32 problems

Lindenstrauss–Tsukamoto conjecture. If

Let $T\colon \mathbb{R}^k\times \mathcal{X}\to \mathcal{X}$ be a continuous action on a compact metrizable space $\mathcal{X}$, and let $\varphi\colon \mathcal{X}\to\mathbb{R}$ be…

Let $G$ be an amenable group acting minimally and topologically freely on a compact metric space $X$. The action has mean dimension zero when its mean dimension, in the sense of Gr…

No-additional-obstructions conjecture. There are no obstructions to shift embeddability beyond mean dimension and the dimension of periodic points.

Let $(X,\mathbb{Z})$ be a dynamical system generated by a homeomorphism $T:X\longrightarrow X$. The system is free when no nonzero power of $T$ has a fixed point, and it has the ma…

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…

Let $(X,T)$ be a dynamical system with finite, nonzero mean dimension, meaning … Let $(Y,S)$ be a factor of $(X,T)$ via a factor map $\pi$, and write $\operatorname{mdim}(\pi,T)$ f…

Let $(X,T)$ be a dynamical system, and let $z\ne z'$ be distinct points of $X$. A factor of $(X,T)$ via a factor map $\pi$ is a dynamical system $(Y,S)$ with $\pi\colon X\to Y$ int…

Metric realization conjecture. For any dynamical system $\phi:N\rightarrow N$, there exists a metric $d$ on $N$ such that

Let $(X,\mathbb{Z})$ be a minimal dynamical system on a compact metric space $X$ with mean dimension ${\rm mdim}\,X=d$, and let $k>d$ be a natural number. For a map $f:X\to[0,1]^k$…

Let $\mathcal{X}$ be a compact metrizable space and let $T\colon\mathbb{R}^d\times\mathcal{X}\to\mathcal{X}$ be a continuous action. Let $\varphi\colon\mathcal{X}\to\mathbb{R}$ be…

Let $z_j\geqslant0$ be independent random variables satisfying the sixth moment condition and the mean condition, and write $z_{1:d}=(z_1,\ldots,z_d)$. For a function $f$, let…

Absolute metric mean dimension conjecture. The absolute metric mean dimension coincides with the topological mean dimension:

Metric characterization conjecture. The topological mean dimension is the infimum of the metric mean dimensions over all distances inducing the topology:

Authors' conjecture. The answers to Questions (i) and (ii) are positive, namely

Metric realization conjecture. There always exists a compatible metric $d$ for which both equalities hold:

Lindenstrauss's conjecture. Every aperiodic dynamical system with zero mean dimension has the small boundary property.

Zero mean dimension conjecture. Every aperiodic dynamical system with zero mean dimension has the marker property.

A free dynamical system is a topological dynamical system $(X,T)$ in which $T^n x\neq x$ for every $x\in X$ and every nonzero integer $n$. Its mean dimension is the dynamical dimen…

A free dynamical system is a topological dynamical system $(X,T)$ in which $T^n x\neq x$ for every $x\in X$ and every nonzero integer $n$. Its mean dimension is the dynamical dimen…

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $(X,T)$ be a topological dynamical system, let $a$ be a positive real parameter, let $\operatorname{mdim}(X,T)$ denote its mean dimension, let $\operatorname{dim}$ denote topol…

Let $(X,\Gamma)$ be a topological dynamical system, let $\mathrm{C}(X)\rtimes\Gamma$ be its crossed product C-algebra, let $\operatorname{rc}$ denote comparison radius, and let…

Let $(\mathcal{X},T)$ be a dynamical system satisfying the hypotheses of Theorem 1.1, except possibly the marker property assumption. Marker property conjecture. The marker propert…

Marker-free embedding conjecture. If, for every subgroup $A\subset \mathbb{Z}^k$, the mean dimension of $X_A$ with respect to the $(\mathbb{Z}^k/A)$-action is smaller than $D/2$, t…