35 problems

Let a closed manifold carry a geodesic flow that is analytically integrable, meaning integrable by real-analytic first integrals in the sense used in the source, and let…

Let $T$ be a trellis and let $f$ be a trellis map for $T$. Write $N_\infty(f)$ for the asymptotic Nielsen number and $h_\mathit{top}$ for topological entropy. Optimality conjecture…

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…

Entropy-or-flatness conjecture. Either $M$ is flat or its geodesic flow has positive topological entropy.

Alternation conjecture. Braids with maximal entropy are alternated. The conjecture concerns the observed structure of entropy-maximizing braids among those considered by word lengt…

Strand-number conjecture. Braids of maximal entropy belong to $\operatorname{Br}(3)$ or $\operatorname{Br}(4)$. The paper presents this as an empirical conjecture obtained from its…

Let $\mathcal{B}$ be the defining set of the $\mathcal{B}$-free system, and let $X_{\mathcal{B},3}$ denote the Möbius-$\mathcal{B}$-free dynamical system with shift map $S$. Möbius…

Homologically undetectable entropy. The minimal entropy of any $g\in\operatorname{Diff}(M)$ isotopic to $f_T$ satisfies

Let $(X,T)$ be a dynamical system. A factor is nontrivial if it is not a one-point dynamical system. Topological entropy measures orbit complexity, while mean dimension measures th…

Positive topological entropy. Every topological Anosov flow $\theta\in\mathcal{TA}\mathcal{F}(M)$ has

Entropy-minimal subshift conjecture. There exists an entropy minimal subshift representing $A$ if and only if, for every arithmetic progression $P$,

Surface-metric entropy stability conjecture. The functional $h_{\rm top}$ is lower semi-continuous with respect to $\overline{d}_{C^0}$.

Reeb-flow entropy stability conjecture. The functional $h_{\rm top}$ is lower semi-continuous with respect to $d_{C^0}$.

Let ${\mathcal P}$ be the parameter space of pairs $(a,b)$, let $f_{a,b}$ denote the corresponding $(a,b)$-continued fraction transformation, and let $h_{\rm top}(f_{a,b})$ denote…

Let ${\mathcal P}$ be the parameter space of pairs $(a,b)$, let $f_{a,b}$ denote the corresponding $(a,b)$-continued fraction transformation, and let $h_{\rm top}(f_{a,b})$ denote…

Let ${\mathcal P}$ be the parameter space of pairs $(a,b)$, let $f_{a,b}$ denote the corresponding $(a,b)$-continued fraction transformation, and let $h_{\rm top}(f_{a,b})$ denote…

Let $k$ be a complete metrized field, let $X$ be a projective variety defined over $k$, and let $f\colon X\dashrightarrow X$ be a dominant rational map. For any complete metrized f…

Let $k$ be any complete non-Archimedean field, and let $X$ be a projective $k$-variety. A morphism $f\colon X\to X$ is polarized if there exists an ample line bundle $L\to X$ and a…

Let $\mathcal{Q}(b)$ be the intersection of two unit disks with center distance $b\in(0,2)$, and let … be the billiard map on the lemon table $\mathcal{Q}(b)$. Lemon billiard entro…

Let $\mathbf{G}$ be the countably infinite set of regular maps on the torus, where each $\mathcal{G}\in\mathbf{G}$ is an embedded graph in the torus that is arc-transitive. For eac…

Let $Q$ be a closed surface, let $g$ be a Riemannian metric on $Q$, and write $h_{\rm top}(g)$ for the topological entropy of its geodesic flow. Say that $h_{\rm top}(g)$ is robust…

Tresser's conjecture. Generically, maps in the boundary of positive topological entropy have a set of periodic orbits which, except for a finite subset, consists of infinitely many…

Let $S$ be a cancellative left amenable monoid acting on the compact abelian group $K$ on the left by $\gamma$. Let $\widehat\gamma$ denote the induced dual action on the Pontryagi…

Let $G$ be a nilpotent locally compact $p$-group, and let $\mathrm{rank}_p(G)$ denote its $p$-rank. The finite-rank conjecture. If … then $G\in\mathfrak E_{<\infty}$, meaning that…

Hyperbolic-density conjecture. In the space of polynomials of degree $d>2$ with all critical points real, there are no isentropes of entropy