20 problems

Let $\mathcal{A}$ be a finite alphabet of Verblunsky coefficients, and let $\Omega\subset \mathcal{A}^{\mathbb{Z}}$ be a minimal subshift that is not periodic. The associated measu…

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

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 $d53c$ be a minimal and aperiodic subshift over an alphabet $d538\subseteq\mathbb{D}$, and let $f:d53c\to\mathbb{D}$ be given by … Let $\Sigma$ denote the associated CMV spectr…

Extender-entropy separation conjecture. The behavior of extender sets, and in particular extender entropy, could provide a way to separate the classes of sofic and effective subshi…

Let $\Sigma_A$ be the shift space associated with the word system, and let $\mathcal{F}$ and $\mathcal{R}$ denote the corresponding sets of forbidden and repetitive words, respecti…

A subshift is a shift-invariant closed subset of a full shift over a finite alphabet, and two subshifts are eventually conjugate if, for all sufficiently large positive integers…

Let $(Y,S)$ be a subshift with a collection $\mathcal C$ having property $F$ and entropy $h(\mathcal C)$. Let $(X,T)$ be a free Borel $\mathbb{Z}^d$ system, and write its Gurevic e…

Let $X\subseteq 2^{\mathbb{Z}}$ be a minimal subshift with zero entropy, and let $\operatorname{Aut}(X,\mathbb{Z})$ denote the group of homeomorphisms of $X$ that commute with the…

Full-shift noncommutativity conjecture. The full vector shift $X$ is not compatible with a nontrivial Lie algebraic subshift structure.

Let $S\subseteq\mathbb{N}\cup\{0\}$. The $S$-gap subshift $X_S$ is the set of bi-infinite sequences over $\{0,1\}$ in which the gap between any two nearest $1$s has length in $S$,…

A decidable subshift is a subshift whose language is decidable, and its irreducibility rate measures its mixing strength. Consider decidable subshifts whose irreducibility rate is…

Let $X$ be a subshift, meaning a closed, shift-invariant subset of $A^{\mathbb{Z}}$ for a finite set $A$. For each $n$, let $F_X(n)$ be the set of distinct follower sets of words o…

Let $\Xi$ be a minimal, aperiodic and uniquely ergodic ${\mathbb Z}^2$ subshift, and let $\Omega$ be its suspension. Let $\lambda=(\lambda_x,\lambda_y)$ be a virtual eigenvalue of…

Let $\mathcal{A}\subset \mathbb{R}$ be finite, and let $\Omega\subset \mathcal{A}^{\mathbb{Z}}$ be a minimal subshift that is not periodic. By minimality, the Schrödinger operators…

Let $X$ be a countable quasiminimal recursive subshift. The model-checking problem asks whether the language of $X$ intersects a given formal language. Locally testable decidabilit…

A universal symbolic system is a symbolic dynamical system capable of simulating every effective symbolic system. Universality conjecture. Every universal symbolic system has infin…

Fix a finite alphabet $S$. For a subshift $X\subset S^\mathbb{X}$, let $h(X)$ denote its topological entropy and let $W(X)$ denote its winning shift. High-entropy winning-shift con…

Weak conjugacy conjecture. The limit set of any stable cellular automaton is weakly conjugate to an SFT.

Equality conjecture. The two quantities are always equal: