11 problems

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…

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

Mean-dimension embedding conjecture. If

Let $(X,\mathbb{Z}^k)$ be a dynamical system, and for each subgroup $A$ of $\mathbb{Z}^k$ define … The quotient group $\mathbb{Z}^k/A$ acts naturally on $X_A$; when the quotient is…

Let $X\subset \mathbb{C} P^N$ be a projective manifold, and let $\mathcal{M}(X)$ be the space of Brody curves in $X$. Gromov's conjecture. … The conjecture relates positivity of th…

For a Brody curve $f$ in $\mathbb{C} P^1$, let $e(f)$ be its mean energy, and let $e(\mathbb{C} P^1)$ be the supremum of $e(f)$ over all Brody curves in $\mathbb{C} P^1$. Let…