Suppose $X$ is a compact metric space and $F$ is a closed relation on $X$. For a classical dynamical property $\mathcal{P}$, we introduce a natural trichotomy of CR-dynamical properties associated with a closed relation $F$: $CR-\mathcal{P}, CR-post\mathcal{P}, CR-pre\mathcal{P}.$ These three notions are designed so that $(X,F)$ satisfies $CR-\mathcal{P}$ exactly when the shift system $(X_F^+,\sigma_F^+)$ satisfies $\mathcal{P}$; whenever the shift system has property $\mathcal{P}$, the relation $F$ has $CR-post\mathcal{P}$; and whenever $F$ has $CR-pre\mathcal{P}$. the shift system has $\mathcal{P}$. We apply this to minimality, dense-orbit transitivity, and transitivity, establishing precise equivalences in each case. Our examples show that, in general, the three CR-versions of a property form a strict hierarchy, with none of the implications reversible without additional assumptions.
This is joint work with Iztok Banic, Matevz Crepnjak, Goran Erceg, Ivan Jelic.