84 problems

Let $m\in\mathbb N$, and let a minimal $\mathbb Z$-system mean a minimal dynamical system with a continuous $\mathbb Z$-action. Its maximal equicontinuous factor is the maximal fac…

The conjecture.

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with…

Leibman's conjecture. The orbit closure

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Projective compactification no-retraction conjecture. Every projective $\Gamma$-compactification with non-trivial boundary $X$ is non-Dirac. In particular, there does not exist a r…

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $(X,S)$ be a bounded speedup of $(X,T)$ with orbit number $c$. For the Kakutani–Rokhlin partition…

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $c$ be a positive integer. A minimal speedup with orbit number $c$ means a minimal bounded speedup $(X,S)$ of…

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…

Characterization of smoothness. The flow $\theta$ is topologically conjugate to a smooth Anosov flow generated by some $Y\in\operatorname{An}(M)$ if and only if $X$ belongs to the…

For each route to chaos $C$, let $[C]$ be its boundary, namely the collection of limit dilatations of sequences drawn from the dilatation sets of the stages of $C$. Let $T_{chaos}$…

Let $C$ and $C'$ be different routes to chaos, and let their Conformal Index be the endpoint invariant associated with each route. Say that $C$ and $C'$ define the same braid types…

A route to chaos is a sequence of dynamical systems connected by an isotopy, and its Index-Invariant is the invariant introduced in the paper for such a route. Index-Invariant comp…

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and let $\pi_{eq}:X\to X_{eq}$ be the factor map to the maximal equicontinuous factor. Suppos…

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and suppose that $(X,G)$ admits no essential $K$-IT-tuples. Let $\pi_{eq}:X\to X_{eq}$ be the…

Let a stable region be a region of stable interval translation maps, and let an infinite-type map be an interval translation map that is not of finite type. Accumulation of infinit…

Let $F$ be a dynamical system with a compact global attractor. An edge of the graph of $F$ is called strong when it satisfies the source's strong-edge relation. Strong-edge conject…

Let $(X,\varphi)$ be a minimal topological dynamical system, and let $(Y,\psi)$ be its maximal equicontinuous factor (MEF). For $m\in\mathbb{N}$, say that $(X,\varphi)$ is mean $m$…

Let $(X,T)$ and $(X,S)$ be minimal systems on the same compact space $X$. Mixed linear-quadratic nonrecurrence conjecture. There are minimal systems $(X,T)$ and $(X,S)$ such that f…

Let $p$ and $q$ be integral polynomials vanishing at $0$, with $\deg p\ge 2$ and $\deg q\ge 2$. Let $(X,T)$ and $(X,S)$ be systems on the same compact space $X$. Polynomial simulta…

A topological dynamical system is a pair $(X,T)$ consisting of a compact space $X$ and a continuous map $T:X\to X$; it is minimal if every orbit is dense. For two transformations…

Let $G$ be a Polish group, and let $C\subseteq G$ be a comeagre conjugacy class, meaning a conjugacy class whose complement is meagre. Say that $G$ has the countable index property…

Let $M$ be a model of an NIP theory, let $G$ be a $0$-definable group in $M$, and let $N\succ M$ be an $|M|^{+}$-saturated elementary extension. Let $S_G^{\operatorname{fs}}(N,M)$…

Let $G$ be a group. A faithful action of $G$ is called chaotic almost minimal (CAM) if it has dense periodic points and satisfies the almost minimality condition defined for CAM sy…

Monotonicity conjecture. If $N_1\thicksim_S N_2$ in $\operatorname{Down}$, then $N_1^*\thicksim_S N_2^*$ in $\operatorname{Down}^*$; equivalently, $\Gamma$ maps into $\Gamma^*$.