88 problems

Zero-entropy conjecture. One has

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…

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 $\alpha$ be a $\mathbb{Z}_+^2$-action on the circle $\mathbb{S}^1$ whose generators are … where $p$ and $q$ are coprime integers. Let $\sigma_\alpha$ be the shift map on the in…

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 a Riemannian manifold carry an integrable geodesic flow, and denote the flow's topological entropy by $h_{\mathrm{top}}$ and the manifold's fundamental group by $\pi_1(M)$. A f…

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…

Contractibility conjecture. The set $\{h=h_0\}$ is contractible.

Monotonicity conjecture. The topological entropy of a real cubic map depends monotonically on its parameters, in the sense that every locus of constant entropy in parameter space i…

Let $M$ be a manifold of dimension $2$, let $f:M\rightarrow M$ be a homeomorphism, and let ${\cal S}(M,f)$ denote the space of strictly ergodic sets associated with $f$. A positive…

Let $g:[0,1]\rightarrow [0,1]$ be a mixing map. For any open set $U\subseteq[0,1]$, suppose there exists an integer $n\in\mathbb N$ such that $g^n(U)=[0,1]$. Let $M_g$ denote the s…

Let $s(A,b)$ denote the growth number of the real cubic map $x\mapsto x^3-3Ax+b$, and fix a level set $\{(A,b):s(A,b)=s\}$. A map is hyperbolic when its critical orbits converge to…

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.

A continuous map $f$ of a compact interval is $2$-fold if every point has at least two preimages under $f$. For a continuous map $f$ of a compact interval that is $2$-fold on the w…

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 $n$ be an integer with $n\ge 2$. For a braid $\beta\in \operatorname{Br}(n)$, let $(c_m)_{m\ge 0}$ be the sequence produced by the stated computation and let $h(\beta)$ denote…

Let $f:X\to X$ be a holomorphic or meromorphic self-map of a compact complex manifold $X$, inducing actions $f^*,f_*:H^*(X,\mathbb{R})\to H^*(X,\mathbb{R})$ on the real cohomology…

Topological entropy conjecture. The topological entropy of $c$ should agree with the logarithm of its first dynamical degree:

Ellipticity conjecture. If $M$ admits a Riemannian metric with zero topological entropy, then its universal covering has the rational homotopy type of a finite elliptic CW complex…

Let $(X,\tau)$ be a compact Hausdorff dynamical system, with $\tau$ a continuous self-map, and let $h_{\mathrm{top}}(\tau)$ denote its topological entropy. For every $\tau$-invaria…

Ochem's exponential conjecture. Every nonempty shift space $X_\beta^d$ has positive topological entropy:

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…