequency of a finite word $w$ occurring in it, when that frequency exists. Word-frequenc…

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 $G$ be a group and let $A$ be a finite set. Let $\tau\colon A^G\to A^G$ be a cellular automaton. It is post-surjective if, for every $x,y\in A^G$ such that $y$ is asymptotic to…

Let $\gamma$ be a periodic trajectory in a cyclic quadrilateral tiling, and let $\Omega_{\gamma}$ be the domain bounded by it. Tree conjecture for cyclic quadrilateral tiling billi…

Minimality conjecture. If every mixed state in $\boldsymbol{\mathcal{R}}$ corresponds to a unique probability distribution over symbols, then the corresponding mixed-state presenta…

Let $A$ be a square matrix over a subring $\mathcal R$ of $\mathbb R$. Its nonzero spectrum is the multiset of its nonzero eigenvalues. Boyle–Handelman's strong generalized spectra…

A cellular automaton is a shift-commuting endomorphism of a one-dimensional full shift. Two cellular automata are conjugate if there is a shift-commuting homeomorphism intertwining…

Aperiodic-word complexity conjecture. For $n$ sufficiently large, the complexity of each individual aperiodic word is