The zero mean dimension marker property conjecture

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 dimension invariant associated with $(X,T)$, and the marker property is the existence, for every prescribed orbit-separation scale, of an open set whose sufficiently long translates are pairwise disjoint and whose translates cover $X$. Zero mean dimension marker property conjecture. If a free dynamical system has zero mean dimension, then it has the marker property. This conjecture proposes that vanishing mean dimension supplies the additional hypothesis needed for the marker property beyond freeness; the source presents it as plausible, while noting that periodic-point freeness alone does not imply the marker property.